Book Series, Vol. 20, 2018 Florentin Smarandache and Surapati Pramanik ISBN 978-1-59973-560-3 ISBN 978-1-59973-560-3 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering Copyright Notice Copyright @ Neutrosophics Sets and Systems All rights reserved. The authors of the articles do hereby grant Neutrosophic Sets and Systems non-exclusive, worldwide, royalty-free license to publish and distribute the articles in accordance with the Budapest Open Initiative: this means that electronic copying, distribution and printing of both full-size version of the book and the individual papers published therein for non-commercial, ac- ademic or individual use can be made by any user without permission or charge. The authors of the articles published in Neutrosophic Sets and Systems retain their rights to use this book as a whole or any part of it in any other publications and in any way they see fit. 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This theory considers every notion or idea <A> together with its opposite or negation <antiA> and with their spectrum of neutralities <neutA> in between them (i.e. notions or ideas supporting neither <A> nor <antiA>). The <neutA> and <antiA> ideas together are referred to as <nonA>. Neutrosophy is a generalization of Hegel's dialectics (the last one is based on <A> and <antiA> only). According to this theory every idea <A> tends to be neutralized and balanced by <antiA> and <nonA> ideas - as a state of equilibrium. In a classical way <A>, <neutA>, <antiA> are disjoint two by two. But, since in many cases the borders between notions are vague, imprecise, Sorites, it is possible that <A>, <neutA>, <antiA> (and <nonA> of course) have common parts two by two, or even all three of them as well. Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic). In neutrosophic logic a proposition has a degree of truth (T), a degree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ] -0, 1+[. The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA. Neutrosophic Probability is a generalization of the classical probability and imprecise probability. Neutrosophic Statistics is a generalization of the classical statistics. What distinguishes the neutrosophics from other fields is the <neutA>, which means neither <A> nor <antiA>. <neutA>, which of course depends on <A>, can be indeterminacy, neutrality, tie game, unknown, contradiction, ignorance, imprecision, etc. All submissions should be designed in MS Word format using our template file: http://fs.gallup.unm.edu/NSS/NSS-paper-template.doc. A variety of scientific books in many languages can be downloaded freely from the Digital Library of Science: http://fs.gallup.unm.edu/eBooks-otherformats.htm. To submit a paper, mail the file to the Editor-in-Chief. To order printed issues, contact the Editor-in-Chief. This book series is a non-commercial, academic edition. It is printed from private donations. Information about the neutrosophics you get from the UNM website: http://fs.gallup.unm.edu/neutrosophy.htm. The home page of the book series can be accessed on http://fs.gallup.unm.edu/NSS. Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, 20/2018 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering (GLWRUsLQ&KLHI (GLWRUV Prof. Florentin Smarandache, PhD, Postdoc, Math Department, University of New Mexico, Gallup, NM 87301, USA. W. B. Vasantha Kandasamy, Indian Institute of Technology, Chennai, Tamil Nadu, India. A. A. Salama, Faculty of Science, Port Said University, Egypt. Yanhui Guo, University of Illinois at Springfield, One Univ. Plaza, Springfield, IL 62703, USA. Young Bae Jun, Gyeongsang National University, South Korea. Francisco Gallego Lupianez, Universidad Complutense, Madrid, Spain. Peide Liu, Shandong University of Finance and Economics, China. Pabitra Kumar Maji, Math Department, K. N. University, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Jun Ye, Shaoxing University, China. Madad Khan, Comsats Institute of Information Technology, Abbottabad, Pakistan. Stefan Vladutescu, University of Craiova, Romania. Valeri Kroumov, Okayama University of Science, Japan. Dmitri Rabounski and Larissa Borissova, independent researchers. Selcuk Topal, Mathematics Department, Bitlis Eren University, Turkey. Luige Vladareanu, Romanian Academy, Bucharest, Romania. Ibrahim El-henawy, Faculty of Computers and Informatics, Zagazig University, Egypt. A. A. A. Agboola, Federal University of Agriculture, Abeokuta, Nigeria. Luu Quoc Dat, Univ. of Economics and Business, Vietnam National Univ., Hanoi, Vietnam. Maikel Leyva-Vazquez, Universidad de Guayaquil, Ecuador. Muhammad Akram, University of the Punjab, New Campus, Lahore, Pakistan. Irfan Deli, Muallim Rifat Faculty of Education, Kilis 7 Aralik University, Turkey. Ridvan Sahin, Faculty of Science, Ataturk University, Erzurum 25240, Turkey. Ibrahim M. Hezam, Faculty of Education, Ibb University, Ibb City, Yemen. Pingping Chi, International College, Dhurakij Pundit University, Bangkok 10210, Thailand. Karina Perez-Teruel, Universidad de las Ciencias Informaticas, La Habana, Cuba. B. Davvaz, Department of Mathematics, Yazd University, Iran. Victor Christianto, Malang Institute of Agriculture (IPM), Malang, Indonesia. Ganeshsree Selvachandran, UCSI University, Jalan Menara Gading, Kuala Lumpur, Malaysia. Saeid Jafari, College of Vestsjaelland South, Slagelse, Denmark. Paul Wang, Pratt School of Engineering, Duke University, USA. Arun Kumar Sangaiah, VIT University, Vellore, India. Kul Hur, Wonkwang University, Iksan, Jeollabukdo, South Korea. Darjan Karabasevic, University Business Academy, Novi Sad, Serbia. Dragisa Stanujkic, John Naisbitt University, Belgrade, Serbia. E. K. Zavadskas, Vilnius Gediminas Technical University, Vilnius, Lithuania. M. Ganster, Graz University of Technology, Graz, Austria. Willem K. M. Brauers, Faculty of Applied Economics, University of Antwerp, Antwerp, Belgium. Dr. Surapati Pramanik, Assistant Professor, Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, Narayanpur, Dist-North 24 Parganas, West Bengal, India-743126 $VVRFLDWH(GLWRUV Said Broumi, University of Hassan II, Casablanca, Morocco. Mohamed Abdel-Baset, Faculty of Computers and Informatics, Zagazig University, Egypt. Huda E. Khalid, University of Telafer, College of Basic Education, Telafer - Mosul, Iraq. Prof. Le Hoang Son, VNU Univ. of Science, Vietnam National Univ. Hanoi, Vietnam. Dr. Mumtaz Ali, University of Southern Queensland, Australia. $GGUHVV ³1HXWURVRSKLF6HWVDQG6\VWHPV´ $Q ,QWHUQDWLRQDO -RXUQDO LQ ,QIRUPDWLRQ 6FLHQFHDQG(QJLQHHULQJ  'HSDUWPHQWRI0DWKHPDWLFVDQG6FLHQFH 8QLYHUVLW\RI1HZ0H[LFR *XUOH\$YHQXH *DOOXS1086$ (PDLOVPDUDQG#XQPHGX +RPHSDJHKWWSIVJDOOXSXQPHGX166 9ROXPH20 Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy ………………..…....… Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assessments ………….....……... Seon Jeong Kim, Seok-Zun Song, Young Bae Jun. Generalizations of Neutrosophic Subalgebras in BCK/BCIAlgebras Based on Neutrosophic Points ……………..… 8 &RQWHQWV 3 12 26 G. Muhiuddin, Hashem Bordbar, F. Smarandache, Young Bae Jun. Further results on (ԑ, ԑ)-neutrosophic subalgebras and ideals in BCK/BCI-algebras ……...…… 36 Young Bae Jun, F. Smarandache, Mehmat Ali Ozturk. Commutative falling neutrosophic ideals in BCKalgebras …………………………………….....………… 44 Tuhin Bera, Nirmal K. Mahapatra. On Neutrosophic Soft Prime Ideal …...…………………............................. 54 Emad Marei. Single Valued Neutrosophic Soft Approach to Rough Sets, Theory and Application …… 76 M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets ………....................................................… 86 S. Pramanik, Shyamal Dalapati, Shariful Alam, T. K. Roy: NC-VIKOR Based MAGDM Strategy under Neutrosophic Cubic Set Environment....…………………….. 95 Surapati Pramanik, Rama Mallick, Anindita Dasgupta: Contributions of Selected Indian Researchers to MultiAttribute Decision Making in Neutrosophic Environment: An Overview ……………………......................… 109 Copyright © Neutrosophic Sets and Systems 3 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Kalyan Mondal1, Surapati Pramanik2, and Bibhas C. Giri3 1 Department of Mathematics, Jadavpur University, Kolkata: 700032, West Bengal, India. E mail:kalyanmathematic@gmail.com ²Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P O - Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West Bengal, India. Email: sura_pati@yahoo.co.in, 3 Department of Mathematics, Jadavpur University, Kolkata: 700032, West Bengal, India. Email: bibhasc.giri@jadavpuruniversity.in Abstract: In this paper, we introduce new type of similarity measures for single valued neutrosophic sets based on hyperbolic sine function. The new similarity measures are namely, single valued neutrosophic hyperbolic sine similarity measure and weighted single valued neutrosophic hyperbolic sine similarity measure. We prove the basic properties of the proposed similarity measures. We also develop a multi-attribute decision- making strategy for single valued neutrosophic set based on the proposed weighted similarity measure. We present a numerical example to verify the practicability of the proposed strategy. Finally, we present a comparison of the proposed strategy with the existing strategies to exhibit the effectiveness and practicality of the proposed strategy. Keywords: Single valued neutrosophic set, Hyperbolic sine function, Similarity measure, MADM, Compromise function 1 Introduction Smarandache [1] introduced the concept of neutrosophic set (NS) to deal with imprecise and indeterminate data. In the concept of NS, truth-membership, indeterminacymembership, and falsity-membership are independent. Indeterminacy plays an important role in many real world decision-making problems. NS generalizes the Cantor set discovered by Smith [2] in 1874 and introduced by German mathematician Cantor [3] in 1883, fuzzy set introduced by Zadeh [4], intuitionistic fuzzy set proposed by Atanassov [5]. Wang et al. [6] introduced the concept of single valued neutrosophic set (SVNS) that is the subclass of a neutrosophic set. SVNS is capable to represent imprecise, incomplete, and inconsistent information that manifest the real world. Neutrosophic sets and its various extensions have been studied and applied in different fields such as medical diagnosis [7, 8, 9], decision making problems [10, 11, 12, 13, 14], social problems [15, 16], educational problem [17, 18], conflict resolution [19], image processing [ 20, 21, 22], etc. The concept of similarity is very important in studying almost every scientific field. Many strategies have been proposed for measuring the degree of similarity between fuzzy sets studied by Chen [23], Chen et al. [24], Hyung et al. [25], Pappis and Karacapilidis [26], Pramanik and Roy [27], etc. Several strategies have been proposed for measuring the degree of similarity between intuitionistic fuzzy sets studied by Xu [28], Papakostas et al. [29], Biswas and Pramanik [30], Mondal and Pramanik [31], etc. However, these strategies are not capable of dealing with the similarity measures involving indeterminacy. SVNS can handle this situation. In the literature, few studies have addressed similarity measures for neutrosophic sets and single valued neutrosophic sets [32, 33, 34, 35]. Ye [36] proposed an MADM method with completely unknown weights based on similarity measures under SVNS environment. Ye [37] proposed vector similarity measures of simplified neutrosophic sets and applied it in multi-criteria decision making problems. Ye [38] developed improved cosine similarity measures of simplified neutrosophic sets for medical diagnosis. Ye [39] also proposed exponential similarity measure of neutrosophic numbers for fault diagnoses of steam turbine. Ye [40] developed clustering algorithms based on similarity measures for SVNSs. Ye and Ye [41] proposed Dice similarity measure between single valued neutrosophic multisets. Ye et al. [42] proposed distancebased similarity measures of single valued neutrosophic multisets for medical diagnosis. Ye and Fu [43] developed a single valued neutrosophic similarity measure based on tangent function for multi-period medical diagnosis. In hybrid environment Pramanik and Mondal [44] proposed cosine similarity measure of rough neutrosophic sets and provided its application in medical diagnosis. Pramanik and Mondal [45] also proposed cotangent Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure based MADM Strategy Neutrosophic Sets and Systems, Vol. 20, 2018 4 similarity measure of rough neutrosophic sets and its application to medical diagnosis. Research gap: MADM strategy using similarity measure based on hyperbolic sine function under single valued neutrosophic environment is yet to appear. Research questions:  Is it possible to define a new similarity measure between single valued neutrosophic sets using hyperbolic sine function? Is it possible to develop a new MADM strategy based on the proposed similarity measures in single valued neutrosophic environment?  Having motivated from the above researches on neutrosophic similarity measures, we have introduced the concept of hyperbolic sine similarity measure for SVNS environment. The new similarity measures called single valued neutrosophic hyperbolic sine similarity measure (SVNHSSM) and single valued neutrosophic weighted hyperbolic sine similarity measure (SVNWHSSM). The properties of hyperbolic sine similarity are established. We have developed a MADM model using the proposed SVNWHSSM. The proposed hyperbolic sine similarity measure is applied to multi-attribute decision making. The objectives of the paper: 2 Neutrosophic preliminaries 2.1 Neutrosophic set (NS) Definition 2.1 [1] Let U be a universe of discourse. Then the neutrosophic set P can be presented of the form: P = {< x:TP(x ), IP(x ), FP(x)> | x  U}, where the functions T, I, F: U→ ]−0,1+[ define respectively the degree of membership, the degree of indeterminacy, and the degree of non-membership of the element x  U to the set P satisfying the following the condition. 0 ≤ supTP(x) + supIP( x) + supFP(x) ≤ 3+ − 2.2 Single valued neutrosophic set (SVNS) Definition 2.2 [6] Let X be a space of points with generic elements in X denoted by x. A SVNS P in X is characterized by a truth-membership function TP(x), an indeterminacy-membership function IP(x), and a falsity membership function FP(x), for each point x in X. TP(x), IP(x), FP(x)  [0, 1]. When X is continuous, a SVNS P can be written as follows:  ( x), I P ( x), F P ( x)  :x X P  X T P x When X is discrete, a SVNS P can be written as follows:  T P ( x i ), I P ( x i ), F P ( x i )  : xi  X P  in1 xi For two SVNSs,  To define hyperbolic sine similarity measures for SVNS environment and prove some of it’s basic properties. PSVNS = {<x: TP(x ), IP(x), FP(x )> | x  X} and QSVNS = {<x, TQ(x), IQ(x), FQ(x)> | x  X } the two relations are defined as follows:  To define conpromise function for determining unknown weight of attributes.  To develop a multi-attribute decision making model based on proposed similarity measures. (1) PSVNS  QSVNS if and only if TP(x)  TQ(x), IP(x)  IQ(x), FP(x)  FQ(x) (2) PSVNS = QSVNS if and only if TP(x) = TQ(x), IP(x) = IQ(x), FP(x) = FQ(x) for any x  X .  To present a numerical example for the efficiency and effectiveness of the proposed strategy. Rest of the paper is structured as follows. Section 2 presents preliminaries of neutrosophic sets and single valued neutrosophic sets. Section 3 is devoted to introduce hyperbolic sine similarity measure for SVNSs and some of its properties. Section 4 presents a method to determine unknown attribute weights. Section 5 presents a novel decision making strategy based on proposed neutrosophic hyperbolic sine similarity measure. Section 6 presents an illustrative example for the application of the proposed method. Section 7 presents a comparison analysis for the applicability of the proposed strategy. Section 8 presents the main contributions of the proposed strategy. Finally, section 9 presents concluding remarks and scope of future research. 3. Hyperbolic sine similarity measures for SVNSs Let A = <x(TA(x), IA(x), FA(x))> and B = <x(TB(x), IB(x), FB(x))> be two SVNSs. Now hyperbolic sine similarity function which measures the similarity between two SVNSs can be presented as follows (see Eqn. 1): SVNHSSM ( A, B)      sinh  T A ( xi )  T B ( xi )  I A ( xi )  I B ( xi )       F A ( xi )  F B ( x i ) 1 n   1   11 n i 1       (1) Theorem 1. The defined hyperbolic sine similarity measure SVNHSSM(A, B) between SVNSs A and B satisfies the following properties: Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy 5 Neutrosophic Sets and Systems, Vol. 20, 2018 1. 2. 3. 4. 0  SVNHSSM(A, B)  1 SVNHSSM(A, B) = 1 if and only if A = B SVNHSSM (A, B) = SVNHSSM(B, A) If R is a SVNS in X and A  B  R then SVNHSSM(A, R)  SVNHSSM(A, B) and SVNHSSM(A, R)  SVNHSSM(B, R). Proofs: 1. For two neutrosophic sets A and B, 0  T A ( xi ), I A ( xi ), F A ( xi ), T B ( xi ), I B ( xi ), F B ( xi )  1  0  T A (xi )  T B (xi )  I A (xi )  I B (xi )  F A (xi )  F B (xi )  3     sinh  T A ( x i )  T B ( x i )  I A ( x i )  I B ( x i )       F A (xi )  F B (xi )  1  0   11       Hence 0  SVNHSSM(A, B)  1 2. For any two SVNSs A and B, if A = B,  TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x)  T A ( x)  T B ( x)  0 , I A ( x )  I B ( x)  0 , F A ( x)  F B ( x )  0 Hence SVNHSSM(A, B) = 1. Conversely, SVNHSSM(A, B) = 1  T A ( x)  T B ( x)  0 , I A ( x)  I B ( x )  0 , F A ( x)  F B ( x)  F A ( x)  F R ( x) , F B ( x)  F R ( x)  F A ( x)  F R ( x) . Thus, SVNHSSM(A, R)  SVNHSSM(A, B) and SVNHSSM(A, R)  SVNHSSM(B, R). 3.1 Weighted hyperbolic sine similarity measures for SVNSs Let A = <x(TA(x), IA(x), FA(x))> and B = <x(TB(x), IB(x), FB(x))> be two SVNSs. Now weighted hyperbolic sine similarity function which measures the similarity between two SVNSs can be presented as follows (see Eqn. 2): SVN WHSSM ( A, B)      sinh  T A ( xi )  T B ( xi )  I A ( xi )  I B ( xi )       F A ( xi )  F B ( xi ) n   1   wi   11 i 1        n Here, 0  wi  1 ,  wi  1. i 1 Theorem 2. The defined weighted hyperbolic sine similarity measure SVNWHSSM(A, B) between SVNSs A and B satisfies the following properties: 1. 2. 3. 4. F A ( x)  F B ( x )  0 . This implies, TA(x) = TB(x) , IA(x) = IB(x), FA(x) = FB(x). Hence A = B. 3. Since, T A ( x)  T B ( x)  T B ( x)  T A ( x) , I A ( x)  I B ( x)  I B ( x)  I A ( x) , 0  SVNWHSSM(A, B)  1 SVNWHSSM (A, B) = 1 if and only if A = B SVNWHSSM (A, B) = SVNWHSSM (B, A) If R is a SVNS in X and A  B  R then SVNWHSSM (A, R)  SVNWHSSM(A, B) and SVNWHSSM (A, R)  SVNWHSSM (B, R). Proofs: 1. For two neutrosophic sets A and B, 0  T A ( xi ), I A ( xi ), F A ( xi ), T B ( xi ), I B ( xi ), F B ( xi )  1  0  T A (xi )  T B (xi )  I A (xi )  I B (xi )  F A (xi )  F B (xi )  3 F A ( x)  F B ( x)  F B ( x)  F A ( x) . We can write, SVNHSSM(A, B) = SVNHSSM(B, A). 4. A  B  R  TA(x)  TB(x)  TR(x), IA(x)  IB(x)  IR(x), FA(x)  FB(x)  FR(x) for x  X. Now we have the following inequalities: T A ( x)  T B ( x)  T A ( x)  T R ( x) , T B ( x)  T R ( x)  T A ( x)  T R ( x) ; I A ( x)  I B ( x)  I A ( x)  I R ( x) , I B ( x)  I R ( x)  I A ( x)  I R ( x) ; (2)     sinh  T A ( x i )  T B ( x i )  I A ( x i )  I B ( x i )      F A (xi )  F B (xi )    1 0   11       n Again, 0  wi  1 ,  wi  1. i 1 Hence 0  SVNWHSSM(A, B)  1 2. For any two SVNSs A and B, if A = B, Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Neutrosophic Sets and Systems, Vol. 20, 2018 6  TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x) The weight of j-th attribute is defined as follows (see Eqn.  T A ( x)  T B ( x)  0 , I A ( x)  I B ( x)  0 , 4). F A ( x)  F B ( x)  0 wj  Hence SVNWHSSM(A, B) = 1. Conversely, C j ( A)  C j ( A) (4) n j 1 n Here,  w j  1. j 1 SVNWHSSM(A, B) = 1  T A ( x)  T B ( x)  0 , I A ( x)  I B ( x)  0 , Theorem 3. The compromise function Cj(A) satisfies the following properties: F A ( x)  F B ( x)  0 . This implies, TA(x) = TB(x) , IA(x) = IB(x), FA(x) = FB(x). P1. C j ( A)  1 , if T ij 1, F ij  I ij  0 . Hence A = B. P2. C j ( A)  0 , if T ij , I ij , F ij  0, 1, 1 . 3. Since, T A ( x)  T B ( x)  T B ( x)  T A ( x) , P3. C j ( A)  E j ( B) , if T ijA  T ijB and I ijA  F ijA  I ijB  F ijB . Proofs. I A ( x)  I B ( x)  I B ( x)  I A ( x) , P1. T ij 1, F ij  I ij  0 F A ( x)  F B ( x)  F B ( x)  F A ( x) .  C j ( A)  We can write, SVNWHSSM(A, B) = SVNWHSSM(B, A). 1 m 1  3 3  .m  1 m i 1 m P2. T ij , I ij , F ij  0, 1, 1 . 4. A  B  R  TA(x)  TB(x)  TR(x), IA(x)  IB(x)  IR(x), FA(x)  FB(x)  FR(x) for x  X.  C j ( A)  1 m 0 3  0 m i 1 P3. C j ( A)  C j ( B) Now we have the following inequalities: T A ( x)  T B ( x)  T A ( x)  T R ( x) ,     I A ( x)  I B ( x)  I A ( x)  I R ( x) , 1 m  1 m  2T ijA  I ijA  F ijA 3   2T ijB  I ijB  F ijB 3   0 m i 1   m i 1 A B A A B  C j ( A)  C j ( B)  0 , Since, T ij  T ij and I ij  F ij  I ij  F ijB . I B ( x)  I R ( x)  I A ( x)  I R ( x) ; Hence, C j ( A)  C j ( B) .   T B ( x)  T R ( x)  T A ( x)  T R ( x) ; F A ( x)  F B ( x)  F A ( x)  F R ( x) , 5. Decision making procedure F B ( x)  F R ( x)  F A ( x)  F R ( x) . Thus SVNWHSSM(A, R)  SVNWHSSM(A, B) and SVNWHSSM(A, R)  SVNWHSSM(B, R). 4. Determination of unknown attribute weights When attribute weights are completely unknown to decision makers, the entropy measure [46] can be used to calculate attribute weights. Biswas et al. [47] employed entropy measure for MADM problems to determine completely unknown attribute weights of SVNSs. 4.1 Compromise function The compromise function of a SVNS A = T ijA , I ijA , F ijA (i = 1, 2, ..., m; j = 1, 2, ..., n) is defined as follows (see Eqn. 3): m   A C j ( A)   2  T ij  I ijA  F ijA 3 i 1 Let A1, A2 , ..., Am be a discrete set of alternatives, C1, C2, ..., Cn be the set of attributes of each alternative. The values associated with the alternatives Ai (i = 1, 2,..., m) against the attribute Cj (j = 1, 2, ..., n) for MADM problem is presented in a SVNS based decision matrix. The steps of decision-making (see Figure 2) based on single valued neutrosophic weighted hyperbolic sine similarity measure (SVNWHSSM) are presented using the following steps. Step 1: Determination of the relation between alternatives and attributes The relation between alternatives Ai (i = 1, 2, ..., m) and the attribute Cj (j = 1, 2, ..., n) is presented in the Eqn. (5). (3) Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy 7 Neutrosophic Sets and Systems, Vol. 20, 2018 D[ A | C ]     A1 A  2     Am C1 T 11, I 11, F 11 T 21, I 21, F 21  T m1, I m1, F 1m1 C2 T 12, I 12, F 12 T 22, I 22, F 22  T m 2, I m 2, F m 2      Cn T 1n, I 1n, F 1n T 2 n, I 2 n, F 2 n  T mn, I mn, F mn     (5)     Here T ij, I ij, F ij (i = 1, 2, ..., m; j = 1, 2, ..., n) be SVNS assessment value. Step 2: Determine the weights of attributes Using the Eqn. (3) and (4), decision-maker calculates the weight of the attribute Cj (j = 1, 2, …, n). Step 3: Determine ideal solution Generally, the evaluation attribute can be categorized into two types: benefit type attribute and cost type attribute. In the proposed decision-making method, an ideal alternative can be identified by using a maximum operator for the benefit type attributes and a minimum operator for the cost type attributes to determine the best value of each attribute among all the alternatives. Therefore, we define an ideal alternative as follows: 𝐴* = {C1*, C2*, … , Cm*}. Here, benefit attribute C *j (6) for j = 1, 2, ..., n. Similarly, the cost attribute C *j can be presented as follows: (A ) (A ) (A )   C*j  min T C j i , max I C j i , max F C j i  i i   i  A1: Airtel  A2: Vodafone  A3: BSNL  A4: Reliance Jio The person must take a decision based on the following five attributes of SIM cards:  C1: Service quality  C2: Cost  C3: Initial talk time  C4: Call rate per second  C5: Internet and other facilities The decision-making strategy is presented using the following steps. Step 1: Determine the relation between alternatives and attributes The relation between alternatives A1, A2, A3, and A4 and the attributes C1, C2, C3, C4, C5 is presented in the Eqn. (8). D[ A |C 1, C 2 , C 3 , C 4 , C 5 ]  can be presented as follows: (A ) (A ) (A )   C*j  max T C j i , min I C j i , min F C j i  i i   i nection. Therefore, it is necessary to select suitable SIM card for his/her mobile connection. After initial screening, there are four possible alternatives (SIM cards) for mobile connection. The alternatives (SIM cards) are presented as follows: (7)    A1 A  2  A3   A4 C1 .7, .3, .3 .5, .3, .1 .8, .2, .2 .6, .1, .3 C2 .6, .4, .3 .7, .1, .3 .6, .4, .3 .5, .1, .2 C3 .8, .1, .1 .7, .3, .1 .6, 0, .1 .6, .3, .1 C4 .5, .4, .4 .6, .1, .1 .7, .3, 0 .5, .1, .2 C5 .5, .3, .2 .5, .2, .3 .5, .3, .4 .9, .1, .1     (8)     Step 2: Determine the weights of attributes for j = 1, 2, ..., n Using the Eq. (3) and (4), we calculate the weight of the attributes C1, C2, C3, C4, C5 as follows: Step 4: Determine the similarity values [w1, w2, w3, w4, w5] = Using Eqns. (2) and (5), calculate SVNWHSSM values for each alternative between positive (or negative) ideal solutions and corresponding single valued neutrosophic from decision matrix D[A|C]. [0.2023, 0.1917, 0.2078, 0.2009, 0.1973] Step 5: Ranking the alternatives Ranking the alternatives is prepared based on the descending order of similarity measures. Highest value indicates the best alternative. Step 6: End 6. Numerical example In this section, we illustrate a numerical example as an application of the proposed approach. We consider a decision-making problem stated as follows. Suppose a person who wants to purchase a SIM card for his/her mobile con- Step 3: Determine ideal solution In this problem, attributes C1, C3, C4, C5 are benefit type attributes and , C2 is the cost type attribute. 𝐴* = {(0.8, 0.1, 0.1), (0.5, 0.4, 0.3), (0.8, 0.0, 0.1), (0.7, 0.1, 0.0), (0.9, 0.1, 0.1)}. Step 4: Determine the weighted similarity values Using Eq. (2) and Eq. (8), we calculate similarity measure values for each alternative as follows. SVNWHSSM( A*, A1 ) = 0 .92422 SVNWHSSM( A*, A2 ) = 0 .95629 SVNWHSSM( A*, A3 ) = 0 .97866 Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Neutrosophic Sets and Systems, Vol. 20, 2018 8 SVNWHSSM( A*, A4 ) = 0 .96795 Step 5: Ranking the alternatives Ranking the alternatives is prepared based on the descending order of similarity measures (see Figure 1). Now the final ranking order will be as follows. A3  A4  A2  A1 Highest value indicates the best alternative. Step 6: End Weighted similarity measure values 1.0 2) We have proposed ‘compromise function’ for calculating unknown weights structure of attributes in SVNS environment. 3) We develop a decision making strategy based on the proposed weighted similarity measure (SVNWHSSM). 4) Steps and calculations of the proposed strategy are easy to use. 5) We have solved a numerical example to show the feasibility, applicability, and effectiveness of the proposed strategy. 9. Conclusion 0.8 0.6 0.4 0.2 0.0 A1 A2 A3 A4 Alternatives FIGURE 1: Graphical representation of alternatives versus weighted similarity measures. 7. Comparison analysis The ranking results calculated from proposed strategy and different existing strategies [38, 48, 49, 50] are furnished in Table 1. We observe that the ranking results obtained from proposed and existing strategies in the literature differ. The proposed strategy reflects that the optimal alternative is A3. The ranking result obtained from Ye [38] is similar to the proposed strategy. The ranking results obtained from Ye and Zhang [48] and Mondal and Pramanik [49] differ from the optimal result of the proposed strategy. In Ye [50], the ranking order differs but the best alternative is the same to the proposed strategy. In the paper, we have proposed hyperbolic sine similarity measure and weighted hyperbolic sine similarity measures for SVNSs and proved their basic properties. We have proposed compromise function to determine unknown weights of the attributes in SVNS environment. We have developed a novel MADM strategy based on the proposed weighted similarity measure to solve decision problems. We have solved a numerical problem and compared the obtained result with other existing strategies to demonstrate the effectiveness of the proposed MADM strategy. The proposed MADM strategy can be applied in other decision-making problem such as supplier selection, pattern recognition, cluster analysis, medical diagnosis, weaver selection [51-53], fault diagnosis [54], brick selection [55-56], data mining [57], logistic centre location selection [58-60], teacher selection [61, 62], etc. Table 1 The ranking results of existing strategies Strategies Ye and Zhang[48] Mondal and Pramanik [49] Ye [38] Ye [50] Proposed strategy Ranking results A4  A2  A3  A1 A4  A3  A2  A1 A3  A4  A2  A1 A3  A2  A4  A1 A3  A4  A2  A1 8. Contributions of the proposed strategy 1) SVNHSSM and SVNWHSSM in SVNS environment are firstly defined in the literature. We have also proved their basic properties. Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy 9 Neutrosophic Sets and Systems, Vol. 20, 2018 Multi attribute decision making problem Decision making analysis phase Determination of the relation between alternatives and attributes Step-1 Determine the weights of attributes Step- 2 Determine ideal solution Step- 3 Determine the similarity values Step-4 Ranking the alternatives Step-5 End Step- 6 FIGURE 2: Phase diagram of the proposed decision making strategy References [4] [1] [5] [2] [3] F. Smarandache, A unifying field in logics, neutrosophy: neutrosophic probability, set and logic. Rehoboth, American Research Press, 1998. H. J. S. Smith. On the integration of discontinuous functions. 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Giri3 1 Department of Mathematics, Jadavpur University, Kolkata: 700032, West Bengal, India. E mail:kalyanmathematic@gmail.com ²Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P O - Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West Bengal, India. Email: sura_pati@yahoo.co.in, 3 Department of Mathematics, Jadavpur University, Kolkata: 700032, West Bengal, India. Email: bibhasc.giri@jadavpuruniversity.in Abstract: Single valued neutrosophic set is an important mathematical tool for tackling uncertainty in scientific and engineering problems because it can handle situation involving indeterminacy. In this research, we introduce new similarity measures for single valued neutrosophic sets based on binary logarithm function. We define two type of binary logarithm similarity measures and weighted binary logarithm similarity measures for single valued neutrosophic sets. Then we define hybrid binary logarithm similarity measure and weighted hybrid binary logarithm similarity measure for single valued neutrosophic sets. We prove the basic properties of the proposed measures. Then, we define a new entropy function for determining unknown attribute weights. We develop a novel multi attribute group decision making strategy for single valued neutrosophic sets based on the weighted hybrid binary logarithm similarity measure. We present an illustrative example to demonstrate the effectiveness of the proposed strategy. We conduct a sensitivity analysis of the developed strategy. We also present a comparison analysis between the obtained results from proposed strategy and different existing strategies in the literature. Keywords: single valued neutrosophic set; binary logarithm function; similarity measure; entropy function; ideal solution; MAGDM 1 Introduction Smarandache [1] introduced neutrosophic sets (NSs) to pave the way to deal with problems involving uncertainty, indeterminacy and inconsistency. Wang et al. [2] grounded the concept of single valued neutrosophic sets (SVNSs), a subclass of NSs to tackle engineering and scientific problems. SVNSs have been applied to solve various problems in different fields such as medical problems [3– 5], decision making problems [6–18], conflict resolution [19], social problems [20–21] engineering problems [2223], image processing problems [24–26] and so on. The concept of similarity measure is very significant in studying almost every practical field. In the literature, few studies have addressed similarity measures for SNVSs [27–30]. Peng et al. [31] developed SVNSs based multi attribute decision making (MADM) strategy employing MABAC (Multi-Attributive Border Approximation area Comparison and similarity measure), TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) and a new similarity measure. Ye [32] proposed cosine similarity measure based neutrosophic multiple attribute decision making (MADM) strategy. In order to overcome some disadvantages in the definition of cosine similarity measure, Ye [33] proposed ‘improved cosine similarity measures’ based on cosine function. Biswas et al. [34] studied cosine similarity measure based MCDM with trapezoidal fuzzy neutrosophic numbers. Pramanik and Mondal [35] proposed weighted fuzzy similarity measure based on tangent function. Mondal and Pramanik [36] proposed intuitionistic fuzzy similarity measure based on tangent function. Mondal and Pramanik [37] developed tangent similarity measure of SVNSs and applied it to MADM. Ye and Fu [38] studied medical diagnosis problem using a SVNSs similarity measure based on tangent function. Can and Ozguven [39] studied a MADM problem for adjusting the proportional-integral-derivative (PID) coefficients based on neutrosophic Hamming, Euclidean, set-theoretic, Dice, and Jaccard similarity measures. Several studies [40–42] have been reported in the literature for multi-attribute group decision making (MAGDM) in neutrosophic environment. Ye [43] studied the similarity measure based on distance function of SVNSs and applied it to MAGDM. Ye [44] developed several clustering methods using distance-based similarity measures for SVNSs. Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments 13 Neutrosophic Sets and Systems, Vol. 20, 2018 Mondal et al. [45] proposed sine hyperbolic similarity measure for solving MADM problems. Mondal et al. [46] also proposed tangent similarity measure to deal with MADM problems for interval neutrosophic environment. Lu and Ye [47] proposed logarithmic similarity measure for interval valued fuzzy set [48] and applied it in fault diagnosis strategy. Research gap: MAGDM strategy using similarity measure based on binary logarithm function under single valued neutrosophic environment is yet to appear. Research questions:    Is it possible to define a new similarity measure between single valued neutrosophic sets using binary logarithm function? Is it possible to define a new entropy function for single valued neutrosophic sets for determining unknown attribute weights? Is it possible to develop a new MAGDM strategy based on the proposed similarity measures in single valued neutrosophic environment? The objectives of the paper:  To define binary logarithm similarity measures for SVNS environment and prove the basic properties.  To define a new entropy function for determining unknown weight of attributes.  To develop a multi-attribute droup decision making model based on proposed similarity measures.  To present a numerical example for the efficiency and effectiveness of the proposed strategy. Having motivated from the above researches on neutrosophic similarity measures, we introduce the concept of binary logarithm similarity measures for SVNS environment. The properties of binary logarithm similarity measures are established. We also propose a new entropy function to determine unknown attribute weights. We develope a MAGDM strategy using the proposed hybrid binary logarithm similarity measures. The proposed similarity measure is applied to a MAGDM problem. The structure of the paper is as follows. Section 2 presents basic concepts of NSs, operations on NSs, SVNSs and operations on SVNSs. Section 3 proposes binary logarithm similarity measures and weighted binary logarithm similarity measures, hybrid binary logarithm similarity measure (HBLSM), weighted hybrid binary logarithm similarity measure (WHBLSM) in SVNSs environment. Section 4 proposes a new entropy measure to calculate unknown attribute weights and proves basic properties of entropy function. Section 5 presents a MAGDM strategy based weighted hybrid binary logarithm similarity measure. Section 6 presents an illustrative example to demonstrate the applicability and feasibility of the proposed strategies. Section 7 presents a sensitivity analysis for the results of the numerical example. Section 8 conducts a comparative analysis with the other existing strategies. Section 9 presents the key contribution of the paper. Section 10 summarizes the paper and discusses future scope of research. 2 Preliminaries In this section, the concepts of NSs, SVNSs, operations on NSs and SVNSs and binary logarithm function are outlined. 2.1 Neutrosophic set (NS) Assume that X be an universe of discourse. Then a neutrosophic sets [1] N can be defined as follows: N = {< x: TN(x), IN(x), FN(x) > | x  X}. Here the functions T, I and F define respectively the membership degree, the indeterminacy degree, and the non-membership degree of the element x  X to the set N. The three functions T, I and F satisfy the following the conditions:  T, I, F: X → ]−0,1+[  − 0 ≤ supTN(x) + supIN( x) + supFN(x) ≤ 3+ For two neutrosophic sets M = {< x: TM (x), IM(x), FM(x) > | x  X} and N = {< x, TN(x), IN(x), FN(x) > | x  X }, the two relations are defined as follows:  M  N if and only if TM(x)  TN(x), IM(x)  IN(x), FM(x )  FN(x)  M = N if and only if TM(x) = TN(x), IM(x) = IN(x), FM(x) = FN(x). 2.2. Single valued Neutrosophic sets (SVNSs) Assume that X be an universe of discourse. A SVNS [2] P in X is formed by a truth-membership function TP(x), an indeterminacy membership function IP(x), and a falsity membership function FP(x). For each point x in X, TP(x), IP(x), and FP(x)  [0, 1]. For continuous case, a SVNS P can be expressed as follows:  ( x), I P ( x), F P ( x)  P  x T P :x  X , x Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments Neutrosophic Sets and Systems, Vol. 20, 2018 14 For discrete case, a SVNS P can be expressed as follows: n  ( x ), ( x ), (x )  : xi  X P   TP i IP i FP i xi i 1 For two SVNSs P = {< x: TP(x), IP(x), FP(x)> | x  X} and Q = {< x: TQ(x), IQ(x), FQ(x)> | x  X}, some definitions are stated below:  P  Q if and only if TP(x)  TQ(x), IP(x)  IQ(x), and    FP(x)  FQ(x). P  Q if and only if TP(x)  TQ(x), IP(x)  IQ(x), and FP(x)  FQ(x). P = Q if and only if TP(x) = TQ(x), IP(x) = IQ(x), and FP(x) = FQ(x) for any x  X. Complement of P i.e. Pc ={< x: FP(x), 1− IP(x), TP(x)> | x  X }. 2.3. Some arithmetic operations on SVNSs 3.1. Binary logarithm similarity measures of SVNSs (type-I) Definition 2. Let A = <x(TA(xi), IP(xi), FP(xi))> and B = <x(TB(xi), IB(xi), FB(xi))> be any two SVNSs. The binary logarithm similarity measure (type-I) between SVNSs A and B are defined as follows: BL1 ( A, B) = 1 n n  i 1   1  TA ( xi )  TB ( xi )  I A ( xi )  I B ( xi )     log 2  2         3   FA ( xi )  FB ( xi )    (1) Theorem 1. The binary logarithm similarity measure BL 1 ( A, B) between any two SVNSs A and B satisfy the following properties: P 1. 0  BL 1 ( A, B)  1 any two SVNSs in a universe of discourse then arithmetic P 2. BL 1 ( A, B) 1 , if and only if A = B P 3. BL 1 ( A, B)  BL 1 ( B, A) P4. If C is a SVNS in X and A  B  C then BL 1 ( A, C )  BL 1 ( A, B) and BL 1 ( A, C )  BL 1 ( B, C ) . operations are stated as follows. Proof 1. Definition 1 [49] Let P  T P( x), I P( x), F P ( x) and Q  T Q( x), I Q( x), F Q( x) be  T P( x)  T Q( x)  T P( x)T Q( x) , I P( x) I Q( x) ,    P  Q    F P ( x) F Q ( x)    T P ( x ) T Q ( x ) , I P ( x )  I Q ( x )  I P ( x ) I Q( x ) ,    P  Q    F P ( x)  F Q ( x)  F P ( x) F Q ( x)        P  1  1 T P( x)  , I P( x)  , F P ( x)  ;   0  P   T P( x)  , 1  1  I P( x)  , 1  1  F P( x)  ;   0 2.4. Binary logarithm function In mathematics, the logarithm of the form log2x , x > 0 is called binary logarithm function [50]. For example, the binary logarithm of 1 is 0, the binary logarithm of 4 is 2, the binary logarithm of 16 is 4, and the binary logarithm of 64 is 6. 3. Binary logarithm similarity measures for SVNSs In this section, we define two types of binary logarithm similarity measures and their hybrid and weighted hybrid similarity measures. From the definition of SVNS, we write, 0 ≤ TA(x) + IA( x) + FA(x) ≤ 3 and 0 ≤ TB(x) + IB(x) + FB(x) ≤ 3  0  TA ( xi )  TB ( xi )  I A ( xi )  I B ( xi )  FA ( xi )  FB ( xi )  3 ,  TA ( xi )  TB ( xi ) , I A ( xi )  I B ( xi ) ,   1 0  max    F (x )  F (x ) A i B i    0  BL 1 ( A, B )  1 . Proof 2. For any two SVNSs A and B, A=B  TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x)  T A ( x)  T B ( x)  0 , I A ( x)  I B ( x)  0 , F A ( x )  F B ( x)  0  BL 1 ( A, B) 1 . Conversely, for BL 1 ( A, B) 1 , we have,  T A ( x)  T B ( x)  0 , I A ( x)  I B ( x)  0 , Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments 15 Neutrosophic Sets and Systems, Vol. 20, 2018 F A ( x)  F B ( x)  0  T A ( x)  T B ( x) , I A ( x)  I B ( x) , F A ( x)  F B ( x)  A = B. Proof 3. We have, T A ( x)  T B ( x)  T B ( x)  T A ( x) , I A ( x)  I B ( x)  I B ( x)  I A ( x) , F A ( x)  F B ( x)  F B ( x)  F A ( x)  BL 1 ( A, B)  BL 1 ( B, A) . Proof 4. For A  B  C, we have, TA(x)  TB(x)  TC(x), IA(x)  IB(x)  IC(x), FA(x)  FB(x)  FC(x) for x  X.  T A ( x)  T B ( x)  T A ( x)  T C ( x) , T B ( x)  T C ( x)  T A ( x)  T C ( x) ; I A ( x)  I B ( x)  I A ( x)  I C ( x) , I B ( x)  I C ( x)  I A ( x)  I C ( x) ; F A ( x)  F B ( x)  F A ( x)  F C ( x) , F B ( x)  F C ( x)  F A ( x)  F C ( x) .  BL 1 ( A, C )  BL 1 ( A, B) and BL 1 ( A, C )  BL 1 ( B, C ) . 3.2. Binary logarithm similarity measures of SVNSs ( type-II) Definition 3. [51] Let A = <x(TA(xi), IP(xi), FP(xi))> and B = <x(TB(xi), IB(xi), FB(xi))> be any two SVNSs. The binary logarithm similarity measure (type-II) between SVNSs A and B are defined as follows: BL 2 ( A, B) = 1 n  n  TA ( xi )  TB ( xi ) , I A ( xi )  I B ( xi ) ,     (2)  A i B i  log  2  max  F ( x )  F ( x ) 2 i 1  Theorem 2. The binary logarithm similarity measure BL 2 ( A, B) between any two SVNSs A and B satisfy the following properties: P 1. 0  BL 2 ( A, B )  1 P 2. BL 2 ( A, B) 1 , if and only if A = B P 3. BL 2 ( A, B)  BL 2 ( B, A) P4. If C is a SVNS in X and A  B  C then BL 2 ( A, C )  BL 2 ( A, B) and BL 2 ( A, C )  BL 2 ( B, C ) . Proof. Proofs of the properties are shown in [51]. 3.3. Weighted binary logarithm similarity measures of SVNSs for type-I Definition 4. Let A = <x(TA(xi), IP(xi), FP(xi))> and B = <x(TB(xi), IB(xi), FB(xi))> be any two SVNSs. Then the weighted binary logarithm similarity measure for type-I between SVNSs A and B are defined as follows: BL1 ( A, B) = w   1  TA ( x i )  TB ( x i )  I A ( x i )  I B ( x i )    n   w i log 2  2      3  FA ( x i )  FB ( x i ) i 1     (3) n Here, 0  wi  1 and  wi 1 . i 1 Theorem 3. The weighted binary logarithm similarity measures BL1w ( A, B) between SVNSs A and B satisfy the following properties: P 1. 0  BL 1w ( A, B )  1 P 2. BL 1w ( A, B)  1 , if and only if A = B P 3. BL 1w ( A, B)  BL 1w ( B, A) P4. If C is a SVNS in X and A  B  C, then BL1w ( A, C )  BL1w ( A, B ) and BL1w ( A, C )  BL1w ( B, C ) ; n  wi 1 . i 1 Proof 1. From the definition of SVNSs A and B, we write, 0 ≤ TA(x) + IA( x) + FA(x) ≤ 3 and 0 ≤ TB(x) + IB( x) + FB(x) ≤ 3  TA ( xi )  TB ( xi ) , I A ( xi )  I B ( xi ) ,   1  0  max    F (x )  F (x ) B i   A i  0  TA ( xi )  TB ( xi )  I A ( xi )  I B ( xi )  FA ( xi )  FB ( xi )  3 , n  0  BL1 ( A, B)  1 . since,  wi  1 . w i 1 Proof 2. For any two SVNSs A and B if A = B, then we have, TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x)  T A ( x )  T B ( x)  0 , I A ( x)  I B ( x)  0 , F A ( x)  F B ( x)  0 Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments Neutrosophic Sets and Systems, Vol. 20, 2018 16 n  BL1w ( A, B )  1 , (t = 1, 2), since  wi  1 . BL 2 ( A, B) = w i 1 Conversely,  n For BL1w ( A, B )  1 , then we have,  T A ( x )  T B ( x )  0 , I A ( x)  I B ( x)  0 , F A ( x)  F B ( x)  0  T A ( x)  T B ( x) , I A ( x )  I B ( x ) , F A ( x )  F B ( x )  TA ( xi )  TB ( xi ) , I A ( xi )  I B ( xi ) ,     A i B i   w log  2  max F ( x )  F ( x ) 2 i i 1  (4) n Here, 0  wi  1 and  wi 1 . i 1 n  A = B, since  wi  1 . i 1 Proof 3. For any two SVNSs A and B, we have, T A ( x)  T B ( x)  T B ( x)  T A ( x) , I A ( x)  I B ( x)  I B ( x )  I A ( x ) , Proof. For proof, see [51]. 3.3. Hybrid binary logarithm similarity measures (HBLSM) for SVNSs F A ( x)  F B ( x)  F B ( x)  F A ( x)  BL1w ( A, B )  BL1w ( B, A) for. Definition 6. Let A = <x(TA(xi), IP(xi), FP(xi))> and B = <x(TB(xi), IB(xi), FB(xi))> be any two SVNSs. The hybrid binary logarithm similarity measure between SVNSs A and B is defined as follows: Proof 4. BL Hyb  A, B  = For A  B  C, we have, TA(x)  TB(x)  TC(x), IA(x)  IB(x)  IC(x), FA(x)  FB(x)  FC(x) for x  X.  T A ( x)  T B ( x)  T A ( x)  T C ( x) , T B ( x)  T C ( x)  T A ( x)  T C ( x) ; I A ( x)  I B ( x)  I A ( x)  I C ( x) , I B (x)  I C (x)  I A (x)  I C (x) ; F A ( x)  F B ( x)  F A ( x)  F C ( x) , F B ( x)  F C ( x)  F A ( x)  F C ( x) . and BL1w ( A, C )  BL1w ( B, C )  BL1w ( A, C )  BL1w ( A, B ) since in1 wi 1 . 3.4. Weighted binary logarithm similarity measures of SVNSs for type-II Definition 5. [51] Let A = <x(TA(xi), IP(xi), FP(xi))> and B = <x(TB(xi), IB(xi), FB(xi))> be any two SVNSs. Then the weighted binary logarithm similarity measure (type-II between SVNSs A and B is defined as follows:      TA ( xi )  TB ( xi )          n       log  2   1   I ( x )  I ( x )     2 A i B i  3   i 1       FA ( xi )  FB ( xi )          1       n   TA ( xi )  TB ( xi ) ,     n    (1  ) log  2  max  I ( x )  I ( x ) ,   2 A i B i      i 1  F ( x )  F ( x )     B i   A i    Here, 0    1 .  (5)  Theorem 4. The hybrid binary logarithm similarity measure BL Hyb  A, B  between any two SVNSs A and B satisfy the following properties: P 1. 0  BL Hyb ( A, B)  1 P 2. BL Hyb ( A, B) 1 , if and only if A = B P 3. BL Hyb ( A, B)  BLHyb ( B, A) P4. If C is a SVNS in X and A  B  C then BL Hyb ( A, C )  BL Hyb ( A, B) and BL Hyb ( A, C )  BL Hyb ( B, C ) . Proof 1. From the definition of SVNS, we write, 0 ≤ TA(x)+ IA( x)+ FA(x) ≤ 3 and 0 ≤ TB(x) + IB(x) + FB(x) ≤ 3 Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri, Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments 17 Neutrosophic Sets and Systems, Vol. 20, 2018  TA ( xi )  TB ( xi ) , I A ( xi )  I B ( xi ) ,   1   F (x )  F (x ) B i   A i 0  TA ( xi )  TB ( xi )  I A ( xi )  I B ( xi ) ;   FA ( xi )  FB ( xi )  3  0  max   0  BL Hyb ( A, B) 1 . Proof 2. For any two SVNSs A and B, for A = B, we have,  TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x)  T A ( x )  T B ( x )  0 , I A ( x)  I B ( x)  0 , F A ( x)  F B ( x)  0  BL Hyb ( A, B)  1 . Conversely, for BL Hyb ( A, B)  1 , we have, T A ( x )  T B ( x )  0 , I A ( x)  I B ( x)  0 , F A ( x)  F B ( x)  0  T A ( x)  T B ( x) , I A ( x )  I B ( x ) , F A ( x )  F B ( x )  A = B. and BL Hyb ( A, C )  BL Hyb ( B, C ) . 3.4. Weighted hybrid binary logarithm similarity measures (WHBLSM) for SVNSs Definition 7. Let A = <x(TA(xi), IP(xi), FP(xi))> and B = <x(TB(xi), IB(xi), FB(xi))> be any two SVNSs. The weighted hybrid binary logarithm similarity measure between SVNSs A and B is defined as follows: BL wHyb  A, B  =      TA ( xi )  TB ( xi )         n       w log  2   1   I ( x )  I ( x )    A i B i 2  3   i 1 i      FA ( xi )  FB ( xi )                 TA ( xi )  TB ( xi ) ,   n   (1  )  wi log 2 2  max  I A ( xi )  I B ( xi ) ,    i 1  F (x )  F (x )   B i  A i     Here, 0    1 . A and B satisfy the following properties: P1. 0  BL wHyb ( A, B) 1 For any two SVNSs A and B, we have, T A ( x)  T B ( x)  T B ( x)  T A ( x) , P 2. BL wHyb ( A, B)  1 , if and only if A = B P 3. BL wHyb ( A, B)  BL wHyb ( B, A) F A ( x)  F B ( x)  F B ( x)  F A ( x)  BL Hyb ( A, B)  BL Hyb ( B, A) . P4. If C is a SVNS in then BL wHyb ( A, C )  BL wHyb ( A, B) and X A  B  C, and BL wHyb ( A, C )  BL wHyb ( B, C ) . Proof 4. Proof 1. For A  B  C, we have, TA(x)  TB(x)  TC(x), IA(x)  IB(x)  IC(x), FA(x)  FB(x)  FC(x) for x  X.  T A ( x)  T B ( x)  T A ( x)  T C ( x) , From the definition of SVNS, we write, 0 ≤ TA(x)+ IA( x)+ FA(x) ≤ 3 and 0 ≤ TB(x) + IB(x) + FB(x) ≤ 3  TA ( xi )  TB ( xi ) , I A ( xi )  I B ( xi ) ,   1  0  max   F (x )  F (x )  A i B i   T B ( x)  T C ( x)  T A ( x)  T C ( x) ; I A ( x)  I B ( x)  I A ( x)  I C ( x) , (6) Theorem 5. The weighted hybrid binary logarithm similarity measure BL wHyb ( A, B) between any two SVNSs Proof 3. I A ( x)  I B ( x)  I B ( x )  I A ( x ) ,                    0  TA ( xi )  TB ( xi )  I A ( xi )  I B ( xi ) I B ( x)  I C ( x)  I A ( x)  I C ( x) ; F A ( x)  F B ( x)  F A ( x)  F C ( x) , F B ( x)  F C ( x)  F A ( x)  F C ( x) . Proof 2.  BL Hyb ( A, C )  BL Hyb ( A, B ) For any two SVNSs A and B,   FA ( xi )  FB ( xi )  3 ; 0  BL wHyb ( A, B) 1 . Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments Neutrosophic Sets and Systems, Vol. 20, 2018 18 for A = B, we have,  TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x)  T A ( x )  T B ( x )  0 , I A ( x)  I B ( x)  0 , F A ( x)  F B ( x)  0  BL wHyb ( A, B )  1 . Conversely, for BL wHyb ( A, B )  1 , we have, T A ( x)  T B ( x)  0 , I A ( x)  I B ( x)  0 , paper, we define an entropy measure for determining unknown attribute weights. The entropy function of a SVNS P Definition 8. = T ijP ( x), I ijP ( x), F ijP ( x) (i = 1, 2, ..., m; j = 1, 2, ..., n) is defined as follows: E j ( P)  1  wj    1 m P  T ( x)  F ijP ( x) 1 2 I ijP ( x) n i 1 ij 2 (7) 1  E j ( P) n  nj1 E j ( P ) (8) F A ( x)  F B ( x)  0  T A ( x)  T B ( x) , I A ( x )  I B ( x ) , F A ( x )  F B ( x )  A = B. Theorem 6. The entropy function E j (P) satisfies the Proof 3. following properties: For any two SVNSs A and B, we have, T A ( x)  T B ( x)  T B ( x)  T A ( x) , P1. E j ( P )  0 , if T ij 1, F ij  I ij  0 . I A ( x)  I B ( x)  I B ( x )  I A ( x ) , F A ( x)  F B ( x)  F B ( x)  F A ( x)  BL wHyb ( A, B )  BL wHyb ( B, A) . n Here,  w j  1 j 1 P2. E j ( P )  1 , if T ij , I ij , F ij  0.5, 0.5, 0.5 . P3. E j ( P)  E j (Q) , if T ijP  F ijP  T Qij  F Qij ; I ijP  I Qij . P4. E j ( P)  E j ( P c ) . Proof 4. Proof 1. For A  B  C, we have, TA(x)  TB(x)  TC(x), IA(x)  IB(x)  IC(x), FA(x)  FB(x)  FC(x) for all x  X.  T A ( x)  T B ( x)  T A ( x)  T C ( x) , T ij 1, F ij  I ij  0 n 1 n  E j ( P )  1   1  0   1   0 n i 1 n Proof 2. T B ( x)  T C ( x)  T A ( x)  T C ( x) ; I A ( x)  I B ( x)  I A ( x)  I C ( x) , I B ( x)  I C ( x)  I A ( x)  I C ( x) ; F A ( x)  F B ( x)  F A ( x)  F C ( x) , F B ( x)  F C ( x)  F A ( x)  F C ( x) .  BL wHyb ( A, C )  BL wHyb ( A, B ) and BL wHyb ( A, C )  BL wHyb ( B, C ) . 4. A new entropy measure for SVNSs Entropy strategy [52] is an important contribution for determining indeterminate information. Zhang et al. [53] introduced the fuzzy entropy. Vlachos and Sergiadis [54] proposed entropy function for intuitionistic fuzzy sets. Majumder and Samanta [55] developed some entropy measures for SVNSs. When attribute weights are completely unknown to decision makers, the entropy measure is used to calculate attribute weights. In this T ij , I ij , F ij  0.5, 0.5, 0.5 .  E j ( P)  1  1 n  0.5  0.5  0  1  0  1 n i 1 Proof 3. P Q P P Q Q T ij  F ij  T ij  F ij , I ij  I ij   T ijP  F ijP 1 2 I ijP    T Qij  F Qij 1 2 I Qij  m i 1 2 m 2 i 1 2 2 1 m P 1 m  T ij  F ijP 1 2 I ijP    T Qij  F Qij 1 2 I Qij   n i 1 n i 1 2 2 1 m P 1 m Q P P Q Q  1   T ij  F ij 1 2 I ij   1   T ij  F ij 1 2 I ij  n i 1 n i 1  E j ( P )  E j (Q) . Proof 4. c Since T ij , I ij , F ij  F ij ,1  I ij , T ij , we have E j ( P)  E j ( P c ) . Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments 19 Neutrosophic Sets and Systems, Vol. 20, 2018 5. MAGDM strategy based on weighted hybrid binary logarithm similarity measure for SVNSs Assume that (P1, P2, ..., Pm) be the alternatives, (C1, C2, ..., Cn) be the attributes of each alternative, and {D1, D2, ..., Dr} be the decision makers. Decision makers provide the rating of alternatives based on the predefined attribute. Each decision maker constructs a neutrosophic decision matrix associated with the alternatives based on each attribute shown in Equation (9). Using the following steps, we present the MAGDM strategy (see figure 1) based on weighted hybrid binary logarithm similarity measure (WHBLSM). Step 1: Determine the relation between the alternatives and the attributes At first, each decision maker prepares decision matrix. The relation between alternatives Pi (i = 1, 2, ..., m) and the attribute Cj (j = 1, 2, ..., n) corresponding to each decision maker is presented in the Equation (9). Dr [ P | C ] =   D P1  T 11r ,  Dr , P 2  T 21    P m  T mD1r , Dr I 11 D I 21r C2 , , Dr F11 D F 21r  Dr , T 12 Dr T 22 , Dr , I 12 Dr I 22 , Dr F12 D F 22r  I mD1r , F mD1r T mD2r , I mD2r , F mD2r      Cn T 1Dnr , D T 2 nr , I 1Dnr , D I 2 nr , F 1Dnr D F 2 nr  Dr Dr Dr , I mn , F mn T mn          (9) Here, I ijDr , F ijDr (i = 1, 2, ..., m; j = 1, 2, ..., n) is the single valued neutrosophic rating value of the alternative Pi with respect to the attribute Cj corresponding to the decision maker Dr. Step 2: Determine the core decision matrix We form a new decision matrix, called core decision matrix to combine all the decision maker’s opinions into a group opinion. Core decision matrix minimizes the biasness which is imposed by different decision makers and hence credibility to the final decision increases. The core decision matrix is presented in Equation (10). D[ P | C ] = C1 r  t 1 Dt T 11 Dt , I 11  C2 Dt , F 11 r  t 1 Dt T 11 Dt , I 11 Dt , F 11 r r Dt , Dt , Dt  T 11 I 11 F 11 D t , Dt , Dt  T 11 I 11 F 11 r t 1 r  r Dt , Dt , Dt  T 11 I 11 F 11 t 1   r  r D t , Dt , Dt  T 11 I 11 F 11 t 1 r r t 1 Dt T 11 Dt , I 11 Dt , F 11 r r r t 1 Cn r    D t , Dt , Dt  T 11 I 11 F 11 t 1 r r  D t , Dt , Dt  T 11 I 11 F 11 t 1 r                (10) Step 3: Determine the ideal solution The evaluation of attributes can be categorized into benefit attribute and cost attribute. An ideal alternative can be determined by using a maximum operator for the benefit attributes and a minimum operator for the cost attributes for determining the best value of each attribute among all the alternatives. An ideal alternative [42] is presented as follows: P* = {C1*, C2*, … , Cm*}. where the benefit attribute is C1 T ijDr ,      P1   P  2     P  m  (P ) (P ) (P ) C*j  maxT C j i , min I C j i , min F C j i i i i (11) and the cost attribute is (P ) (P ) (P ) C*j  min T C j i , max I C j i , max F C j i i i i (12) Step 4: Determine the attribute weights Using Equation (8), determine the weights of the attribute. Step 5: Determine the WHBLSM values Using Equation (6), calculate the weighted similarity measures for each alternative. Step 6: Ranking the priority All the alternatives are preference ranked based on the decreasing order of calculated measure values. The highest value reflects the best alternative. Step 7: End. 6. An illustrative example Suppose that a state government wants to construct an ecotourism park for the development of state tourism and especially for mental refreshment of children. After initial screening, three potential spots namely, spot-1 (P1), spot-2 (P2), and spot-3 (P3) remain for further selection. A team Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments Neutrosophic Sets and Systems, Vol. 20, 2018 20 of three decision makers, namely, D1, D2, and D3 has been constructed for selecting the most suitable spot with respect to the following attributes.  Ecology (C1),  Costs (C2),  Technical facility (C3),  Transport (C4),  Risk factors (C5) The steps of decision-making strategy to select the best potential spot to construct an eco-tourism park based on the proposed strategy are stated below: D3 [ P | C ]  C1   0.7,   P1 0.4,  0 .3   0.6,  0.2,  P2  0 .3   0.6,  0.2,  P3  0 .3  6.1. Steps of MAGDM strategy Step 2: Determine the core decision matrix We present MAGDM strategy based on the proposed Using Equation (10), we construct the core decision matrix for all decision makers shown in Equation (16). WHBLSM using the following steps. Step 1: Determine the relation between alternatives and attributes The relation between alternatives P1, P2 and P3 and the attribute set {C1, C2, C3, C4, C5} corresponding to the set of decision makers {D1, D2, D3} are presented in Equations (13), (14), and (15). D1[ P | C ]  C1   0.7,   P1 0.4,  0 .4   0.4,  0.3,  P2  0 .6   0.4,  0.2,  P3  0.3  C2 0.7, 0.4, 0.3 0.5, 0.2, 0.5 0.8, 0.1, 0.3 C3 0.8, 0.1, 0.1 0.6, 0.2, 0.2 0.5, 0.4, 0.4 C4 0.7, 0.2, 0,1 0.7, 0.3, 0.3 0.5, 0.2, 0.2 C5 0.6, 0.5, 0.5 0.4, 0.3, 0.4 0.7, 0.3, 0.2                 C2 0.7, 0.4, 0.4 0.5, 0.2, 0.4 0.8, 0.2, 0.2 C3 0.8, 0.2, 0.2 0.5, 0.3, 0.3 0.5, 0.3, 0.3 C4 0.5, 0.2, 0,2 0.8, 0.3, 0.3 0.7, 0.2, 0.2 C5 0.5, 0.5, 0.4 0.4, 0.1, 0.4 0.7, 0.4, 0.2                 (13) C1 0.5, 0.2, 0.3 0.5, 0.4, 0.4 0.4, 0.2, 0.5 C3 0.6, 0.3, 0.3 0.7, 0.4, 0.4 0.5, 0.3, 0.3 C4 0.7, 0.2, 0,5 0.5, 0.3, 0.4 0.7, 0.4, 0.2                 C5 0.5, 0.6, 0.5 0.3, 0.4, 0.4 0.5, 0.6, 0.4 (15) D[ P | C ]    P  1     P2      P3   C1 0.984, 0.324, 0.332 0.938, 0.292, 0.420 0.949, 0.203, 0.359 C2 0.988, 0.324, 0.232 0.956, 0.162, 0.395 0.994, 0.203, 0.232 C3 0.989, 0.184, 0.184 0.979, 0.292, 0.292 0.956, 0.334, 0.334 C4 0.956, 0.203, 0.219 0.989, 0.304, 0.334 0.984, 0.255, 0.203 C5 0.961, 0.452, 0.219 0.908, 0.232, 0.404 0.984, 0.420, 0.255                 (16) Step 3: Determine the ideal solution Here, C3 and C4 denote benefit attributes and C1, C2 and C5 denote cost attributes. Using Equations (11) and (12), we calculate the ideal solutions as follows:  0.938, 0.324, 0.420 , 0.956 , 0.324, 0.395   P *   0.989, 0.184, 0.184 , 0.989, 0.203, 0.203    0.908, 0.452, 0.404 D2 [ P | C ]      P1      P2      P3   C2 0.8, 0.2, 0.1 0.5, 0.1, 0.3 0.6, 0.4, 0.2 ,   ,.    Step 4: Determine the attribute weights (14) Using Equation (8), we calculate the attribute weights as follows: [w1, w2, w3, w4, w5] = [0.1680, 0.3300, 0.2285, 0.2485, 0.0250] Step 5: Determine the weighted hybrid binary logarithm similarity measures Using Equation (6), we calculate similarity values for alternatives shown in Table 1. Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments 21 Neutrosophic Sets and Systems, Vol. 20, 2018 Step 6: Ranking the alternatives Ranking order of alternatives is prepared as the descending order of similarity values. Highest value indicates the best alternative. Ranking results are shown in Table 1 for different values of  . 10. Conclusion Conclusions in the paper are concise as follows: 1. Step 7. End. 7. Sensitivity analysis In this section, we discuss the variation of ranking results (see Table 1) for different values of  . From the results shown in Tables 1, we observe that the proposed strategy provides the same ranking order for different values of  . 2. 8. Comparison analysis 4. In this section, we solve the problem with different existing strategies [33, 37, 38, 56]. Outcomes are furnished in the Table 2 and figure 2. 5. 6. 3. 9. Contributions of the proposed strategy     We propose two types of binary logarithm similarity measures and their hybrid similarity measure for SVNS environment. We have proved their basic properties. To calculate unknown weights structure of attributes in SVNS environment, we have proposed a new entropy function. We develop a decision making strategy based on the proposed weighted hybrid binary logarithm similarity measure (WHBLSM). We have solved a illustrative example to show the feasibility, applicability, and effectiveness of the proposed strategy. 7. We have proposed hybrid binary logarithm similarity measure and weighted hybrid binary logarithm similarity measure for dealing indeterminacy in decision making situation. We have defined a new entropy function to determine unknown attribute weights. We have developed a new MAGDM strategy based on the proposed weighted hybrid binary logarithm similarity measure. We have presented a numerical example to illustrate the proposed strategy. We have conducted a sensitivity analysis We have presented comparative analyses between the obtained results from the proposed strategies and different existing strategies in the literature. The proposed weighted hybrid binary logarithm similarity measure can be applied to solve MAGDM problems in clustering analysis, pattern recognition, personnel selection, etc. Future research can be continued to investigate the proposed similarity measures in neutrosophic hybrid environment for tackling uncertainty, inconsistency and indeterminacy in decision making. The concept of the paper can be applied in practical decisionmaking, supply chain management, data mining, cluster analysis, teacher selection etc. Table 1 Ranking order for different values of  . Similarity measures () BLwHyb ( P*, Pi ) 0.10 0.25 0.40 0.55 0.70 0.90 BLwHyb ( P*, Pi ) BLwHyb ( P*, Pi ) BLwHyb ( P*, Pi ) BLwHyb ( P*, Pi ) BLwHyb ( P*, Pi ) Measure values BL wHyb ( P*, P1)  0.9426 ; BL wHyb ( P*, P 2)  0.9233 ; BL wHyb ( P*, P 3)  0.9101 BL wHyb ( P*, P1)  0.9479 ; BL wHyb ( P*, P 2)  0.9296 ; BL wHyb ( P*, P 3)  0.9153 BL wHyb ( P*, P1)  0.9532 ; BL wHyb ( P*, P 2)  0.9357 ; BL wHyb ( P*, P 3)  0.9207 BL wHyb ( P*, P1)  0.9585 ; BL wHyb ( P*, P 2)  0.9419 ; BLwHyb ( P*, P3)  0.9260 BL wHyb ( P*, P1)  0.9638 ; BLwHyb ( P*, P2)  0.9482 ; BL wHyb ( P*, P 3)  0.9313 BL wHyb ( P*, P1)  0.9708 ; BL wHyb ( P*, P 2)  0.9565 ; BL wHyb ( P*, P 3)  0.9384 Ranking order P1  P2  P3 P1  P2  P3 P1  P2  P3 P1  P2  P3 P1  P2  P3 P1  P2  P3 Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments Neutrosophic Sets and Systems, Vol. 20, 2018 22 Table 2 Ranking order for different existing strategies Similarity measures Mondal and Pramanik [37] Ye [33] Biswas et al. [56] (  0.55) Ye and Fu [38] Proposed strategy (  0.55) Measure values for P1, P2 and P3 0.8901, 0.8679, 0.8093 0.8409, 0.8189, 0.7766 0.9511, 0.9219, 0.9007 0.9161, 0.8758, 0.7900 0.9585, 0.9419, 0.9260 Ranking order P1  P2  P3 P1  P2  P3 P1  P2  P3 P1  P2  P3 P1  P2  P3 WHBLSM based decision making strategy Decision making analysis phase Determination of the relation between alternatives and attributes Determine the core decision matrix Step-1 Step- 2 Determine ideal solution Step- 3 Determine the attribute weights Step-4 Calculate the WHBLSM values Ranking the alternatives Step-5 Step- 6 Fig. 1: Decision making phases of the proposed approach Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. 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[64] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116. Received : March 19, 2018. Accepted : April 9, 2018. Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assesments 26 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico Generalizations of Neutrosophic Subalgebras in BCK/BCI-Algebras Based on Neutrosophic Points Seon Jeong Kim1 , Seok-Zun Song2 and Young Bae Jun1 1 Department of Mathematics, Natural Science of College, Gyeongsang National University, Jinju 52828, Korea. E-mail: skim@gnu.ac.kr 2 Department 3 Department of Mathematics, Jeju National University, Jeju 63243, Korea. E-mail: szsong@jejunu.ac.kr of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea. E-mail: skywine@gmail.com ∗ Correspondence: Y.B. Jun (skywine@gmail.com) Abstract: Saeid and Jun introduced the notion of neutrosophic points, and studied neutrosophic subalgebras of several types in BCK/BCI-algebras by using the notion of neutrosophic points (see [4] and [6]). More general form of neutrosophic points is considered in this paper, and generalizations of Saeid and Jun’s results are discussed. The concepts of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra, (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra and (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra are introduced, and several properties are investigated. Characterizations of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra are discussed. Keywords: (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra; (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra; (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra. 1 Introduction terizations of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. We consider relations between (∈, ∈)-neutrosophic subalgebra, (∈, As a generalization of fuzzy sets, Atanassov [1] introduced the q(kT ,kI ,kF ) )-neutrosophic subalgebra and (∈, ∈ ∨q(kT ,kI ,kF ) )degree of nonmembership/falsehood (f) in 1986 and defined the neutrosophic subalgebra. intuitionistic fuzzy set. As a more general platform which extends the notions of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued (intuitionistic) fuzzy set, Smaran- 2 Preliminaries dache introduced the notion of neutrosophic sets (see [7, 8]), which is useful mathematical tool for dealing with incomplete, By a BCI-algebra, we mean a set X with a binary operation ∗ inconsistent and indeterminate information. For further particu- and the special element 0 satisfying the conditions (see [3, 5]): lars on neutrosophic set theory, we refer the readers to the site (a1) (∀x, y, z ∈ X)(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), http://fs.gallup.unm.edu/FlorentinSmarandache.htm Jun [4] introduced the notion of (Φ, Ψ)-neutrosophic subalgebra of a BCK/BCI-algebra X for Φ, Ψ ∈ {∈, q, ∈ ∨ q}, and investigated related properties. He provided characterizations of an (∈, ∈)-neutrosophic subalgebra and an (∈, ∈ ∨ q)-neutrosophic subalgebra, and considered conditions for a neutrosophic set to be a (q, ∈ ∨ q)-neutrosophic subalgebra. Saeid and Jun [6] gave relations between an (∈, ∈ ∨ q)-neutrosophic subalgebra and a (q, ∈ ∨ q)-neutrosophic subalgebra, and investigated properties on neutrosophic q-subsets and neutrosophic ∈ ∨ q-subsets. The purpose of this article is to give an algebraic tool of neutrosophic set theory which can be used in applied sciences, for example, decision making problems, medical sciences etc. We consider a general form of neutrosophic points, and then we discuss generalizations of the papers [4] and [6]. As a generalization of (∈, ∈ ∨ q)-neutrosophic subalgebras, we introduce the notions of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra, and (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra in BCK/BCIalgebras, and investigate several properties. We discuss charac- (a2) (∀x, y ∈ X)((x ∗ (x ∗ y)) ∗ y = 0), (a3) (∀x ∈ X)(x ∗ x = 0), (a4) (∀x, y ∈ X)(x ∗ y = y ∗ x = 0 ⇒ x = y). If a BCI-algebra X satisfies the axiom (a5) 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra (see [3, 5]). A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X (see [3, 5]) if x ∗ y ∈ S for all x, y ∈ S. The collection of all BCK-algebras and all BCI-algebras are denoted by BK (X) and BI (X), respectively. Also B(X) := BK (X) ∪ BI (X). We refer the reader to the books [3] and [5] for further information regarding BCK/BCI-algebras. Let X be a non-empty set. A neutrosophic set (NS) in X (see [7]) is a structure of the form: A := {hx; AT (x), AI (x), AF (x)i | x ∈ X} (2.1) S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points 27 Neutrosophic Sets and Systems, Vol. 20, 2018 where AT , AI and AF are a truth membership function, an inde- (0, 1] and γ ∈ [0, 1), we consider the following sets: terminate membership function and a false membership function, TqkT (A; α) := {x ∈ X | AT (x) + α + kT > 1}, respectively, from X into the unit interval [0, 1]. The neutroIqkI (A; β) := {x ∈ X | AI (x) + β + kI > 1}, sophic set (2.1) will be denoted by A = (AT , AI , AF ). FqkF (A; γ) := {x ∈ X | AF (x) + γ + kF < 1}, Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ T∈∨ qkT (A; α) := {x ∈ X | AT (x) ≥ α or (0, 1] and γ ∈ [0, 1), we consider the following sets (see [4]): AT (x) + α + kT > 1}, (A; β) := {x ∈ X | AI (x) ≥ β or I ∈∨ q kI T∈ (A; α) := {x ∈ X | AT (x) ≥ α}, AI (x) + β + kI > 1}, I∈ (A; β) := {x ∈ X | AI (x) ≥ β}, F∈∨ qkF (A; γ) := {x ∈ X | AF (x) ≤ γ or F∈ (A; γ) := {x ∈ X | AF (x) ≤ γ}, AF (x) + γ + kF < 1}. Tq (A; α) := {x ∈ X | AT (x) + α > 1}, We say TqkT (A; α), IqkI (A; β) and FqkF (A; γ) are neuIq (A; β) := {x ∈ X | AI (x) + β > 1}, trosophic qk -subsets; and T∈∨ qkT (A; α), I∈∨ qkI (A; β) and Fq (A; γ) := {x ∈ X | AF (x) + γ < 1}, F∈∨ qkF (A; γ) are neutrosophic (∈ ∨ qk )-subsets. For Φ ∈ {∈, T∈∨ q (A; α) := {x ∈ X | AT (x) ≥ α or AT (x) + α > 1}, q, qk , qkT , qkI , qkF , ∈ ∨ q, ∈ ∨ qk , ∈ ∨ qkT , ∈ ∨ qkI , ∈ ∨ qkF }, I∈∨ q (A; β) := {x ∈ X | AI (x) ≥ β or AI (x) + β > 1}, the element of TΦ (A; α) (resp., IΦ (A; β) and FΦ (A; γ)) is called a neutrosophic TΦ -point (resp., neutrosophic IΦ -point and neuF∈∨ q (A; γ) := {x ∈ X | AF (x) ≤ γ or AF (x) + γ < 1}. trosophic FΦ -point) with value α (resp., β and γ). We say T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are neutrosophic ∈-subsets; Tq (A; α), Iq (A; β) and Fq (A; γ) are neutrosophic qsubsets; and T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are neutrosophic ∈ ∨ q-subsets. It is clear that T∈∨ q (A; α) = T∈ (A; α) ∪ Tq (A; α), I∈∨ q (A; β) = I∈ (A; β) ∪ Iq (A; β), F∈∨ q (A; γ) = F∈ (A; γ) ∪ Fq (A; γ). T∈∨ qkT (A; α) = T∈ (A; α) ∪ TqkT (A; α), (3.1) I∈∨ qkI (A; β) = I∈ (A; β) ∪ IqkI (A; β), (3.2) F∈∨ qkF (A; γ) = F∈ (A; γ) ∪ FqkF (A; γ). (3.3) (2.2) (2.3) Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ (2.4) (0, 1] and γ ∈ [0, 1), we consider the following sets: Given Φ, Ψ ∈ {∈, q, ∈ ∨ q}, a neutrosophic set A = (AT , AI , AF ) in X ∈ B(X) is called a (Φ, Ψ)-neutrosophic subalgebra of X (see [4]) if the following assertions are valid. x ∈ TΦ (A; αx ), y ∈ TΦ (A; αy ) ⇒ x ∗ y ∈ TΨ (A; αx ∧ αy ), x ∈ IΦ (A; βx ), y ∈ IΦ (A; βy ) ⇒ x ∗ y ∈ IΨ (A; βx ∧ βy ), x ∈ FΦ (A; γx ), y ∈ FΦ (A; γy ) ⇒ x ∗ y ∈ FΨ (A; γx ∨ γy ) It is clear that T∈∗ (A; α) := {x ∈ X | AT (x) > α}, I∈∗ (A; β) := {x ∈ X | AI (x) > β}, F∈∗ (A; γ) := {x ∈ X | AF (x) < γ}. (3.4) (3.5) (3.6) Proposition 3.1. For any neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ (0, 1] and γ ∈ [0, 1), we have (2.5) for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). α≤ β≤ γ≥ α> β> γ< 1−k 2 1−k 2 1−k 2 1−k 2 1−k 2 1−k 2 ⇒ Tqk (A; α) ⊆ T∈∗ (A; α), (3.7) Iqk (A; β) ⊆ I∈∗ (A; β), Fqk (A; γ) ⊆ F∈∗ (A; γ), (3.8) (3.9) ⇒ T∈ (A; α) ⊆ Tqk (A; α), (3.10) ⇒ I∈ (A; β) ⊆ Iqk (A; β), (3.11) ⇒ F∈ (A; γ) ⊆ Fqk (A; γ). (3.12) ⇒ ⇒ Generalizations of (∈, ∈ ∨q)-neutrosophic subalgebras 1+k Proof. If α ≤ 1−k 2 , then 1 − α ≥ 2 and α ≤ 1 − α. Assume that x ∈ Tqk (A; α). Then AT (x) + k > 1 − α ≥ 1+k 2 , and 1−k 1+k ∗ so AT (x) > 2 − k = 2 ≥ α. Hence x ∈ T∈ (A; α). Similarly, we have the result (3.8). Suppose that γ ≥ 1−k 2 and let x ∈ F (A; γ). Then A (x) + γ + k < 1, and thus q F k In what follows, let kT , kI and kF denote arbitrary elements of [0, 1) unless otherwise specified. If kT , kI and kF are the same 1−k AF (x) < 1 − γ − k ≤ 1 − 1−k 2 − k = 2 ≤ γ. number in [0, 1), then it is denoted by k, i.e., k = kT = kI = kF . 3 Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ Hence x ∈ F∈∗ (A; γ). Suppose that α > 1−k 2 . If x ∈ T∈ (A; α), S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points Neutrosophic Sets and Systems, Vol. 20, 2018 28 Corollary 3.5. If A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subalgebra of X ∈ B(X), then neutrosophic qk -subsets Tqk (A; α), Iqk (A; β) and Fqk (A; γ) are subalgebras of X for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty. then AT (x) + α + k ≥ 2α + k > 2 · 1−k 2 +k =1 and so x ∈ Tqk (A; α). Hence T∈ (A; α) ⊆ Tqk (A; α). Similarly, we can verify that if β > 1−k If we take kT = kI = kF = 0 in Theorem 3.4, then we have 2 , then I∈ (A; β) ⊆ Iqk (A; β). Supthe following corollary. . If x ∈ F (A; γ), then A (x) ≤ γ, and pose that γ < 1−k ∈ F 2 thus Corollary 3.6 ([4]). If A = (AT , AI , AF ) is an (∈, ∈)1−k neutrosophic subalgebra of X ∈ B(X), then neutrosophic qAF (x) + γ + k ≤ 2γ + k < 2 · 2 + k = 1, subsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subalgebras of X for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty. that is, x ∈ Fqk (A; γ). Hence F∈ (A; γ) ⊆ Fqk (A; γ). Definition 3.2. A neutrosophic set A = (AT , AI , AF ) in X ∈ Definition 3.7. A neutrosophic set A = (AT , AI , AF ) in X ∈ B(X) is called an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra B(X) is called a (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subof X if algebra of X if x ∈ T∈ (A; αx ), y ∈ T∈ (A; αy ) ⇒ x ∗ y ∈ T∈∨qkT (A; αx ∧ αy ), x ∈ I∈ (A; βx ), y ∈ I∈ (A; βy ) ⇒ x ∗ y ∈ I∈∨qkI (A; βx ∧ βy ), x ∈ F∈ (A; γx ), y ∈ F∈ (A; γy ) ⇒ x ∗ y ∈ F∈∨qkF (A; γx ∨ γy ) (3.13) for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). x ∈ TqkT (A; αx ), y ∈ TqkT (A; αy ) ⇒ x ∗ y ∈ T∈∨qkT (A; αx ∧ αy ), x ∈ IqkI (A; βx ), y ∈ IqkI (A; βy ) ⇒ x ∗ y ∈ I∈∨qkI (A; βx ∧ βy ), x ∈ FqkF (A; γx ), y ∈ FqkF (A; γy ) ⇒ x ∗ y ∈ F∈∨qkF (A; γx ∨ γy ) (3.15) for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). An (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra with kT = kI = kF = k is called an (∈, ∈ ∨qk )-neutrosophic subalgebra. A (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra with kT = kI = kF = k is called a (qk , ∈ ∨qk )-neutrosophic subalLemma 3.3 ([4]). A neutrosophic set A = (AT , AI , AF ) in gebra. X ∈ B(X) is an (∈, ∈)-neutrosophic subalgebra of X if and Theorem 3.8. If A = (AT , AI , AF ) is a (q(kT ,kI ,kF ) , ∈ only if it satisfies: ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X ∈ B(X), then neu  AT (x ∗ y) ≥ AT (x) ∧ AT (y) trosophic qk -subsets TqkT (A; α), IqkI (A; β) and FqkF (A; γ) are   1−kI T (∀x, y ∈ X)  AI (x ∗ y) ≥ AI (x) ∧ AI (y)  . (3.14) subalgebras of X for all α ∈ ( 1−k 2 , 1], β ∈ ( 2 , 1] and 1−kF γ ∈ [0, 2 ) whenever they are nonempty. AF (x ∗ y) ≤ AF (x) ∨ AF (y) T Proof. Let x, y ∈ TqkT (A; α) for α ∈ ( 1−k 2 , 1]. Then x ∗ y ∈ T∈∨ qkT (A; α), that is, x ∗ y ∈ T∈ (A; α) or x ∗ y ∈ TqkT (A; α). If x ∗ y ∈ T∈ (A; α), then x ∗ y ∈ TqkT (A; α) by (3.10). Therefore TqkT (A; α) is a subalgebra of X. Similarly, we prove that IqkI (A; β) is a subalgebra of X. Let x, y ∈ FqkF (A; γ) Proof. Let x, y ∈ TqkT (A; α). Then AT (x) + α + kT > 1 and F for γ ∈ [0, 1−k 2 ). Then x ∗ y ∈ F∈∨ qkF (A; γ), and so AT (y) + α + kT > 1. It follows from Lemma 3.3 that x∗y ∈ F∈ (A; γ) or x∗y ∈ FqkF (A; γ). If x∗y ∈ F∈ (A; γ), then x ∗ y ∈ FqkF (A; γ) by (3.12). Hence FqkF (A; γ) is a subalgebra AT (x ∗ y) + α + kT ≥ (AT (x) ∧ AT (y)) + α + kT of X. = (AT (x) + α + kT ) ∧ (AT (y) + α + kT ) > 1 Theorem 3.4. If A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subalgebra of X ∈ B(X), then neutrosophic qk -subsets TqkT (A; α), IqkI (A; β) and FqkF (A; γ) are subalgebras of X for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty. Taking kT = kI = kF = 0 in Theorem 3.8 induces the foland so that x∗y ∈ TqkT (A; α). Hence TqkT (A; α) is a subalgebra lowing corollary. of X. Similarly, we can prove that IqkI (A; β) is a subalgebra of X. Now let x, y ∈ FqkF (A; γ). Then AF (x) + γ + kF < 1 and Corollary 3.9 ([4]). If A = (AT , AI , AF ) is a (q, ∈ ∨ q)AF (y) + γ + kF < 1, which imply from Lemma 3.3 that neutrosophic subalgebra of X ∈ B(X), then neutrosophic qsubsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subalgebras of X AF (x ∗ y) + γ + kF ≤ (AF (x) ∨ AF (y)) + γ + kF for all α, β ∈ (0.5, 1] and γ ∈ [0, 0, 5) whenever they are = (AF (x) + γ + kF ) ∨ (AF (y) + γ + kF ) < 1. nonempty. Hence x ∗ y ∈ FqkF (A; γ) and so FqkF (A; γ) is a subalgebra of We provide characterizations of an (∈, ∈ ∨q(kT ,kI ,kF ) )-neuX. trosophic subalgebra. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points 29 Neutrosophic Sets and Systems, Vol. 20, 2018 Theorem 3.10. Given a neutrosophic set A = (AT , AI , AF ) in that is, a ∗ b ∈ / FqkF (A; γF ). Thus a ∗ b ∈ / F∈∨ qkF (A; γF ), F X ∈ B(X), the following are equivalent. which is a contradiction. If AF (a) ∨ AF (b) < 1−k 2 , then a, b ∈ 1−kF 1−kF / F∈ (A; 2 ). Also, (1) A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic F∈ (A; 2 ) and a ∗ b ∈ subalgebra of X. F F F > 1−k + 1−k = 1 − kF AF (a ∗ b) + 1−k 2 2 2 (2) A = (AT , AI , AF ) satisfies the following assertion. F F and so a∗b ∈ / FqkF (A; 1−k / F∈∨ qkF (A; 1−k V 2 ). Hence a∗b ∈ 2 ), T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 } a contradiction. Therefore V I (3.16) AI (x ∗ y) ≥ {AI (x), AI (y), 1−k _ 2 } F W AF (x ∗ y) ≤ {AF (x), AF (y), 1−k F 2 } AF (x ∗ y) ≤ {AF (x), AF (y), 1−k } 2 for all x, y ∈ X. Conversely, let A = (AT , AI , AF ) be a neutrosophic set in X Proof. Let A = (AT , AI , AF ) be an (∈, ∈ ∨q(kT ,kI ,kF ) )- which satisfies the condition (3.16). Let x, y ∈ X and β , β ∈ x y neutrosophic subalgebra of X. Assume that there exist a, b ∈ X (0, 1] be such that x ∈ I (A; β ) and y ∈ I (A; β ). Then ∈ x ∈ y such that ^ ^ ^ I I } ≥ {βx , βy , 1−k AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 1−kT 2 2 }. AT (a ∗ b) < {AT (a), AT (b), 2 }. for all x, y ∈ X. If AT (a) ∧ AT (b) < Hence 1−kT 2 , then AT (a ∗ b) < AT (a) ∧ AT (b). AT (a ∗ b) < αt ≤ AT (a) ∧ AT (b) for some αt ∈ (0, 1]. It follows that a ∈ T∈ (A; αt ) and b ∈ T∈ (A; αt ) but a ∗ b ∈ / T∈ (A; αt ). Moreover, I I Suppose that βx ≤ 1−k or βy ≤ 1−k 2 2 . Then AI (x ∗ y) ≥ βx ∧ βy , and so x ∗ y ∈ I∈ (A; βx ∧ βy ). Now, assume that 1−kI I I and βy > 1−k βx > 1−k 2 2 . Then AI (x ∗ y) ≥ 2 , and so AI (x ∗ y) + βx ∧ βy > 1−kI 2 + 1−kI 2 = 1 − kI , that is, x ∗ y ∈ IqkI (A; βx ∧ βy ). Hence AT (a ∗ b) + αt < 2αt < 1 − kT , x ∗ y ∈ I∈∨ qkI (A; βx ∧ βy ). and so a ∗ b ∈ / TqkT (A; αt ). Thus a ∗ b ∈ / T∈∨ qkT (A; αt ), a conT T tradiction. If AT (a) ∧ AT (b) ≥ 1−k , then a ∈ T∈ (A; 1−k 2 2 ), 1−kT 1−kT / T∈ (A; 2 ). Also, b ∈ T∈ (A; 2 ) and a ∗ b ∈ Similarly, we can verify that if x ∈ T∈ (A; αx ) and y ∈ T∈ (A; αy ), then x ∗ y ∈ T∈∨ qkT (A; αx ∧ αy ). Finally, let x, y ∈ X and γx , γy ∈ [0, 1) be such that x ∈ F∈ (A; γx ) and y ∈ F∈ (A; γy ). Then _ _ F F {γx , γy , 1−k AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 }≤ 2 }. AT (a ∗ b) + 1−kT 2 < 1−kT 2 + 1−kT 2 = 1 − kT , T T i.e., a ∗ b ∈ / TqkT (A; 1−k / T∈∨ qkT (A; 1−k 2 ). Hence a ∗ b ∈ 2 ), a contradiction. Consequently, ^ T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 } F F or γy ≥ 1−k If γx ≥ 1−k 2 2 , then AF (x ∗ y) ≤ γx ∨ γy and thus F F x ∗ y ∈ F∈ (A; γx ∨ γy ). If γx < 1−k and γy < 1−k 2 2 , then 1−kF AF (x ∗ y) ≤ 2 . Hence AF (x ∗ y) + γx ∨ γy < 1−kF 2 + 1−kF 2 = 1 − kF , for all x, y ∈ X. Similarly, we know that ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } that is, x ∗ y ∈ FqkF (A; γx ∨ γy ). Thus for all x, y ∈ X. Suppose that there exist a, b ∈ X such that _ F AF (a ∗ b) > {AF (a), AF (b), 1−k 2 }. Therefore A = (AT , AI , AF ) is an (∈, ∈ ∨ qkF )-neutrosophic subalgebra of X. x ∗ y ∈ F∈∨ qkF (A; γx ∨ γy ). W F Then AF (a ∗ b) > γF ≥ {AF (a), AF (b), 1−k 2 } for some Corollary 3.11 ([4]). A neutrosophic set A = (AT , AI , AF ) in 1−kF γF ∈ [0, 1). If AF (a) ∨ AF (b) ≥ 2 , then X ∈ B(X) is an (∈, ∈ ∨ q)-neutrosophic subalgebra of X if and only if it satisfies: AF (a ∗ b) > γF ≥ AF (a) ∨ AF (b)   V AT (x ∗ y) ≥ {AT (x), AT (y), 0.5} V   which implies that a, b ∈ F∈ (A; γF ) and a ∗ b ∈ / F∈ (A; γF ). (∀x, y ∈ X)  AI (x ∗ y) ≥ {AI (x), AI (y).0.5}  . Also, W AF (x ∗ y) ≤ {AF (x), AF (y), 0.5} A (a ∗ b) + γ > 2γ ≥ 1 − k , F F F F S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points Neutrosophic Sets and Systems, Vol. 20, 2018 30 Proof. It follows from taking kT = kI = kF = 0 in Theorem AF (a ∗ b) ≤ γF which is a contradiction. Thus 3.10. _ F AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 } Theorem 3.12. Let A = (AT , AI , AF ) be a neutrosophic set in X ∈ B(X). Then A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )- for all x, y ∈ X. Therefore A = (AT , AI , AF ) is an (∈, ∈ neutrosophic subalgebra of X if and only if neutrosophic ∈- ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X by Theorem 3.10. subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras of X 1−kI 1−kF T for all α ∈ (0, 1−k 2 ], β ∈ (0, 2 ] and γ ∈ [ 2 , 1) when- Corollary 3.13. Let A = (A , A , A ) be a neutrosophic set T I F ever they are nonempty. in X ∈ B(X). Then A = (AT , AI , AF ) is an (∈, ∈ ∨ q)neutrosophic subalgebra of X if and only if neutrosophic ∈Proof. Assume that A = (AT , AI , AF ) is an (∈, ∈ subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras of 1−kI ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X. Let β ∈ (0, 2 ] X for all α, β ∈ (0, 0.5] and γ ∈ [0.5, 1) whenever they are and x, y ∈ I∈ (A; β). Then AI (x) ≥ β and AI (y) ≥ β. It nonempty. follows from Theorem 3.10 that ^ Proof. It follows from taking kT = kI = kF = 0 in Theorem 1−kI I =β AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 }≥β∧ 2 3.12. and so that x ∗ y ∈ I∈ (A; β). Hence I∈ (A; β) is a subalgebra I of X for all β ∈ (0, 1−k 2 ]. Similarly, we know that T∈ (A; α) 1−kF T is a subalgebra of X for all α ∈ (0, 1−k 2 ]. Let γ ∈ [ 2 , 1) and x, y ∈ F∈ (A; γ). Then AF (x) ≤ γ and AF (y) ≤ γ. Using Theorem 3.10 implies that _ 1−kF F = γ. AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 }≤γ∨ 2 Theorem 3.14. Every (∈, ∈)-neutrosophic subalgebra is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. Proof. Straightforward. The converse of Theorem 3.14 is not true as seen in the following example. Example 3.15. Consider a BCI-algebra X = {0, a, b, c} with Hence x ∗ y ∈ F∈ (A; γ), and therefore F∈ (A; γ) is a subalgebra the binary operation ∗ which is given in Table 1 (see [5]). F of X for all γ ∈ [ 1−k 2 , 1). Conversely, suppose that the nonempty neutrosophic ∈-subsets Table 1: Cayley table for the binary operation “∗” T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras of X for all 1−kI 1−kF 1−kT α ∈ (0, 2 ], β ∈ (0, 2 ] and γ ∈ [ 2 , 1). If there exist a, b ∈ X such that ∗ 0 a b c ^ 0 0 a b c T AT (a ∗ b) < {AT (a), AT (b), 1−k 2 }, a a 0 c b b b c 0 a then a, b ∈ T∈ (A; αT ) by taking c c b a 0 ^ T αT := {AT (a), AT (b), 1−k 2 }. Let A = (AT , AI , AF ) be a neutrosophic set in X ∈ BI (X) Since T∈ (A; αT ) is a subalgebra of X, it follows that a ∗ b ∈ defined by Table 2 T∈ (A; αT ), that is, AT (a ∗ b) ≥ αT . This is a contradiction, and hence ^ Table 2: Tabular representation of “A = (AT , AI , AF )” T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 } for all x, y ∈ X. Similarly, we can verify that ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } for all x, y ∈ X. Now, assume that there exist a, b ∈ X such that _ F AF (a ∗ b) > {AF (a), AF (b), 1−k 2 }. W F Then AF (a ∗ b) > γF ≥ {AF (a), AF (b), 1−k 2 } for some 1−kF γF ∈ [ 2 , 1). Hence a, b ∈ F∈ (A; γF ), and so a ∗ b ∈ F∈ (A; γF ) since F∈ (A; γF ) is a subalgebra of X. It follows that X 0 a b c AT (x) 0.6 0.7 0.3 0.3 AI (x) 0.5 0.3 0.6 0.3 AF (x) 0.2 0.6 0.6 0.4 If kT = 0.36, then T∈ (A; α) =  X {0, a} if α ∈ (0, 0.3], if α ∈ (0.3, 0.32]. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points 31 Neutrosophic Sets and Systems, Vol. 20, 2018 If kI = 0.32, then and  X {0, b} if β ∈ (0, 0.3], if β ∈ (0.3, 0.34].   {0} {0, c} F∈ (A; γ) =  X if γ ∈ [0.32, 0.4), if γ ∈ [0.4, 0.6), if γ ∈ [0.6, 1]. I∈ (A; β) =   {0} {0, c} F∈ (A; γ) =  X If kF = 0.36, then if β ∈ [0.38, 0.4), if β ∈ [0.4, 0.6), if β ∈ [0.6, 1). Since X, {0}, {0, a}, {0, b} and {0, c} are subalgebras of X, we know from Theorem 3.12 that A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X for kT = 0.2, kI = 0.3 and kF = 0.24. Note that We know that T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras a∗b∈ / Tq0.2 (A; 0.25 ∧ 0.4) of X for all α ∈ (0, 0.32], β ∈ (0, 0.34] and γ ∈ [0.32, 1). It follows from Theorem 3.12 that A = (AT , AI , AF ) is an (∈, for a ∈ T∈ (A; 0.4) and b ∈ T∈ (A; 0.25), ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X for kT = 0.36, b∗c∈ / Iq0.3 (A; 0.5 ∧ 0.27) kI = 0.32 and kF = 0.36. Since for b ∈ I∈ (A; 0.5) and c ∈ I∈ (A; 0.27), and/or AT (0) = 0.6 < 0.7 = AT (a) ∧ AT (a) a∗c∈ / Fq0.24 (A; 0.6 ∨ 0.44) and/or AI (0) = 0.5 > 0.3 = AI (c) ∨ AI (c), for a ∈ F∈ (A; 0.6) and c ∈ F∈ (A; 0.44). Hence A = (AT , AI , we know that A = (AT , AI , AF ) is not an (∈, ∈)-neutrosophic AF ) is not an (∈, q(0.2,0.3,0.24) )-neutrosophic subalgebra of X. subalgebra of X by Lemma 3.3. Theorem 3.19. If 0 ≤ kT < jT < 1, 0 ≤ kI < jI < 1 and Definition 3.16. A neutrosophic set A = (AT , AI , AF ) in X ∈ 0 ≤ jF < kF < 1, then every (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic B(X) is called an (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra of subalgebra is an (∈, ∈ ∨q(jT ,jI ,jF ) )-neutrosophic subalgebra. X if x ∈ T∈ (A; αx ), y ∈ T∈ (A; αy ) ⇒ x ∗ y ∈ TqkT (A; αx ∧ αy ), x ∈ I∈ (A; βx ), y ∈ I∈ (A; βy ) ⇒ x ∗ y ∈ IqkI (A; βx ∧ βy ), x ∈ F∈ (A; γx ), y ∈ F∈ (A; γy ) ⇒ x ∗ y ∈ FqkF (A; γx ∨ γy ) Proof. Straightforward. (3.17) for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). An (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra with kT = kI = kF = k is called an (∈, qk )-neutrosophic subalgebra. Theorem 3.17. Every (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. The following example shows that if 0 ≤ kT < jT < 1, 0 ≤ kI < jI < 1 and 0 ≤ jF < kF < 1, then an (∈, ∈ ∨q(jT ,jI ,jF ) )-neutrosophic subalgebra may not be an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. Example 3.20. Let X be the BCI-algebra given in Example 3.15 and let A = (AT , AI , AF ) be a neutrosophic set in X defined by Table 3 Table 3: Tabular representation of “A = (AT , AI , AF )” Proof. Straightforward. The converse of Theorem 3.17 is not true as seen in the following example. Example 3.18. Consider the BCI-algebra X = {0, a, b, c} and the neutrosophic set A = (AT , AI , AF ) which are given in Example 3.15. Taking kT = 0.2, kI = 0.3 and kF = 0.24 imply that  X if α ∈ (0, 0.3], T∈ (A; α) = {0, a} if α ∈ (0.3, 0.4], I∈ (A; β) =  X {0, b} if β ∈ (0, 0.3], if β ∈ (0.3, 0.35], X 0 a b c AT (x) 0.42 0.40 0.48 0.40 AI (x) 0.40 0.44 0.36 0.36 AF (x) 0.44 0.66 0.66 0.33 If kT = 0.04, then   X {0, b} T∈ (A; α) =  {b} if α ∈ (0, 0.40], if α ∈ (0.40, 0.42], if α ∈ (0.42, 0.48]. Note that T∈ (A; α) is not a subalgebra of X for α ∈ (0.42, 0.48]. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points Neutrosophic Sets and Systems, Vol. 20, 2018 32 If kI = 0.08, then  X    {0, a} I∈ (A; β) = {a}    ∅ if β if β if β if β ∈ (0, 0.36], ∈ (0.36, 0.40], ∈ (0.40, 0.44], ∈ (0.44, 0.46]. subalgebra of X if and only if the neutrosophic set AS = (AST , ASI , ASF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X. Proof. Let S be a subalgebra of X. Then neutrosophic ∈subsets T∈ (AST ; α), I∈ (AST ; β) and F∈ (AST ; γ) are obviously 1−kI T subalgebras of X for all α ∈ (0, 1−k 2 ], β ∈ (0, 2 ] and Note that I∈ (A; β) is not a subalgebra of X for β ∈ (0.40, 0.44]. γ ∈ [ 1−kF , 1). Hence A = (A , A , A ) is an (∈, S ST SI SF 2 If kF = 0.42, then ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X by Theorem 3.12.  Conversely, assume that AS = (AST , ASI , ASF ) is an (∈, ∅ if γ ∈ [0.29, 0.33),    ∈ ∨q (kT ,kI ,kF ) )-neutrosophic subalgebra of X. Let x, y ∈ S. {c} if γ ∈ [0.33, 0.44), F∈ (A; γ) = Then {0, c} if γ ∈ [0.44, 0.66),    ^ X if γ ∈ [0.66, 1). T AST (x ∗ y) ≥ {AST (x), AST (y), 1−k 2 } Note that F∈ (A; γ) is not a subalgebra of X for γ ∈ [0.33, 0.44). Therefore A = (AT , AI , AF ) is not an (∈, ∈ ∨q(kT ,kI ,kF ) )neutrosophic subalgebra of X for kT = 0.04, kI = 0.08 and kF = 0.42. If jT = 0.16, then  X if α ∈ (0, 0.40], T∈ (A; α) = and {0, b} if α ∈ (0.40, 0.42].  X {0, a} if β ∈ (0, 0.36], if β ∈ (0.36, 0.40]. If jF = 0.12, then F∈ (A; γ) = ASI (x ∗ y) ≥ ^ _ 1−kT 2 = 1−kT 2 , I {ASI (x), ASI (y), 1−k 2 } =1∧ ASF (x ∗ y) ≤ If jI = 0.20, then I∈ (A; β) = =1∧ 1−kI 2 = 1−kI 2 F {ASF (x), ASF (y), 1−k 2 } =0∨ 1−kF 2 = 1−kF 2 , which imply that AST (x ∗ y) = 1, ASI (x ∗ y) = 1 and ASF (x ∗ y) = 0.  {0, c} X if γ ∈ [0.44, 0.66), if γ ∈ [0.66, 1). Hence x ∗ y ∈ S, and so S is a subalgebra of X. Theorem 3.22. Let S be a subalgebra of X ∈ B(X). For every Therefore A = (AT , AI , AF ) is an (∈, ∈ ∨q(jT ,jI ,jF ) )1−kI 1−kF T α ∈ (0, 1−k 2 ], β ∈ (0, 2 ] and γ ∈ [ 2 , 1), there exneutrosophic subalgebra of X for jT = 0.16, jI = 0.20 and ists an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra A = (AT , jF = 0.12. AI , AF ) of X such that T∈ (A; α) = S, I∈ (A; β) = S and Given a subset S of X, consider a neutrosophic set AS = F∈ (A; γ) = S. (AST , ASI , ASF ) in X defined by Proof. Let A = (AT , AI , AF ) be a neutrosophic set in X de fined by (1, 1, 0) if x ∈ S, AS (x) :=  (0, 0, 1) otherwise, (α, β, γ) if x ∈ S, A(x) := (0, 0, 1) otherwise, that is,  that is, 1 if x ∈ S, AST (x) :=  0 otherwise, α if x ∈ S, AT (x) := 0 otherwise,  1 if x ∈ S, ASI (x) :=  0 otherwise, β if x ∈ S, AI (x) := 0 otherwise, and  and 0 if x ∈ S, ASF (x) :=  1 otherwise. γ if x ∈ S, AF (x) := 1 otherwise. Theorem 3.21. A nonempty subset S of X ∈ B(X) is a S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points 33 Neutrosophic Sets and Systems, Vol. 20, 2018 Obviously, T∈ (A; α) = S, I∈ (A; β) = S and F∈ (A; γ) = S. Suppose that ^ T AT (a ∗ b) < {AT (a), AT (b), 1−k 2 } Case 1. AI (x) ≥ β and AI (y) ≥ β. If β > AI (x ∗ y) ≥ ^ 1−kI 2 , I {AI (x), AI (y), 1−k 2 }= then 1−kI 2 , I I and so AI (x ∗ y) + β > 1−k + 1−k = 1 − kI . Hence x ∗ y ∈ 2 2 for some a, b ∈ X. Since #Im(A ) = 2, it follows that 1−kI T I (A; β). If β ≤ , then V q kI 2 T {AT (a), AT (b), 1−k 2 } = α and AT (a ∗ b) = 0. Hence ^ AT (a) = α = AT (b), and so a, b ∈ S. Since S is a subalgebra I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } ≥ β, of X, we have a ∗ b ∈ S. Thus AT (a ∗ b) = α, a contradiction. Therefore and thus x ∗ y ∈ I∈ (A; β). Hence ^ 1−kT AT (x ∗ y) ≥ {AT (x), AT (y), 2 } x ∗ y ∈ I∈ (A; β) ∪ IqkI (A; β) = I∈∨ qkI (A; β). for all x, y ∈ X. Similarly, we can verify that ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } for all x, y ∈ X. Assume that there exist a, b ∈ X such that _ F AF (a ∗ b) > {AF (a), AF (b), 1−k 2 }. Case 2. AI (x) ≥ β and AI (y) + β + kI > 1. If β > then ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } I = AI (y) ∧ 1−k > (1 − β − kI ) ∧ 2 = 1 − β − kI , 1−kI 2 , 1−kI 2 W F Then AF (a ∗ b) = 1 and {AF (a), AF (b), 1−k 2 } = γ since #Im(AF ) = 2. It follows that AF (a) = γ = AF (b) and so that and so x ∗ y ∈ I (A; β). If β ≤ 1−kI , then q kI 2 a, b ∈ S. Hence a ∗ b ∈ S, and so AF (a ∗ b) = γ, which is a ^ contradiction. Thus I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } _ ^ 1−kF AF (x ∗ y) ≤ {AF (x), AF (y), 2 } ≥ {β, 1 − β − k , 1−kI } = β, I 2 for all x, y ∈ X. Therefore A = (AT , AI , AF ) is an (∈, ∈ and thus x ∗ y ∈ I (A; β). Therefore x ∗ y ∈ I ∈ ∈∨ qkI (A; β). ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X by Theorem 3.10. Case 3. AI (x) + β + kI > 1 and AI (y) ≥ β. We have Corollary 3.23. Let S be a subalgebra of X ∈ B(X). For every x ∗ y ∈ I∈∨ qkI (A; β) by the similar way to the Case 2. α ∈ (0, 0.5], β ∈ (0, 0.5] and γ ∈ [0.5, 1), there exists an (∈, ∈ ∨q)-neutrosophic subalgebra A = (AT , AI , AF ) of X such Case 4. AI (x) + β + kI > 1 and AI (y) + β + kI > 1. If 1−kI I that T∈ (A; α) = S, I∈ (A; β) = S and F∈ (A; γ) = S. β > 1−k 2 , then 1 − β − kI < 2 , and so Proof. It follows from taking kT = kI = kF = 0 in Theorem 3.22. AI (x ∗ y) ≥ ^ I {AI (x), AI (y), 1−k 2 } > 1 − β − kI , I Theorem 3.24. Given a neutrosophic set A = (AT , AI , AF ) in i.e., x ∗ y ∈ IqkI (A; β). If β ≤ 1−k 2 , then X ∈ B(X), the following are equivalent. ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } (1) A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic I subalgebra of X. ≥ (1 − β − kI ) ∧ 1−k 2 I ≥ β, = 1−k 2 (2) The neutrosophic (∈ ∨ qk )-subsets T∈∨ qkT (A; α), I∈∨ qkI (A; β) and F∈∨ qkF (A; γ) are subalgebras of X for i.e., x ∗ y ∈ I∈ (A; β). Hence x ∗ y ∈ I∈∨ qkI (A; β). Conall α, β ∈ (0, 1] and γ ∈ [0, 1). sequently, I∈∨ qkI (A; β) is a subalgebra of X. Similarly, we Proof. Assume that A = (AT , AI , AF ) is an (∈, ∈ can prove that if x, y ∈ T∈∨ qkT (A; α) for α ∈ (0, 1], then ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X. Let x, y ∈ x ∗ y ∈ T∈∨ qkT (A; α), that is, T∈∨ qkT (A; α) is a subalgebra I∈∨ qkI (A; β) for β ∈ (0, 1]. Then AI (x) ≥ β or AI (x) + β + of X. Let x, y ∈ F∈∨ qkF (A; γ) for γ ∈ [0, 1). Then AF (x) ≤ γ kI > 1, and AI (y) ≥ β or AI (y) + β + kI > 1. Using Theorem or AF (x)+γ +kF < 1, and AF (y) ≤ γ or AF (y)+γ +kF < 1. Using Theorem 3.10, we have 3.10, we have ^ _ I F AI (x ∗ y) ≥ {AI (x), AI (y), 1−k AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 }. 2 }. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points Neutrosophic Sets and Systems, Vol. 20, 2018 34 Case 1. AF (x) ≤ γ and AF (y) ≤ γ. If γ < AF (x ∗ y) ≤ _ F {AF (x), AF (y), 1−k 2 }= and so AF (x ∗ y) + γ < x ∗ y ∈ FqkF (A; γ). If γ ≥ AF (x ∗ y) ≤ 1−kF 2 _ 1−kF 2 1−kF 2 , or AT (x ∗ y) + α + kT > 1, a contradiction. Hence ^ T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 } , then 1−kF 2 , F + 1−k = 1 − kF . Hence for all x, y ∈ X. Similarly, we can verify that 2 ^ then I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } F {AF (x), AF (y), 1−k 2 } ≤ γ, for all x, y ∈ X. Now assume that there exist a, b ∈ X and F γ ∈ ( 1−k 2 , 1) such that and thus x ∗ y ∈ F∈ (A; γ). Hence AF (a ∗ b) > γ ≥ x ∗ y ∈ F∈ (A; γ) ∪ FqkF (A; γ) = F∈∨ qkF (A; γ). Case 2. AF (x) ≤ γ and AF (y) + γ + kF < 1. If γ < then _ F AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 } F = AF (y) ∨ 1−k < (1 − γ − kF ) ∨ 2 = 1 − γ − kF , and so x ∗ y ∈ FqkF (A; γ). If γ ≥ AF (x ∗ y) ≤ ≤ _ _ 1−kF 2 1−kF 2 _ F {AF (a), AF (b), 1−k 2 }. Then a, b ∈ F∈ (A; γ) ⊆ F∈∨ qkF (A; γ), which implies that , 1−kF 2 , then F {AF (x), AF (y), 1−k 2 } a ∗ b ∈ F∈∨ qkF (A; γ). Thus AF (a ∗ b) ≤ γ or AF (a ∗ b) + γ + kF < 1, which is a contradiction. Hence _ F AF (x ∗ y) ≥ {AF (x), AF (y), 1−k 2 } for all x, y ∈ X. Using Theorem 3.10, we conclude that A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X. F {γ, 1 − γ − kF , 1−k 2 } = γ, and thus x ∗ y ∈ F∈ (A; γ). Therefore x ∗ y ∈ F∈∨ qkF (A; γ). 4 Conclusions Similarly, if AI (x) + β + kI < 1 and AI (y) ≤ β, then x ∗ y ∈ F∈∨ qkF (A; γ). Neutrosophic set theory is a nice mathematical tool which can be applied to several fields. The aim of this paper is to consider a general form of neutrosophic points, and to discuss generalFinally, assume that AF (x) + γ + kF < 1 and AF (y) + γ + izations of the papers [4] and [6]. We have introduce the no1−kF 1−kF kF < 1. If γ < 2 , then 1 − γ − kF > 2 , and so tions of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra, and (∈, _ q(kT ,kI ,kF ) )-neutrosophic subalgebra in BCK/BCI-algebras, 1−kF AF (x ∗ y) ≤ {AF (x), AF (y), 2 } < 1 − γ − kF , and have investigated several properties. We have discussed characterizations of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgeF i.e., x ∗ y ∈ FqkF (A; γ). If γ ≥ 1−k , then bra. We have considered relations between (∈, ∈)-neutrosophic 2 subalgebra, (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra and (∈, _ F } AF (x ∗ y) ≤ {AF (x), AF (y), 1−k ∈ ∨q (kT ,kI ,kF ) )-neutrosophic subalgebra. We hope the idea and 2 result in this paper can be a mathematical tool for dealing with F ≤ (1 − γ − kF ) ∨ 1−k 2 several informations containing uncertainty such as medical diagF ≤ γ, = 1−k nosis, decision making, graph theory, etc. So, based on the results 2 in this article, our future research will be focused to solve real-life i.e., x ∗ y ∈ F∈ (A; γ). Hence x ∗ y ∈ F∈∨ qkF (A; γ). Therefore problems under the opinions of experts in a neutrosophic set enviF∈∨ qkF (A; γ) is a subalgebra of X. ronment such as medical diagnosis, decision making, graph theory etc. In particular, Bucolo et al. [2] suggested a generalization Conversely, suppose that (2) is valid. If it is possible, let of the synchronization principles for the class of array of fuzzy ^ 1−kT logic chaotic based dynamical systems and evaluated as alternaAT (x ∗ y) < α ≤ {AT (x), AT (y), 2 } tive approach to build locally connected fuzzy complex systems 1−kT by manipulating both the rules driving the cells and the architecfor some α ∈ (0, 2 ). Then ture of the system. We will also try to study complex dynamics through neutrosophic environment. The future works also may x, y ∈ T∈ (A; α) ⊆ T∈∨ qkT (A; α), use the study neutrosophic set environment on several related alwhich implies that x ∗ y ∈ T∈∨ qkT (A; α). Thus AT (x ∗ y) ≥ α gebraic structures, for example, M V -algebras, BL-algebras, R0 algebras, EQ-algebras, equality algebras, M T L-algebras etc. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points Neutrosophic Sets and Systems, Vol. 20, 2018 35 References [1] K. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20 (1986), 87–96. [9] Abdel-Basset, M., Mohamed, M., Smarandache, F., & Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis. Symmetry, 10(4), 106. [2] M. Bucolo, L. Fortuna and M. La Rosa, Complex dynamics through fuzzy chains, IEEE Trans. Fuzzy Syst. 12 (2004), no. 3, 289–295. [10] Abdel-Basset, M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Sets in Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages 47-55 [3] Y.S. Huang, BCI-algebra, Science Press: Beijing, China, 2006. 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A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22. [13] Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloud computing services. Future Generation Computer Systems, 86, 12-29. [14] Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066. [15] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116. Received : March 23, 2018. Accepted : April 13, 2018. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points 36 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras G. Muhiuddin1 , Hashem Bordbar2 , Florentin Smarandache3 , Young Bae Jun4,∗ 1 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia. e-mail: chishtygm@gmail.com 2 Postdoctoral 3 Mathematics Research Fellow, Shahid Beheshti University, Tehran, Iran. e-mail: bordbar.amirh@gmail.com & Science Department, University of New Mexico. 705 Gurley Ave., Gallup, NM 87301, USA. e-mail: fsmarandache@gmail.com 4 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea. e-mail: skywine@gmail.com ∗ Correspondence: Y.B. Jun (skywine@gmail.com) Abstract: Characterizations of an (∈, ∈)-neutrosophic ideal are considered. Any ideal in a BCK/BCI-algebra will be realized as level neutrosophic ideals of some (∈, ∈)-neutrosophic ideal. The relation between (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic subalgebra in a BCK-algebra is discussed. Conditions for an (∈, ∈)-neutrosophic subalgebra to be a (∈, ∈)-neutrosophic ideal are provided. Using a collection of ideals in a BCK/BCI-algebra, an (∈, ∈)-neutrosophic ideal is established. Equivalence relations on the family of all (∈, ∈)-neutrosophic ideals are introduced, and related properties are investigated. Keywords: (∈, ∈)-neutrosophic subalgebra, (∈, ∈)-neutrosophic ideal. 1 Introduction vestigated by several researchers. By a BCI-algebra, we mean a set X with a special element 0 Neutrosophic set (NS) developed by Smarandache [8, 9, 10] in- and a binary operation ∗ that satisfies the following conditions: troduced neutrosophic set (NS) as a more general platform which (I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), extends the concepts of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued intuitionistic fuzzy set. Neutro- (II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), sophic set theory is applied to various part which is refered to the (III) (∀x ∈ X) (x ∗ x = 0), site http://fs.gallup.unm.edu/neutrosophy.htm. (IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). Jun et al. studied neutrosophic subalgebras/ideals in BCK/BCI-algebras based on neutrosophic points (see [1], [5] and [7]). In this paper, we characterize an (∈, ∈)-neutrosophic ideal in a BCK/BCI-algebra. We show that any ideal in a BCK/BCIalgebra can be realized as level neutrosophic ideals of some (∈, ∈)-neutrosophic ideal. We investigate the relation between (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic subalgebra in a BCK-algebra. We provide conditions for an (∈, ∈)neutrosophic subalgebra to be a (∈, ∈)-neutrosophic ideal. Using a collection of ideals in a BCK/BCI-algebra, we establish an (∈, ∈)-neutrosophic ideal. We discuss equivalence relations on the family of all (∈, ∈)-neutrosophic ideals, and investigate related properties. If a BCI-algebra X satisfies the following identity: 2 Preliminaries A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki (see [2] and [3]) and was extensively in- (V) (∀x ∈ X) (0 ∗ x = 0), then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following conditions: (∀x ∈ X) (x ∗ 0 = x) ,   x≤y ⇒ x∗z ≤y∗z (∀x, y, z ∈ X) , x≤y ⇒ z∗y ≤z∗x (2.1) (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) , (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.3) (2.4) (2.2) where x ≤ y if and only if x ∗ y = 0. A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. A subset I of a BCK/BCI-algebra X is called an ideal of X if it satisfies: 0 ∈ I, (∀x ∈ X) (∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) . (2.5) (2.6) G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras 37 Neutrosophic Sets and Systems, Vol. 20, 2018 We refer the reader to the books [4, 6] for further information and regarding BCK/BCI-algebras. For any family {ai | i ∈ Λ} of real numbers, we define _ {ai | i ∈ Λ} := sup{ai | i ∈ Λ} and ^     (∀x, y ∈ X)     x ∗ y ∈ T∈ (A∼ ; αx ), y ∈ T∈ (A∼ ; αy ) ⇒ x ∈ T∈ (A∼ ; αx ∧ αy ) x ∗ y ∈ I∈ (A∼ ; βx ), y ∈ I∈ (A∼ ; βy ) ⇒ x ∈ I∈ (A∼ ; βx ∧ βy ) x ∗ y ∈ F∈ (A∼ ; γx ), y ∈ F∈ (A∼ ; γy ) ⇒ x ∈ F∈ (A∼ ; γx ∨ γy ) Let X be a non-empty set. A neutrosophic set (NS) in X (see [9]) is a structure of the form: A∼ := {hx; AT (x), AI (x), AF (x)i | x ∈ X}        (2.9) {ai | i ∈ Λ} := inf{ai | i ∈ Λ}. W If Λ = {1, 2}, weVwill also use a1 ∨ a2 and a1 ∧ a2 instead of {ai | i ∈ Λ} and {ai | i ∈ Λ}, respectively.  for all αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). 3 (∈, ∈)-neutrosophic subalgebras and ideals We first provide characterizations of an (∈, ∈)-neutrosophic ideal. where AT : X → [0, 1] is a truth membership function, Theorem 3.1. Given a neutrosophic set A∼ = (AT , AI , AF ) in AI : X → [0, 1] is an indeterminate membership function, and a BCK/BCI-algebra X, the following assertions are equivaAF : X → [0, 1] is a false membership function. For the sake of lent. simplicity, we shall use the symbol A∼ = (AT , AI , AF ) for the neutrosophic set (1) A∼ = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal of X. A∼ := {hx; AT (x), AI (x), AF (x)i | x ∈ X}. Given a neutrosophic set A∼ = (AT , AI , AF ) in a set X, α, β ∈ (0, 1] and γ ∈ [0, 1), we consider the following sets: T∈ (A∼ ; α) := {x ∈ X | AT (x) ≥ α}, I∈ (A∼ ; β) := {x ∈ X | AI (x) ≥ β}, F∈ (A∼ ; γ) := {x ∈ X | AF (x) ≤ γ}. We say T∈ (A∼ ; α), I∈ (A∼ ; β) and F∈ (A∼ ; γ) are neutrosophic ∈-subsets. A neutrosophic set A∼ = (AT , AI , AF ) in a BCK/BCIalgebra X is called an (∈, ∈)-neutrosophic subalgebra of X (see [5]) if the following assertions are valid.   x ∈ T∈ (A∼ ; αx ), y ∈ T∈ (A∼ ; αy )  ⇒ x ∗ y ∈ T∈ (A∼ ; αx ∧ αy ),      x ∈ I∈ (A∼ ; βx ), y ∈ I∈ (A∼ ; βy )  (2.7)  (∀x, y ∈ X)   ⇒ x ∗ y ∈ I (A ; β ∧ β ), ∈ ∼ x y    x ∈ F∈ (A∼ ; γx ), y ∈ F∈ (A∼ ; γy )  ⇒ x ∗ y ∈ F∈ (A∼ ; γx ∨ γy ) (2) A∼ = (AT , AI , AF ) satisfies the following assertions.   AT (0) ≥ AT (x), (∀x ∈ X)  AI (0) ≥ AI (x),  (3.1) AF (0) ≤ AF (x) and   AT (x) ≥ AT (x ∗ y) ∧ AT (y) (∀x, y ∈ X)  AI (x) ≥ AI (x ∗ y) ∧ AI (y)  (3.2) AF (x) ≤ AF (x ∗ y) ∨ AF (y) Proof. Assume that A∼ = (AT , AI , AF ) is an (∈, ∈)neutrosophic ideal of X. Suppose there exist a, b, c ∈ X be such that AT (0) < AT (a), AI (0) < AI (b) and AF (0) > AF (c). Then a ∈ T∈ (A∼ ; AT (a)), b ∈ I∈ (A∼ ; AI (b)) and c ∈ F∈ (A∼ ; AF (c)). But 0∈ / T∈ (A∼ ; AT (a)) ∩ I∈ (A∼ ; AI (b)) ∩ F∈ (A∼ ; AF (c)). This is a contradiction, and thus AT (0) ≥ AT (x), AI (0) ≥ AI (x) and AF (0) ≤ AF (x) for all x ∈ X. Suppose that AT (x) < AT (x ∗ y) ∧ AT (y), AI (a) < AI (a ∗ b) ∧ AI (b) and AF (c) > AF (c ∗ d) ∨ AF (d) for some x, y, a, b, c, d ∈ X. for all αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). Taking α := AT (x∗y)∧AT (y), β := AI (a∗b)∧AI (b) and γ := A neutrosophic set A∼ = (AT , AI , AF ) in a BCK/BCI- AF (c∗d)∨AF (d) imply that x∗y ∈ T∈ (A∼ ; α), y ∈ T∈ (A∼ ; α), algebra X is called an (∈, ∈)-neutrosophic ideal of X (see [7]) a ∗ b ∈ I∈ (A∼ ; β), b ∈ I∈ (A∼ ; β), c ∗ d ∈ F∈ (A∼ ; γ) and if the following assertions are valid. d ∈ F∈ (A∼ ; γ). But x ∈ / T∈ (A∼ ; α), a ∈ / I∈ (A∼ ; β) and c∈ / F∈ (A∼ ; γ). This is impossible, and so (3.2) is valid.   x ∈ T∈ (A∼ ; αx ) ⇒ 0 ∈ T∈ (A∼ ; αx ) Conversely, suppose A∼ = (AT , AI , AF ) satisfies two con(∀x ∈ X)  x ∈ I∈ (A∼ ; βx ) ⇒ 0 ∈ I∈ (A∼ ; βx )  (2.8) ditions (3.1) and (3.2). For any x, y, z ∈ X, let α, β ∈ (0, 1] x ∈ F∈ (A∼ ; γx ) ⇒ 0 ∈ F∈ (A∼ ; γx ) and γ ∈ [0, 1) be such that x ∈ T∈ (A∼ ; α), y ∈ I∈ (A∼ ; β) and G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras Neutrosophic Sets and Systems, Vol. 20, 2018 38 z ∈ F∈ (A∼ ; γ). It follows from (3.1) that AT (0) ≥ AT (x) ≥ α, that AI (0) ≥ AI (y) ≥ β and AF (0) ≤ AF (z) ≤ γ and so that AT (x) < AT (x ∗ y) ∧ AT (y), 0 ∈ T∈ (A∼ ; α)∩I∈ (A∼ ; β)∩F∈ (A∼ ; γ). Let a, b, c, d, x, y ∈ X A I (a) < AI (a ∗ b) ∧ AI (b), be such that a ∗ b ∈ T∈ (A∼ ; αa ), b ∈ T∈ (A∼ ; αb ), c ∗ d ∈ A (u) > AF (u ∗ v) ∨ AF (v) F I∈ (A∼ ; βc ), d ∈ I∈ (A∼ ; βd ), x ∗ y ∈ F∈ (A∼ ; γx ), and y ∈ F∈ (A∼ ; γy ) for αa , αb , βc , βd ∈ (0, 1] and γx , γy ∈ [0, 1). Usfor some x, y, a, b, u, v ∈ X. Taking α := AT (x ∗ y) ∧ AT (y), ing (3.2), we have β := AI (a ∗ b) ∧ AI (b) and γ := AF (u ∗ v) ∨ AF (v) imply that α, β ∈ (0, 1], γ ∈ [0, 1), x ∗ y ∈ T∈ (A∼ ; α), y ∈ T∈ (A∼ ; α), AT (a) ≥ AT (a ∗ b) ∧ AT (b) ≥ αa ∧ αb a ∗ b ∈ I∈ (A∼ ; β), b ∈ I∈ (A∼ ; β), u ∗ v ∈ F∈ (A∼ ; γ) and AI (c) ≥ AI (c ∗ d) ∧ AI (d) ≥ βc ∧ βd v ∈ F∈ (A∼ ; γ). But x ∈ / T∈ (A∼ ; α), a ∈ / I∈ (A∼ ; β) and u ∈ / AF (x) ≤ AF (x ∗ y) ∨ AF (y) ≤ γx ∨ γy . F∈ (A∼ ; γ). This is a contradiction since T∈ (A∼ ; α), I∈ (A∼ ; β) Hence a ∈ T∈ (A∼ ; αa ∧ αb ), c ∈ I∈ (A∼ ; βc ∧ βd ) and x ∈ and F∈ (A∼ ; γ) are ideals of X. Thus F∈ (A∼ ; γx ∨ γy ). Therefore A∼ = (AT , AI , AF ) is an (∈, ∈)AT (x) ≥ AT (x ∗ y) ∧ AT (y), neutrosophic ideal of X. AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y) Theorem 3.2. Let A∼ = (AT , AI , AF ) be a neutrosophic set in a BCK/BCI-algebra X. Then the following assertions are for all x, y ∈ X. Therefore A∼ = (AT , AI , AF ) is an (∈, equivalent. ∈)-neutrosophic ideal of X by Theorem 3.1. (1) A∼ = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal of X. (2) The nonempty neutrosophic ∈-subsets T∈ (A∼ ; α), I∈ (A∼ ; β) and F∈ (A∼ ; γ) are ideals of X for all Proposition 3.3. Every (∈, ∈)-neutrosophic ideal A = ∼ α, β ∈ (0, 1] and γ ∈ [0, 1). (AT , AI , AF ) of a BCK/BCI-algebra X satisfies the followProof. Let A∼ = (AT , AI , AF ) be an (∈, ∈)-neutrosophic ideal ing assertions.    of X and assume that T∈ (A∼ ; α), I∈ (A∼ ; β) and F∈ (A∼ ; γ) are  AT (x) ≥ AT (y) nonempty for α, β ∈ (0, 1] and γ ∈ [0, 1). Then there exist (∀x, y ∈ X) x ≤ y ⇒ AI (x) ≥ AI (y)  , (3.3)  x, y, z ∈ X such that x ∈ T∈ (A∼ ; α), y ∈ I∈ (A∼ ; β) and z ∈ AF (x) ≤ AF (y) F∈ (A∼ ; γ). It follows from (2.8) that     AT (x) ≥ AT (y) ∧ AT (z) AI (x) ≥ AI (y) ∧ AI (z)  . (∀x, y, z ∈ X) x ∗ y ≤ z ⇒ 0 ∈ T∈ (A∼ ; α) ∩ I∈ (A∼ ; β) ∩ F∈ (A∼ ; γ).  AF (x) ≤ AF (y) ∨ AF (z) Let x, y, a, b, u, v ∈ X be such that x ∗ y ∈ T∈ (A∼ ; α), (3.4) y ∈ T∈ (A∼ ; α), a ∗ b ∈ I∈ (A∼ ; β), b ∈ I∈ (A∼ ; β), u ∗ v ∈ F∈ (A∼ ; γ) and v ∈ F∈ (A∼ ; γ). Then Proof. Let x, y ∈ X be such that x ≤ y. Then x ∗ y = 0, and so AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ α ∧ α = α AI (a) ≥ AI (a ∗ b) ∧ AI (b) ≥ β ∧ β = β AF (u) ≤ AF (u ∗ v) ∨ AF (v) ≤ γ ∨ γ = γ AT (x) ≥ AT (x ∗ y) ∧ AT (y) = AT (0) ∧ AT (y) = AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y) = AI (0) ∧ AI (y) = AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y) = AF (0) ∨ AF (y) = AF (y) by (3.2), and so x ∈ T∈ (A∼ ; α), a ∈ I∈ (A∼ ; β) and u ∈ F∈ (A∼ ; γ). Hence the nonempty neutrosophic ∈-subsets by Theorem 3.1. Hence (3.3) is valid. Let x, y, z ∈ X be such T∈ (A∼ ; α), I∈ (A∼ ; β) and F∈ (A∼ ; γ) are ideals of X for all that x ∗ y ≤ z. Then (x ∗ y) ∗ z = 0, and thus α, β ∈ (0, 1] and γ ∈ [0, 1). AT (x) ≥ AT (x ∗ y) ∧ AT (y) Conversely, let A∼ = (AT , AI , AF ) be a neutrosophic ≥ (AT ((x ∗ y) ∗ z) ∧ AT (z)) ∧ AT (y) set in X for which T∈ (A∼ ; α), I∈ (A∼ ; β) and F∈ (A∼ ; γ) are nonempty and are ideals of X for all α, β ∈ (0, 1] and ≥ (AT (0) ∧ AT (z)) ∧ AT (y) γ ∈ [0, 1). Assume that AT (0) < AT (x), AI (0) < AI (y) ≥ AT (z) ∧ AT (y), and AF (0) > AF (z) for some x, y, z ∈ X. Then x ∈ T∈ (A∼ ; AT (x)), y ∈ I∈ (A∼ ; AI (y)) and z ∈ F∈ (A∼ ; AF (z)), AI (x) ≥ AI (x ∗ y) ∧ AI (y) that is, T∈ (A∼ ; α), I∈ (A∼ ; β) and F∈ (A∼ ; γ) are nonempty. But 0 ∈ / T∈ (A∼ ; AT (x)) ∩ I∈ (A∼ ; AI (y)) ∩ F∈ (A∼ ; AF (z)), ≥ (AI ((x ∗ y) ∗ z) ∧ AI (z)) ∧ AI (y) which is a contradiction since T∈ (A∼ ; AT (x)), I∈ (A∼ ; AI (y)) ≥ (AI (0) ∧ AI (z)) ∧ AI (y) and F∈ (A∼ ; AF (z)) are ideals of X. Hence AT (0) ≥ AT (x), ≥ AI (z) ∧ AI (y) AI (0) ≥ AI (x) and AF (0) ≤ AF (x) for all x ∈ X. Suppose G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras 39 Neutrosophic Sets and Systems, Vol. 20, 2018 and If x ∗ y ∈ I and y ∈ / I, then AF (x) ≤ AF (x ∗ y) ∨ AF (y) ≤ (AF ((x ∗ y) ∗ z) ∨ AF (z)) ∨ AF (y) ≤ (AF (0) ∨ AF (z)) ∨ AF (y) ≤ AF (z) ∨ AF (y) by Theorem 3.1. AT (x ∗ y) = α and AT (y) = 0, AI (x ∗ y) = β and AI (y) = 0, AF (x ∗ y) = γ and AF (y) = 1, It follows that AT (x) ≥ 0 = AT (x ∗ y) ∧ AT (y), AI (x) ≥ 0 = AI (x ∗ y) ∧ AI (y), AF (x) ≤ 1 = AF (x ∗ y) ∨ AF (y). Theorem 3.4. Any ideal of a BCK/BCI-algebra X can be reSimilarly, if x ∗ y ∈ / I and y ∈ I, then alized as level neutrosophic ideals of some (∈, ∈)-neutrosophic ideal of X. AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), Proof. Let I be an ideal of a BCK/BCI-algebra X and let AF (x) ≤ AF (x ∗ y) ∨ AF (y). A∼ = (AT , AI , AF ) be a neutrosophic set in X given as follows: Therefore A∼ = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal  of X by Theorem 3.1. This completes the proof. α if x ∈ I, AT : X → [0, 1], x 7→ 0 otherwise, Lemma 3.5 ([5]). A neutrosophic set A∼ = (AT , AI , AF ) in a  BCK/BCI-algebra X is an (∈, ∈)-neutrosophic subalgebra of β if x ∈ I, AI : X → [0, 1], x 7→ X if and only if it satisfies: 0 otherwise,    γ if x ∈ I, AT (x ∗ y) ≥ AT (x) ∧ AT (y) AF : X → [0, 1], x 7→ 1 otherwise (∀x, y ∈ X)  AI (x ∗ y) ≥ AI (x) ∧ AI (y)  . (3.5) AF (x ∗ y) ≤ AF (x) ∨ AF (y) where (α, β, γ) is a fixed ordered triple in (0, 1] × (0, 1] × [0, 1). Then T∈ (A∼ ; α) = I, I∈ (A∼ ; β) = I and F∈ (A∼ ; γ) = I. Theorem 3.6. In a BCK-algebra, every (∈, ∈)-neutrosophic Obviously, AT (0) ≥ AT (x), AI (0) ≥ AI (x) and AF (0) ≤ ideal is an (∈, ∈)-neutrosophic subalgebra. AF (x) for all x ∈ X. Let x, y ∈ X. If x ∗ y ∈ I and y ∈ I, then x ∈ I. Hence Proof. Let A∼ = (AT , AI , AF ) be an (∈, ∈)-neutrosophic ideal of a BCK-algebra X. Since x∗y ≤ x for all x, y ∈ X, it follows AT (x ∗ y) = AT (y) = AT (x) = α, from Proposition 3.3 and (3.2) that AI (x ∗ y) = AI (y) = AI (x) = β, AT (x ∗ y) ≥ AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ AT (x) ∧ AT (y), AF (x ∗ y) = AF (y) = AF (x) = γ, AI (x ∗ y) ≥ AI (x) ≥ AI (x ∗ y) ∧ AI (y) ≥ AI (x) ∧ AI (y), AF (x ∗ y) ≤ AF (x) ≤ AF (x ∗ y) ∨ AF (y) ≤ AF (x) ∨ AF (y). and so AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y). If x ∗ y ∈ / I and y ∈ / I, then AT (x ∗ y) = AT (y) = 0, AI (x ∗ y) = AI (y) = 0, AF (x ∗ y) = AF (y) = 1. Thus AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y). Therefore A∼ = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subalgebra of X by Lemma 3.5. The following example shows that the converse of Theorem 3.6 is not true in general. Example 3.7. Consider a set X = {0, 1, 2, 3} with the binary operation ∗ which is given in Table 1. Then (X; ∗, 0) is a BCK-algebra (see [6]). Let A∼ = (AT , AI , AF ) be a neutrosophic set in X defined by Table 2 It is routine to verify that A∼ = (AT , AI , AF ) is an (∈, ∈)neutrosophic subalgebra of X. We know that I∈ (A∼ ; β) is an ideal of X for all β ∈ (0, 1]. If α ∈ (0.3, 0.7], then T∈ (A∼ ; α) = {0, 1, 3} is not an ideal of X. Also, if γ ∈ [0.2, 0.8), then F∈ (A∼ ; γ) = {0, 1, 3} is not an ideal of X. Therefore A∼ = (AT , AI , AF ) is not an (∈, ∈)-neutrosophic ideal of X by Theorem 3.2. G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras Neutrosophic Sets and Systems, Vol. 20, 2018 40 Table 1: Cayley table for the binary operation “∗” ∗ 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 1 2 0 ing two cases: _ _ α = {i ∈ ΛT | i < α} and α 6= {i ∈ ΛT | i < α}. First case implies that x ∈ T∈ (A∼ ; α) ⇔ x ∈ Di for all i < α ⇔ x ∈ ∩{Di | i < α}. (3.9) Hence T∈ (A∼ ; α) = ∩{Di | i < α}, which is an ideal of X. For the second case, we claim that T∈ (A∼ ; α) = ∪{Di | i ≥ α}. Table 2: Tabular representation of A∼ = (AT , AI , AF ) If x ∈ ∪{Di | i ≥ α}, then x ∈ Di for some i ≥ α. Thus AT (x) ≥ i ≥ α, and so x ∈ T∈ (A∼ ; α). If / ∪{Di | i ≥ α}, X AT (x) AI (x) AF (x) Wx ∈ then x ∈ / Di for all i ≥ α. Since α 6= {i ∈ ΛT | i < α}, 0 0.7 0.9 0.2 there exists ε > 0 such that (α − ε, α) ∩ ΛT = ∅. Hence x ∈ / Di 1 0.7 0.6 0.2 for all i > α − ε, which means that if x ∈ Di then i ≤ α − ε. 2 0.3 0.6 0.8 Thus AT (x) ≤ α − ε < α, and so x ∈ / T∈ (A∼ ; α). Therefore 3 0.7 0.4 0.2 T∈ (A∼ ; α) = ∪{Di | i ≥ α} which is an ideal of X since {Dk } forms a chain. Similarly, we can verify that I∈ (A∼ ; β) is an ideal of X. Finally, we consider the following two cases: We give a condition for an (∈, ∈)-neutrosophic subalgebra to ^ ^ be an (∈, ∈)-neutrosophic ideal. γ = {j ∈ ΛF | γ < j} and γ 6= {j ∈ ΛF | γ < j}. Theorem 3.8. Let A∼ = (AT , AI , AF ) be a neutrosophic set For the first case, we have in a BCK-algebra X. If A∼ = (AT , AI , AF ) is an (∈, ∈)neutrosophic subalgebra of X that satisfies the condition (3.4), x ∈ F∈ (A∼ ; γ) ⇔ x ∈ Dj for all j > γ then it is an (∈, ∈)-neutrosophic ideal of X. ⇔ x ∈ ∩{Dj | j > γ}, (3.10) Proof. Taking x = y in (3.5) and using (III) induce the condition and thus F∈ (A∼ ; γ) = ∩{Dj | j > γ} which is an ideal of X. (3.1). Since x ∗ (x ∗ y) ≤ y for all x, y ∈ X, it follows from (3.4) The second case implies that F∈ (A∼ ; γ) = ∪{Dj | j ≤ γ}. In that fact, if x ∈ ∪{Dj | j ≤ γ}, then x ∈ Dj for some j ≤ γ. Thus AF (x) ≤ j ≤ γ, that is, x ∈ F∈ (A∼ ; γ). Hence ∪{Dj | j ≤ AT (x) ≥ AT (x ∗ y) ∧ AT (y), γ} ⊆ F∈ (A∼ ; γ). NowVif x ∈ / ∪{Dj | j ≤ γ}, then x ∈ / Dj for AI (x) ≥ AI (x ∗ y) ∧ AI (y), all j ≤ γ. Since γ 6= {j ∈ ΛF | γ < j}, there exists ε > 0 AF (x) ≤ AF (x ∗ y) ∨ AF (y) such that (γ, γ+ε)∩ΛF is empty. Hence x ∈ / Dj for all j < γ+ε, for all x, y ∈ X. Therefore A∼ = (AT , AI , AF ) is an (∈, and so if x ∈ Dj , then j ≥ γ + ε. Thus AF (x) ≥ γ + ε > γ, and ∈)-neutrosophic ideal of X by Theorem 3.1. hence x ∈ / F∈ (A∼ ; γ). Thus F∈ (A∼ ; γ) ⊆ ∪{Dj | j ≤ γ}, and therefore F∈ (A∼ ; γ) = ∪{Dj | j ≤ γ} which is an ideal of X. Theorem 3.9. Let {Dk | k ∈ ΛT ∪ ΛI ∪ ΛF } be a collection of Consequently, A = (A , A , A ) is an (∈, ∈)-neutrosophic ∼ T I F ideals of a BCK/BCI-algebra X, where ΛT , ΛI and ΛF are ideal of X by Theorem 3.2. nonempty subsets of [0, 1], such that A mapping f : X → Y of BCK/BCI-algebras is called X = {Dα | α ∈ ΛT } ∪ {Dβ | β ∈ ΛI } ∪ {Dγ | γ ∈ ΛF }, a homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ X. (3.6) Note that if f : X → Y is a homomorphism of BCK/BCIT I F (∀i, j ∈ Λ ∪ Λ ∪ Λ ) (i > j ⇔ Di ⊂ Dj ) . (3.7) algebras, then f (0) = 0. Given a homomorphism f : X → Y of BCK/BCI-algebras and a neutrosophic set A∼ = (AT , AI , Let A∼ = (AT , AI , AF ) be a neutrosophic set in X defined as A ) in Y , we define a neutrosophic set Af = (Af , Af , Af ) in F ∼ T I F follows: X, which is called the induced neutrosophic set, as follows: W AT : X → [0, 1], x 7→ W {α ∈ ΛT | x ∈ Dα }, AfT : X → [0, 1], x 7→ AT (f (x)), AI : X → [0, 1], x 7→ V{β ∈ ΛI | x ∈ Dβ }, (3.8) AfI : X → [0, 1], x 7→ AI (f (x)), AF : X → [0, 1], x 7→ {γ ∈ ΛF | x ∈ Dγ }. AfF : X → [0, 1], x 7→ AF (f (x)). Then A∼ = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal of X. Theorem 3.10. Let f : X → Y be a homomorphism of Proof. Let α, β ∈ (0, 1] and γ ∈ [0, 1) be such that T∈ (A∼ ; α) 6= BCK/BCI-algebras. If A∼ = (AT , AI , AF ) is an (∈, ∅, I∈ (A∼ ; β) 6= ∅ and F∈ (A∼ ; γ) 6= ∅. We consider the follow- ∈)-neutrosophic ideal of Y , then the induced neutrosophic set G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras 41 Neutrosophic Sets and Systems, Vol. 20, 2018 Af∼ = (AfT , AfI , AfF ) in X is an (∈, ∈)-neutrosophic ideal of X. AI (x) = AI (f (a)) = AfI (a) ≥ AfI (a ∗ b) ∧ AfI (b) = AI (f (a ∗ b)) ∧ AI (f (b)) = AI (f (a) ∗ f (b)) ∧ AI (f (b)) = AI (x ∗ y) ∧ AI (y), Proof. For any x ∈ X, we have AfT (x) = AT (f (x)) ≤ AT (0) = AT (f (0)) = AfT (0), AfI (x) = AI (f (x)) ≤ AI (0) = AI (f (0)) = AfI (0), AfF (x) = AF (f (x)) ≥ AF (0) = AF (f (0)) = AfF (0). Let x, y ∈ X. Then AfT (x AfT (y) = AT (f (x ∗ y)) ∧ AT (f (y)) ∗ y) ∧ = AT (f (x) ∗ f (y)) ∧ AT (f (y)) ≤ AT (f (x)) = AfT (x), AfI (x ∗ y) ∧ AfI (y) = AI (f (x ∗ y)) ∧ AI (f (y)) = AI (f (x) ∗ f (y)) ∧ AI (f (y)) ≤ AI (f (x)) = AfI (x), and AfF (x ∗ y) ∨ AfF (y) = AF (f (x ∗ y)) ∨ AF (f (y)) = AF (f (x) ∗ f (y)) ∨ AF (f (y)) ≥ AF (f (x)) = AfF (x). and AF (x) = AF (f (a)) = AfF (a) ≤ AfF (a ∗ b) ∨ AfF (b) = AF (f (a ∗ b)) ∨ AF (f (b)) = AF (f (a) ∗ f (b)) ∨ AF (f (b)) = AF (x ∗ y) ∨ AF (y). Therefore A∼ = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal of Y by Theorem 3.1. Let N(∈,∈) (X) be the collection of all (∈, ∈)-neutrosophic ideals of X and let α, β ∈ (0, 1] and γ ∈ [0, 1). Define binary β γ relations Rα T , RI and RF on N(∈,∈) (X) as follows: AT Rα T BT ⇔ T∈ (A∼ ; α) = T∈ (B∼ ; α) β AI RI BI ⇔ I∈ (A∼ ; β) = I∈ (B∼ ; β) AF RγF BF ⇔ F∈ (A∼ ; γ) = F∈ (B∼ ; γ) (3.11) Therefore Af∼ = (AfT , AfI , AfF ) is an (∈, ∈)-neutrosophic ideal for all A∼ = (AT , AI , AF ) and B∼ = (BT , BI , BF ) in of X by Theorem 3.1. N(∈,∈) (X). γ β Clearly Rα T , RI and RF are equivalence relations on N(∈,∈) (X). For any A∼ = (AT , AI , AF ) ∈ N(∈,∈) (X), Theorem 3.11. Let f : X → Y be an onto homomorphism of let [A∼ ]T (resp., [A∼ ]I and [A∼ ]F ) denote the equivalence BCK/BCI-algebras and let A∼ = (AT , AI , AF ) be a neutroclass of A∼ = (AT , AI , AF ) in N(∈,∈) (X) under Rα T (resp., sophic set in Y . If the induced neutrosophic set Af∼ = (AfT , AfI , β γ β α R and R ). Denote by N (X)/R , N (X)/R (∈,∈) (∈,∈) T I F I and AfF ) in X is an (∈, ∈)-neutrosophic ideal of X, then A∼ = (AT , γ N(∈,∈) (X)/RF the collection of all equivalence classes under AI , AF ) is an (∈, ∈)-neutrosophic ideal of Y . β γ Rα T , RI and RF , respectively, that is, Proof. Assume that the induced neutrosophic set Af∼ = (AfT , AfI , AfF ) in X is an (∈, ∈)-neutrosophic ideal of X. For any x ∈ Y , there exists a ∈ X such that f (a) = x since f is onto. Using (3.1), we have N(∈,∈) (X)/Rα T = {[A∼ ]T | A∼ = (AT , AI , AF ) ∈ N(∈,∈) (X), β N(∈,∈) (X)/RI = {[A∼ ]I | A∼ = (AT , AI , AF ) ∈ N(∈,∈) (X), N(∈,∈) (X)/RγF = {[A∼ ]F | A∼ = (AT , AI , AF ) ∈ N(∈,∈) (X). Now let I(X) denote the family of all ideals of X. Define AT (x) = AT (f (a)) = AfT (a) ≤ AfT (0) = AT (f (0)) = AT (0), maps fα , gβ and hγ from N(∈,∈) (X) to I(X) ∪ {∅} by AI (x) = AI (f (a)) = AfI (a) ≤ AfI (0) = AI (f (0)) = AI (0), fα (A∼ ) = T∈ (A∼ ; α), gβ (A∼ ) = I∈ (A∼ ; β) and AF (x) = AF (f (a)) = AfF (a) ≥ AfF (0) = AF (f (0)) = AF (0). hγ (A∼ ) = F∈ (A∼ ; γ), respectively, for all A∼ = (AT , AI , AF ) in N(∈,∈) (X). Then Let x, y ∈ Y . Then f (a) = x and f (b) = y for some a, b ∈ X. fα , gβ and hγ are clearly well-defined. It follows from (3.2) that Theorem 3.12. For any α, β ∈ (0, 1] and γ ∈ [0, 1), the maps fα , gβ and hγ are surjective from N(∈,∈) (X) to I(X) ∪ {∅}. AT (x) = AT (f (a)) = AfT (a) ≥ AfT (a ∗ b) ∧ AfT (b) = AT (f (a ∗ b)) ∧ AT (f (b)) = AT (f (a) ∗ f (b)) ∧ AT (f (b)) = AT (x ∗ y) ∧ AT (y), Proof. Let 0∼ := (0T , 0I , 1F ) be a neutrosophic set in X where 0T , 0I and 1F are fuzzy sets in X defined by 0T (x) = 0, 0I (x) = 0 and 1F (x) = 1 for all x ∈ X. Obviously, 0∼ := (0T , 0I , 1F ) is an (∈, ∈)-neutrosophic ideal of X. Also, fα (0∼ ) = T∈ (0∼ ; α) = ∅, gβ (0∼ ) = I∈ (0∼ ; β) = ∅ G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras Neutrosophic Sets and Systems, Vol. 20, 2018 42 and hγ (0∼ ) = F∈ (0∼ ; γ) = ∅. For any ideal I of X, let Proof. Consider the (∈, ∈)-neutrosophic ideal 0∼ := (0T , 0I , A∼ = (AT , AI , AF ) be the (∈, ∈)-neutrosophic ideal of X 1F ) of X which is given in the proof of Theorem 3.12. Then in the proof of Theorem 3.4. Then fα (A∼ ) = T∈ (A∼ ; α) = I, ϕα (0∼ ) = fα (0∼ ) ∩ hα (0∼ ) = T∈ (0∼ ; α) ∩ F∈ (0∼ ; α) = ∅, gβ (A∼ ) = I∈ (A∼ ; β) = I and hγ (A∼ ) = F∈ (A∼ ; γ) = I. ϕβ (0∼ ) = gβ (0∼ ) ∩ hβ (0∼ ) = I∈ (0∼ ; β) ∩ F∈ (0∼ ; β) = ∅. Therefore fα , gβ and hγ are surjective. For any ideal I of X, consider the (∈, ∈)-neutrosophic ideal Theorem 3.13. The quotient sets N(∈,∈) (X)/Rα T , A = (A , A , A ) of X in the proof of Theorem 3.4. Then ∼ T I F N(∈,∈) (X)/RβI and N(∈,∈) (X)/RγF are equivalent to I(X) ∪ {∅} for any α, β ∈ (0, 1] and γ ∈ [0, 1). ϕα (A∼ ) = fα (A∼ ) ∩ hα (A∼ ) = T∈ (A∼ ; α) ∩ F∈ (A∼ ; α) = I Proof. Let A∼ = (AT , AI , AF ) ∈ N(∈,∈) (X). For any α, β ∈ (0, 1] and γ ∈ [0, 1), define and fα∗ : N(∈,∈) (X)/Rα T → I(X) ∪ {∅}, [A∼ ]T 7→ fα (A∼ ), β ∗ gβ : N(∈,∈) (X)/RI → I(X) ∪ {∅}, [A∼ ]I 7→ gβ (A∼ ), h∗γ : N(∈,∈) (X)/RγF → I(X) ∪ {∅}, [A∼ ]F 7→ hγ (A∼ ). Assume that fα (A∼ ) = fα (B∼ ), gβ (A∼ ) = gβ (B∼ ) and hγ (A∼ ) = hγ (B∼ ) for B∼ = (BT , BI , BF ) ∈ N(∈,∈) (X). Then T∈ (A∼ ; α) = T∈ (B∼ ; α), I∈ (A∼ ; β) = I∈ (B∼ ; β) and β F∈ (A∼ ; γ) = F∈ (B∼ ; γ) which imply that AT Rα T BT , AI RI BI γ and AF RF BF . Hence [A∼ ]T = [B∼ ]T , [A∼ ]I = [B∼ ]I and [A∼ ]F = [B∼ ]F . Therefore fα∗ , gβ∗ and h∗γ are injective. Consider the (∈, ∈)-neutrosophic ideal 0∼ := (0T , 0I , 1F ) of X which is given in the proof of Theorem 3.12. Then fα∗ ([0∼ ]T ) = fα (0∼ ) = T∈ (0∼ ; α) = ∅, gβ∗ ([0∼ ]I ) = gβ (0∼ ) = I∈ (0∼ ; β) = ∅, and h∗γ ([0∼ ]F ) = hγ (0∼ ) = F∈ (0∼ ; γ) = ∅. For any ideal I of X, consider the (∈, ∈)-neutrosophic ideal A∼ = (AT , AI , AF ) of X in the proof of Theorem 3.4. Then fα∗ ([A∼ ]T ) = fα (A∼ ) = T∈ (A∼ ; α) = I, gβ∗ ([A∼ ]I ) = gβ (A∼ ) = I∈ (A∼ ; β) = I, and h∗γ ([A∼ ]F ) = hγ (A∼ ) = F∈ (A∼ ; γ) = I. Hence fα∗ , gβ∗ and h∗γ are surjective, and the proof is over. ϕβ (A∼ ) = gβ (A∼ ) ∩ hβ (A∼ ) = I∈ (A∼ ; β) ∩ F∈ (A∼ ; β) = I. Therefore ϕα and ϕβ are surjective. Theorem 3.15. For any α, β ∈ (0, 1), the quotient sets N(∈,∈) (X)/ϕα and N(∈,∈) (X)/ϕβ are equivalent to I(X) ∪ {∅}. Proof. Given α, β ∈ (0, 1), define two maps ϕ∗α and ϕ∗β as follows: ϕ∗α : N(∈,∈) (X)/ϕα → I(X) ∪ {∅}, [A∼ ]Rα 7→ ϕα (A∼ ), ϕ∗β : N(∈,∈) (X)/ϕβ → I(X) ∪ {∅}, [A∼ ]Rβ 7→ ϕβ (A∼ ).
= If ϕ∗α ([A∼ ]Rα ) = ϕ∗α ([B∼ ]Rα ) and ϕ∗β [A∼ ]Rβ
ϕ∗β [B∼ ]Rβ for all [A∼ ]Rα , [B∼ ]Rα ∈ N(∈,∈) (X)/ϕα and [A∼ ]Rβ , [B∼ ]Rβ ∈ N(∈,∈) (X)/ϕβ , then For any α, β ∈ [0, 1], we define another relations Rα and Rβ on N(∈,∈) (X) as follows: (A∼ , B∼ ) ∈ Rα ⇔ T∈ (A∼ ; α) ∩ F∈ (A∼ ; α) = T∈ (B∼ ; α) ∩ F∈ (B∼ ; α), (A∼ , B∼ ) ∈ Rβ ⇔ I∈ (A∼ ; β) ∩ F∈ (A∼ ; β) = I∈ (B∼ ; β) ∩ F∈ (B∼ ; β) fα (A∼ ) ∩ hα (A∼ ) = fα (B∼ ) ∩ hα (B∼ ) and gβ (A∼ ) ∩ hβ (A∼ ) = gβ (B∼ ) ∩ hβ (B∼ ), (3.12) that is, T∈ (A∼ ; α) ∩ F∈ (A∼ ; α) = T∈ (B∼ ; α) ∩ F∈ (B∼ ; α) for all A∼ = (AT , AI , AF ) and B∼ = (BT , BI , BF ) in N(∈,∈) (X). Then the relations Rα and Rβ are also equivalence and relations on N(∈,∈) (X). I∈ (A∼ ; β) ∩ F∈ (A∼ ; β) = I∈ (B∼ ; β) ∩ F∈ (B∼ ; β). Theorem 3.14. Given α, β ∈ (0, 1), we define two maps ϕα : N(∈,∈) (X) → I(X) ∪ {∅}, A∼ 7→ fα (A∼ ) ∩ hα (A∼ ), ϕβ : N(∈,∈) (X) → I(X) ∪ {∅}, A∼ 7→ gβ (A∼ ) ∩ hβ (A∼ ) Hence (A∼ , B∼ ) ∈ Rα and (A∼ , B∼ ) ∈ Rβ . It follows that [A∼ ]Rα = [B∼ ]Rα and [A∼ ]Rβ = [B∼ ]Rβ . Thus ϕ∗α and ϕ∗β (3.13) are injective. Consider the (∈, ∈)-neutrosophic ideal 0∼ := (0T , 0I , 1F ) of X which is given in the proof of Theorem 3.12. Then for each A∼ = (AT , AI , AF ) ∈ N(∈,∈) (X). Then ϕα and ϕβ are surjective. ϕ∗α ([0∼ ]Rα ) = ϕα (0∼ ) = fα (0∼ ) ∩ hα (0∼ ) = T∈ (0∼ ; α) ∩ F∈ (0∼ ; α) = ∅ G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras 43 Neutrosophic Sets and Systems, Vol. 20, 2018 and ϕ∗β [0∼ ]Rβ = ϕβ (0∼ ) = gβ (0∼ ) ∩ hβ (0∼ )
= I∈ (0∼ ; β) ∩ F∈ (0∼ ; β) = ∅. For any ideal I of X, consider the (∈, ∈)-neutrosophic ideal A∼ = (AT , AI , AF ) of X in the proof of Theorem 3.4. Then ϕ∗α ([A∼ ]Rα ) = ϕα (A∼ ) = fα (A∼ ) ∩ hα (A∼ ) = T∈ (A∼ ; α) ∩ F∈ (A∼ ; α) = I and ϕ∗β [A∼ ]Rβ = ϕβ (A∼ ) = gβ (A∼ ) ∩ hβ (A∼ )
= I∈ (A∼ ; β) ∩ F∈ (A∼ ; β) = I. Therefore ϕ∗α and ϕ∗β are surjective. This completes the proof. References [1] A. Borumand Saeid and Y.B. Jun, Neutrosophic subalgebras of BCK/BCI-algebras based on neutrosophic points, Ann. Fuzzy Math. Inform. 14 (2017), no. 1, 87–97. [11] Abdel-Basset, M., Mohamed, M., Smarandache, F., & Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis. Symmetry, 10(4), 106. [12] Abdel-Basset, M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Sets in Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages 47-55 [13] Abdel-Basset, M., Gunasekaran, M., Mohamed, M., & Smarandache, F. A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications, 1-11. [14] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22. [15] Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloud computing services. Future Generation Computer Systems, 86, 12-29. [16] Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066. [17] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116. [2] K. Iséki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125–130. [3] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. Received : March 26, 2018. Accepted : April 16, 2018. [4] Y. Huang, BCI-algebra, Science Press, Beijing, 2006. [5] Y.B. Jun, Neutrosophic subalgebras of several types in BCK/BCI-algebras, Ann. Fuzzy Math. Inform. 14 (2017), no. 1, 75–86. [6] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoonsa Co. Seoul, Korea 1994. [7] M.A. Öztürk and Y.B. Jun, Neutrosophic ideals in BCK/BCI-algebras based on neutrosophic points, J. Inter. Math. Virtual Inst. 8 (2018), 1–17. [8] F. Smarandache, Neutrosophy, Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 105 p., 1998. http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (last edition online). [9] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Reserch Press, Rehoboth, NM, 1999. [10] F. Smarandache, Neutrosophic set-a generalization of the intuitionistic fuzzy set, Int. J. Pure Appl. Math. 24 (2005), no.3, 287–297. G. Muhiuddin, H. Bordbar, F. Smarandache, Y.B. Jun, Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras 44 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico Commutative falling neutrosophic ideals in BCK-algebras Young Bae Jun1 , Florentin Smarandache2 , Mehmat Ali Öztürk3 1 Department 2 Mathematics of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea. E-mail: skywine@gmail.com & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: fsmarandache@gmail.com 3 Faculty of Arts and Sciences, Adiyaman University, 02040 Adiyaman, Turkey. E-mail: mehaliozturk@gmail.com ∗ Correspondence: Abstract: The notions of a commutative (∈, ∈)-neutrosophic ideal and a commutative falling neutrosophic ideal are introduced, and several properties are investigated. Characterizations of a commutative (∈, ∈)-neutrosophic ideal are obtained. Relations between commutative (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic ideal are discussed. Conditions for an (∈, ∈)-neutrosophic ideal to skywine@gmail.com be a commutative (∈, ∈)-neutrosophic ideal are established. Relations between commutative (∈, ∈)-neutrosophic ideal, falling neutrosophic ideal and commutative falling neutrosophic ideal are considered. Conditions for a falling neutrosophic ideal to be commutative are provided. Keywords: (commutative) (∈, ∈)-neutrosophic ideal; neutrosophic random set; neutrosophic falling shadow; (commutative) falling neutrosophic ideal. 1 Introduction zations of a commutative (∈, ∈)-neutrosophic ideal, and discuss relations between a commutative (∈, ∈)-neutrosophic ideal and an (∈, ∈)-neutrosophic ideal. We provide conditions for an (∈, Neutrosophic set (NS) developed by Smarandache [11, 12, ∈)-neutrosophic ideal to be a commutative (∈, ∈)-neutrosophic 13] is a more general platform which extends the concepts ideal, and consider relations between a commutative (∈, ∈)of the classic set and fuzzy set, intuitionistic fuzzy set and neutrosophic ideal, a falling neutrosophic ideal and a commuinterval valued intuitionistic fuzzy set. Neutrosophic set tative falling neutrosophic ideal. We give conditions for a falling theory is applied to various part which is refered to the neutrosophic ideal to be commutative. site http://fs.gallup.unm.edu/neutrosophy.htm. Jun, Borumand Saeid and Öztürk studied neutrosophic subalgebras/ideals in BCK/BCI-algebras based on neutrosophic points (see [1], [6] and [10]). Goodman [2] pointed out the equivalence of a fuzzy 2 Preliminaries set and a class of random sets in the study of a unified treatment of uncertainty modeled by means of combining probability and A BCK/BCI-algebra is an important class of logical algebras fuzzy set theory. Wang and Sanchez [16] introduced the theory of introduced by K. Iséki (see [3] and [4]) and was extensively infalling shadows which directly relates probability concepts with vestigated by several researchers. the membership function of fuzzy sets. The mathematical strucBy a BCI-algebra, we mean a set X with a special element 0 ture of the theory of falling shadows is formulated in [17]. Tan et and a binary operation ∗ that satisfies the following conditions: al. [14, 15] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of (I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), falling shadows. Jun and Park [7] considered a fuzzy subalgebra and a fuzzy ideal as the falling shadow of the cloud of the subalgebra and ideal. Jun et al. [8] introduced the notion of neutro- (II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), sophic random set and neutrosophic falling shadow. Using these notions, they introduced the concept of falling neutrosophic sub- (III) (∀x ∈ X) (x ∗ x = 0), algebra and falling neutrosophic ideal in BCK/BCI-algebras, and investigated related properties. They discussed relations be- (IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). tween falling neutrosophic subalgebra and falling neutrosophic ideal, and established a characterization of falling neutrosophic If a BCI-algebra X satisfies the following identity: ideal. In this paper, we introduce the concepts of a commutative (∈, (V) (∀x ∈ X) (0 ∗ x = 0), ∈)-neutrosophic ideal and a commutative falling neutrosophic ideal, and investigate several properties. We obtain characteri- then X is called a BCK-algebra. Any BCK/BCI-algebra X Y.B. Jun, F. Smarandache, M.A. Ozturk, Commutative falling neutrosophic ideals in BCK-algebras. 45 Neutrosophic Sets and Systems, Vol. 20, 2018 satisfies the following conditions: (0, 1] and γ ∈ [0, 1), we consider the following sets: (∀x ∈ X) (x ∗ 0 = x) ,   x≤y ⇒ x∗z ≤y∗z , (∀x, y, z ∈ X) x≤y ⇒ z∗y ≤z∗x (2.1) (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) , (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.3) We say T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are neutrosophic ∈(2.4) subsets. (2.2) T∈ (A; α) := {x ∈ X | AT (x) ≥ α}, I∈ (A; β) := {x ∈ X | AI (x) ≥ β}, F∈ (A; γ) := {x ∈ X | AF (x) ≤ γ}. A neutrosophic set A = (AT , AI , AF ) in a BCK/BCIwhere x ≤ y if and only if x ∗ y = 0. A nonempty subset S of a algebra X is called an (∈, ∈)-neutrosophic subalgebra of X (see BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S if the following assertions are valid. [6]) for all x, y ∈ S. A subset I of a BCK/BCI-algebra X is called an ideal of X if it satisfies:   x ∈ T∈ (A; αx ), y ∈ T∈ (A; αy )  ⇒ x ∗ y ∈ T∈ (A; αx ∧ αy ),  0 ∈ I, (2.5)    x ∈ I∈ (A; βx ), y ∈ I∈ (A; βy )   (2.8) (∀x ∈ X) (∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) . (2.6) (∀x, y ∈ X)   ⇒ x ∗ y ∈ I∈ (A; βx ∧ βy ),     x ∈ F∈ (A; γx ), y ∈ F∈ (A; γy )  A subset I of a BCK-algebra X is called a commutative ideal ⇒ x ∗ y ∈ F∈ (A; γx ∨ γy ) of X if it satisfies (2.5) and for all αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). (x ∗ y) ∗ z ∈ I, z ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I (2.7) A neutrosophic set A = (AT , AI , AF ) in a BCK/BCIalgebra X is called an (∈, ∈)-neutrosophic ideal of X (see [10]) for all x, y, z ∈ X. if the following assertions are valid.   Observe that every commutative ideal is an ideal, but the conx ∈ T∈ (A; αx ) ⇒ 0 ∈ T∈ (A; αx ) verse is not true (see [9]). (∀x ∈ X)  x ∈ I∈ (A; βx ) ⇒ 0 ∈ I∈ (A; βx )  (2.9) x ∈ F∈ (A; γx ) ⇒ 0 ∈ F∈ (A; γx ) We refer the reader to the books [5, 9] for further information regarding BCK/BCI-algebras. and   For any family {ai | i ∈ Λ} of real numbers, we define x ∗ y ∈ T∈ (A; αx ), y ∈ T∈ (A; αy ) _   ⇒ x ∈ T∈ (A; αx ∧ αy )   {ai | i ∈ Λ} := sup{ai | i ∈ Λ}  x ∗ y ∈ I∈ (A; βx ), y ∈ I∈ (A; βy )   (2.10) (∀x, y ∈ X)    ⇒ x ∈ I∈ (A; βx ∧ βy )   and  x ∗ y ∈ F∈ (A; γx ), y ∈ F∈ (A; γy )  ^ ⇒ x ∈ F∈ (A; γx ∨ γy ) {ai | i ∈ Λ} := inf{ai | i ∈ Λ}. for all αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). In what follows, let X and P(X) denote a BCK/BCIW If Λ = {1, 2}, weVwill also use a1 ∨ a2 and a1 ∧ a2 instead of {ai | i ∈ Λ} and {ai | i ∈ Λ}, respectively. algebra and the power set of X, respectively, unless otherwise specified. Let X be a non-empty set. A neutrosophic set (NS) in X (see For each x ∈ X and D ∈ P(X), let [12]) is a structure of the form: A := {hx; AT (x), AI (x), AF (x)i | x ∈ X} x̄ := {C ∈ P(X) | x ∈ C}, (2.11) where AT : X → [0, 1] is a truth membership function, and AI : X → [0, 1] is an indeterminate membership function, and D̄ := {x̄ | x ∈ D}. (2.12) AF : X → [0, 1] is a false membership function. For the sake of simplicity, we shall use the symbol A = (AT , AI , AF ) for the An ordered pair (P(X), B) is said to be a hyper-measurable neutrosophic set structure on X if B is a σ-field in P(X) and X̄ ⊆ B. Given a probability space (Ω, A, P ) and a hyper-measurable structure (P(X), B) on X, a neutrosophic random set on X (see [8]) is defined to be a triple ξ := (ξT , ξI , ξF ) in which ξT , ξI and Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ ξF are mappings from Ω to P(X) which are A-B measurables, A := {hx; AT (x), AI (x), AF (x)i | x ∈ X}. Y.B. Jun, F. Smarandache, M.A. Öztürk, Commutative falling neutrosophic ideals in BCK-algebras. Neutrosophic Sets and Systems, Vol. 20, 2018 46 for all x, y, z ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). that is, Example 3.2. Consider a set X = {0, 1, 2, 3} with the binary  −1  operation ∗ which is given in Table 1. (∀C ∈ B)  ξI (C) = {ωI ∈ Ω | ξI (ωI ) ∈ C} ∈ A  . ξF−1 (C) = {ωF ∈ Ω | ξF (ωF ) ∈ C} ∈ A Table 1: Cayley table for the binary operation “∗” (2.13)  ξT−1 (C) = {ωT ∈ Ω | ξT (ωT ) ∈ C} ∈ A  Given a neutrosophic random set ξ := (ξT , ξI , ξF ) on X, consider functions: H̃T : X → [0, 1], xT 7→ P (ωT | xT ∈ ξT (ωT )), H̃I : X → [0, 1], xI 7→ P (ωI | xI ∈ ξI (ωI )), ∗ 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 1 2 0 H̃F : X → [0, 1], xF 7→ 1 − P (ωF | xF ∈ ξF (ωF )). Then (X; ∗, 0) is a BCK-algebra (see [9]). Let A = Then H̃ := (H̃T , H̃I , H̃F ) is a neutrosophic set on X, and we (AT , AI , AF ) be a neutrosophic set in X defined by Table 2 call it a neutrosophic falling shadow (see [8]) of the neutrosophic random set ξ := (ξT , ξI , ξF ), and ξ := (ξT , ξI , ξF ) is called a Table 2: Tabular representation of A = (AT , AI , AF ) neutrosophic cloud (see [8]) of H̃ := (H̃T , H̃I , H̃F ). For example, consider a probability space (Ω, A, P ) = X AT (x) AI (x) AF (x) ([0, 1], A, m) where A is a Borel field on [0, 1] and m is the usual 0 0.7 0.9 0.2 Lebesgue measure. Let H̃ := (H̃T , H̃I , H̃F ) be a neutrosophic 1 0.3 0.6 0.8 set in X. Then a triple ξ := (ξT , ξI , ξF ) in which 2 0.3 0.6 0.8 3 0.5 0.4 0.7 ξT : [0, 1] → P(X), α 7→ T∈ (H̃; α), ξI : [0, 1] → P(X), β 7→ I∈ (H̃; β), It is routine to verify that A = (AT , AI , AF ) is a commutative (∈, ∈)-neutrosophic ideal of X. ξF : [0, 1] → P(X), γ 7→ F∈ (H̃; γ) is a neutrosophic random set and ξ := (ξT , ξI , ξF ) is a neuTheorem 3.3. For a neutrosophic set A = (AT , AI , AF ) in a trosophic cloud of H̃ := (H̃T , H̃I , H̃F ). We will call ξ := BCK-algebra X, the following are equivalent. (ξT , ξI , ξF ) defined above as the neutrosophic cut-cloud (see [8]) of H̃ := (H̃T , H̃I , H̃F ). (1) The non-empty ∈-subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) Let (Ω, A, P ) be a probability space and let ξ := (ξT , ξI , ξF ) are commutative ideals of X for all α, β ∈ (0, 1] and γ ∈ be a neutrosophic random set on X. If ξT (ωT ), ξI (ωI ) and [0, 1). ξF (ωF ) are subalgebras (resp., ideals) of X for all ωT , ωI , ωF ∈ Ω, then the neutrosophic falling shadow H̃ := (H̃T , H̃I , H̃F ) (2) A = (AT , AI , AF ) satisfies the following assertions.   of ξ := (ξT , ξI , ξF ) is called a falling neutrosophic subalgebra AT (0) ≥ AT (x) (resp., falling neutrosophic ideal) of X (see [8]). (∀x ∈ X)  AI (0) ≥ AI (x)  (3.2) AF (0) ≤ AF (x) 3 and for all x, y, z ∈ X, Commutative (∈, ∈)-neutrosophic ideals Definition 3.1. A neutrosophic set A = (AT , AI , AF ) in a BCK-algebra X is called a commutative (∈, ∈)-neutrosophic ideal of X if it satisfies the condition (2.9) and (x ∗ y) ∗ z ∈ T∈ (A; αx ), z ∈ T∈ (A; αy ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ T∈ (A; αx ∧ αy ) (x ∗ y) ∗ z ∈ I∈ (A; βx ), z ∈ I∈ (A; βy ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I∈ (A; βx ∧ βy ) (x ∗ y) ∗ z ∈ F∈ (A; γx ), z ∈ F∈ (A; γy ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ F∈ (A; γx ∨ γy ) AT (x ∗ (y ∗ (y ∗ x))) ≥ AT ((x ∗ y) ∗ z) ∧ AT (z) AI (x ∗ (y ∗ (y ∗ x))) ≥ AI ((x ∗ y) ∗ z) ∧ AI (z) AF (x ∗ (y ∗ (y ∗ x))) ≤ AF ((x ∗ y) ∗ z) ∨ AF (z) (3.3) Proof. Assume that the non-empty ∈-subsets T∈ (A; α), (3.1) I∈ (A; β) and F∈ (A; γ) are commutative ideals of X for all α, β ∈ (0, 1] and γ ∈ [0, 1). If AT (0) < AT (a) for some a ∈ X, then a ∈ T∈ (A; AT (a)) and 0 ∈ / T∈ (A; AT (a)). This is a contradiction, and so AT (0) ≥ AT (x) for all x ∈ X. Similarly, Y.B. Jun, F. Smarandache, M.A. Ozturk, Commutative falling neutrosophic ideals in BCK-algebras. 47 Neutrosophic Sets and Systems, Vol. 20, 2018 AI (0) ≥ AI (x) for all x ∈ X. Suppose that AF (0) > AF (a) for some a ∈ X. Then a ∈ F∈ (A; AF (a)) and 0 ∈ / F∈ (A; AF (a)). This is a contradiction, and thus AF (0) ≤ AF (x) for all x ∈ X. Therefore (3.2) is valid. Assume that there exist a, b, c ∈ X such that AT (a ∗ (b ∗ (b ∗ a))) < AT ((a ∗ b) ∗ c) ∧ AT (c). subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are commutative ideals of X for all α, β ∈ (0, 1] and γ ∈ [0, 1). Theorem 3.4. Let A = (AT , AI , AF ) be a neutrosophic set in a BCK-algebra X. Then A = (AT , AI , AF ) is a commutative (∈, ∈)-neutrosophic ideal of X if and only if the non-empty neutrosophic ∈-subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are commutative ideals of X for all α, β ∈ (0, 1] and γ ∈ [0, 1). Taking α := AT ((a ∗ b) ∗ c) ∧ AT (c) implies that (a ∗ b) ∗ c ∈ T∈ (A; α) and c ∈ T∈ (A; α) but a ∗ (b ∗ (b ∗ a)) ∈ / T∈ (A; α), Proof. Let A = (AT , AI , AF ) be a commutative (∈, ∈)neutrosophic ideal of X and assume that T∈ (A; α), I∈ (A; β) and which is a contradiction. Hence F∈ (A; γ) are nonempty for α, β ∈ (0, 1] and γ ∈ [0, 1). Then AT (x ∗ (y ∗ (y ∗ x))) ≥ AT ((x ∗ y) ∗ z) ∧ AT (z) there exist x, y, z ∈ X such that x ∈ T∈ (A; α), y ∈ I∈ (A; β) and z ∈ F∈ (A; γ). It follows from (2.9) that 0 ∈ T∈ (A; α), for all x, y, z ∈ X. By the similar way, we can verify that 0 ∈ I∈ (A; β) and 0 ∈ F∈ (A; γ). Let x, y, z, a, b, c, u, v, w ∈ X be such that AI (x ∗ (y ∗ (y ∗ x))) ≥ AI ((x ∗ y) ∗ z) ∧ AI (z) (x ∗ y) ∗ z ∈ T∈ (A; α), z ∈ T∈ (A; α), for all x, y, z ∈ X. Now suppose there are x, y, z ∈ X such that (a ∗ b) ∗ c ∈ I∈ (A; β), c ∈ I∈ (A; β), (u ∗ v) ∗ w ∈ F∈ (A; γ), w ∈ F∈ (A; γ). AF (x ∗ (y ∗ (y ∗ x))) > AF ((x ∗ y) ∗ z) ∨ AF (z) := γ. Then Then (x∗y)∗z ∈ F∈ (A; γ) and z ∈ F∈ (A; γ) but x∗(y∗(y∗x)) ∈ / F∈ (A; γ), a contradiction. Thus x ∗ (y ∗ (y ∗ x)) ∈ T∈ (A; α ∧ α) = T∈ (A; α), a ∗ (b ∗ (b ∗ a)) ∈ I∈ (A; β ∧ β) = I∈ (A; β), AF (x ∗ (y ∗ (y ∗ x))) ≤ AF ((x ∗ y) ∗ z) ∨ AF (z) u ∗ (v ∗ (v ∗ u)) ∈ F∈ (A; γ ∨ γ) = F∈ (A; γ) for all x, y, z ∈ X. by (2.10). Hence the non-empty neutrosophic ∈-subsets T (A; α), I (A; β) and F∈ (A; γ) are commutative ideals of X ∈ ∈ Conversely, let A = (AT , AI , AF ) be a neutrosophic set in X for all α, β ∈ (0, 1] and γ ∈ [0, 1). satisfying two conditions (3.2) and (3.3). Assume that T∈ (A; α), Conversely, let A = (AT , AI , AF ) be a neutrosophic set in X I∈ (A; β) and F∈ (A; γ) are nonempty for α, β ∈ (0, 1] and γ ∈ for which T (A; α), I∈ (A; β) and F∈ (A; γ) are nonempty and ∈ [0, 1). Let x ∈ T∈ (A; α), a ∈ I∈ (A; β) and u ∈ F∈ (A; γ) are commutative ideals of X for all α, β ∈ (0, 1] and γ ∈ [0, 1). for α, β ∈ (0, 1] and γ ∈ [0, 1). Then AT (0) ≥ AT (x) ≥ α, is valid. Let x, y, z ∈ X and αx , αy ∈ (0, 1] Obviously, (2.9) AI (0) ≥ AI (a) ≥ β, and AF (0) ≤ AF (u) ≤ γ by (3.2). It be such that (x ∗ y) ∗ z ∈ T∈ (A; αx ) and z ∈ T∈ (A; αy ). Then follows that 0 ∈ T∈ (A; α), 0 ∈ I∈ (A; β) and 0 ∈ F∈ (A; γ). Let a, b, c ∈ X be such that (a ∗ b) ∗ c ∈ T∈ (A; α) and c ∈ T∈ (A; α) (x ∗ y) ∗ z ∈ T∈ (A; α) and z ∈ T∈ (A; α) where α = αx ∧ αy . Since T∈ (A; α) is a commutative ideal of X, it follows that for α ∈ (0, 1]. Then AT (a ∗ (b ∗ (b ∗ a))) ≥ AT ((a ∗ b) ∗ c) ∧ AT (c) ≥ α x ∗ (y ∗ (y ∗ x)) ∈ T∈ (A; α) = T∈ (A; αx ∧ αy ). by (3.3), and so a ∗ (b ∗ (b ∗ a)) ∈ T∈ (A; α). If (x ∗ y) ∗ z ∈ Similarly, if (x ∗ y) ∗ z ∈ I∈ (A; βx ) and z ∈ I∈ (A; βy ) for all I∈ (A; β) and z ∈ I∈ (A; β) for all x, y, z ∈ X and β ∈ (0, 1], x, y, z ∈ X and βx , βy ∈ (0, 1], then then AI ((x ∗ y) ∗ z) ≥ β and AI (z) ≥ β. Hence the condition x ∗ (y ∗ (y ∗ x)) ∈ I∈ (A; βx ∧ βy ). (3.3) implies that AI (x ∗ (y ∗ (y ∗ x))) ≥ AI ((x ∗ y) ∗ z) ∧ AI (z) ≥ β, that is, x ∗ (y ∗ (y ∗ x)) ∈ I∈ (A; β). Finally, suppose that (x ∗ y) ∗ z ∈ F∈ (A; γ) and z ∈ F∈ (A; γ) Now, suppose that (x∗y)∗z ∈ F∈ (A; γx ) and z ∈ F∈ (A; γy ) for all x, y, z ∈ X and γx , γy ∈ [0, 1). Then (x ∗ y) ∗ z ∈ F∈ (A; γ) and z ∈ F∈ (A; γ) where γ = γx ∨ γy . Hence x ∗ (y ∗ (y ∗ x)) ∈ F∈ (A; γ) = F∈ (A; γx ∨ γy ) for all x, y, z ∈ X and γ ∈ (0, 1]. Then AF ((x ∗ y) ∗ z) ≤ γ and since F∈ (A; γ) is a commutative ideal of X. Therefore A = (AT , AI , AF ) is a commutative (∈, ∈)-neutrosophic ideal of X. AF (z) ≤ γ, which imply from the condition (3.3) that AF (x ∗ (y ∗ (y ∗ x))) ≤ AF ((x ∗ y) ∗ z) ∨ AF (z) ≤ γ. Corollary 3.5. Let A = (AT , AI , AF ) be a neutrosophic set in Hence x ∗ (y ∗ (y ∗ x)) ∈ F∈ (A; γ). Therefore the non-empty ∈- a BCK-algebra X. Then A = (AT , AI , AF ) is a commutaY.B. Jun, F. Smarandache, M.A. Öztürk, Commutative falling neutrosophic ideals in BCK-algebras. Neutrosophic Sets and Systems, Vol. 20, 2018 48 tive (∈, ∈)-neutrosophic ideal of X if and only if it satisfies two conditions (3.2) and (3.3). Proposition 3.6. Every commutative (∈, ∈)-neutrosophic ideal A = (AT , AI , AF ) of a BCK-algebra X satisfies:   x ∗ y ∈ T∈ (A; α)  ⇒ x ∗ (y ∗ (y ∗ x)) ∈ T∈ (A; α)     x ∗ y ∈ I∈ (A; β)   (3.4) (∀x, y ∈ X)   ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I∈ (A; β)     x ∗ y ∈ F∈ (A; γ)  ⇒ x ∗ (y ∗ (y ∗ x)) ∈ F∈ (A; γ) Then (X; ∗, 0) is a BCK-algebra (see [9]). Let A = (AT , AI , AF ) be a neutrosophic set in X defined by Table 4 Table 4: Tabular representation of A = (AT , AI , AF ) X 0 1 2 3 4 AT (x) 0.66 0.55 0.33 0.33 0.33 AI (x) 0.77 0.45 0.66 0.45 0.45 AF (x) 0.27 0.37 0.47 0.67 0.67 for all α, β ∈ (0, 1] and γ ∈ [0, 1). Routine calculations show that A = (AT , AI , AF ) is an (∈, ∈)neutrosophic ideal of X. But it is not a commutative (∈, ∈)neutrosophic ideal of X since (2 ∗ 3) ∗ 0 ∈ T∈ (A; 0.6) and 0 ∈ Theorem 3.7. Every commutative (∈, ∈)-neutrosophic ideal of T∈ (A; 0.5) but 2 ∗ (3 ∗ (3 ∗ 2)) ∈ / T∈ (A; 0.5 ∧ 0.6), (1 ∗ 3) ∗ a BCK-algebra X is an (∈, ∈)-neutrosophic ideal of X. 2 ∈ I∈ (A; 0.55) and 2 ∈ I∈ (A; 0.63) but 1 ∗ (3 ∗ (3 ∗ 1)) ∈ / Proof. Let A = (AT , AI , AF ) be a commutative (∈, ∈)- I∈ (A; 0.55 ∧ 0.63), and/or (2 ∗ 3) ∗ 0 ∈ F∈ (A; 0.43) and 0 ∈ F∈ (A; 0.39) but 2 ∗ (3 ∗ (3 ∗ 2)) ∈ / F∈ (A; 0.43 ∨ 0.39). neutrosophic ideal of a BCK-algebra X. Assume that Proof. It is induced by taking z = 0 in (3.1). We provide conditions for an (∈, ∈)-neutrosophic ideal to be a commutative (∈, ∈)-neutrosophic ideal. x ∗ y ∈ T∈ (A; αx ), y ∈ T∈ (A; αy ), a ∗ b ∈ I∈ (A; βa ), b ∈ I∈ (A; βb ), c ∗ d ∈ F∈ (A; γc ), d ∈ F∈ (A; γd ) Theorem 3.9. Let A = (AT , AI , AF ) be an (∈, ∈)-neutrosophic ideal of a BCK-algebra X in which the condition (3.4) is valid. Then A = (AT , AI , AF ) is a commutative (∈, ∈)-neutrosophic ideal of X. for all x, y, a, b, c, d ∈ X. Using (2.1), we have (x ∗ 0) ∗ y = x ∗ y ∈ T∈ (A; αx ), (a ∗ 0) ∗ b = a ∗ b ∈ I∈ (A; βa ), (c ∗ 0) ∗ d = c ∗ d ∈ F∈ (A; γc ). Proof. Let A = (AT , AI , AF ) be an (∈, ∈)-neutrosophic ideal of X and x, y, z ∈ X be such that (x ∗ y) ∗ z ∈ T∈ (A; αx ) and It follows from (3.1), (2.1) and (V) that z ∈ T∈ (A; αy ) for αx , αy ∈ (0, 1]. Then x ∗ y ∈ T∈ (A; αx ∧ αy ) since A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal of X. x = x ∗ 0 = x ∗ (0 ∗ (0 ∗ x)) ∈ T∈ (A; αx ∧ αy ), It follows from (3.4) that x ∗ (y ∗ (y ∗ x)) ∈ T∈ (A; αx ∧ αy ). a = a ∗ 0 = a ∗ (0 ∗ (0 ∗ a)) ∈ I∈ (A; βa ∧ βb ), Similarly, if (x ∗ y) ∗ z ∈ I∈ (A; βx ) and z ∈ I∈ (A; βy ), then c = c ∗ 0 = c ∗ (0 ∗ (0 ∗ c)) ∈ F∈ (A; γc ∨ γd ). x ∗ (y ∗ (y ∗ x)) ∈ I∈ (A; βx ∧ βy ). Let a, b, c ∈ X and γa , γb ∈ [0, 1) be such that (a ∗ b) ∗ c ∈ F∈ (A; γa ) and c ∈ F∈ (A; γa ). Therefore A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic ideal of Then a ∗ b ∈ F∈ (A; γa ∨ γb ), which implies from (3.4) that X. a ∗ (b ∗ (b ∗ a)) ∈ F∈ (A; γa ∨ γb ). Therefore A = (AT , AI , AF ) The converse of Theorem 3.7 is not true as seen in the follow- is a commutative (∈, ∈)-neutrosophic ideal of X. ing example. Lemma 3.10. Every (∈, ∈)-neutrosophic ideal A = Example 3.8. Consider a set X = {0, 1, 2, 3, 4} with the binary (AT , AI , AF ) of a BCK-algebra X satisfies: operation ∗ which is given in Table 3 y, z ∈ T∈ (A; α) ⇒ x ∈ T∈ (A; α) y, z ∈ I∈ (A; β) ⇒ x ∈ I∈ (A; β) (3.5) Table 3: Cayley table for the binary operation “∗” y, z ∈ F∈ (A; γ) ⇒ x ∈ F∈ (A; γ) ∗ 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 4 2 0 1 0 3 4 3 0 0 0 0 3 4 0 0 0 0 0 for all α, β ∈ [0, 1), γ ∈ (0, 1] and x, y, z ∈ X with x ∗ y ≤ z. Proof. For any α, β ∈ [0, 1), γ ∈ (0, 1] and x, y, z ∈ X with x ∗ y ≤ z, let y, z ∈ T∈ (A; α), y, z ∈ I∈ (A; β) and y, z ∈ F∈ (A; γ). Then (x ∗ y) ∗ z = 0 ∈ T∈ (A; α) ∩ I∈ (A; β) ∩ F∈ (A; γ) Y.B. Jun, F. Smarandache, M.A. Ozturk, Commutative falling neutrosophic ideals in BCK-algebras. 49 Neutrosophic Sets and Systems, Vol. 20, 2018 by (2.9). It follows from (2.10) that Table 5: Cayley table for the binary operation “∗” x ∗ y ∈ T∈ (A; α) ∩ I∈ (A; β) ∩ F∈ (A; γ) ∗ 0 1 2 3 4 and so that x ∈ T∈ (A; α) ∩ I∈ (A; β) ∩ F∈ (A; γ). Thus (3.5) is valid. 0 0 1 2 3 4 1 0 0 1 3 4 2 0 0 0 3 4 3 0 1 2 0 4 4 0 1 2 3 0 Theorem 3.11. In a commutative BCK-algebra, every (∈, ∈)neutrosophic ideal is a commutative (∈, ∈)-neutrosophic ideal. Proof. Let A = (AT , AI , AF ) be an (∈, ∈)-neutrosophic ideal of a commutative BCK-algebra X. Let x, y, z ∈ X be such that trosophic random set on X which is given as follows:  {0, 3} if t ∈ [0, 0.25),  (x ∗ y) ∗ z ∈ T∈ (A; αx ) ∩ I∈ (A; βx ) ∩ F∈ (A; γx )   {0, 4} if t ∈ [0.25, 0.55), ξT : [0, 1] → P(X), x 7→ and {0, 1, 2} if t ∈ [0.55, 0.85),    {0, 3, 4} if t ∈ [0.85, 1], z ∈ T∈ (A; αy ) ∩ I∈ (A; βy ) ∩ F∈ (A; γy ) for αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1). Note that ((x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z)) ∗ z = ((x ∗ (y ∗ (y ∗ x))) ∗ z) ∗ ((x ∗ y) ∗ z) ≤ (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ y) = (x ∗ (x ∗ y)) ∗ (y ∗ (y ∗ x)) =0 by (2.3), (2.4) and (III), which implies that (x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z) ≤ z.   {0, 1, 2} {0, 1, 2, 3} ξI : [0, 1] → P(X), x 7→  {0, 1, 2, 4} if t ∈ [0, 0.45), if t ∈ [0.45, 0.75), if t ∈ [0.75, 1], and  {0}      {0, 3} {0, 4} ξF : [0, 1] → P(X), x 7→   {0, 1, 2, 3}    X if t ∈ (0.9, 1], if t ∈ (0.7, 0.9], if t ∈ (0.5, 0.7], if t ∈ (0.3, 0.5], if t ∈ [0, 0.3]. Then ξT (t), ξI (t) and ξF (t) are commutative ideals of X for all t ∈ [0, 1]. Hence the neutrosophic falling shadow H̃ := (H̃T , H̃I , H̃F ) of ξ := (ξT , ξI , ξF ) is a commutative falling neux ∗ (y ∗ (y ∗ x)) ∈ T∈ (A; αx ) ∩ I∈ (A; βx ) ∩ F∈ (A; γx ). trosophic ideal of X, and it is given as follows:  Therefore A = (AT , AI , AF ) is a commutative (∈, ∈)1 if x = 0,    neutrosophic ideal of X. 0.3 if x ∈ {1, 2}, H̃T (x) = 0.4 if x = 3,    0.45 if x = 4, It follows from Lemma 3.10 that 4 Commutative falling neutrosophic ideals Definition 4.1. Let (Ω, A, P ) be a probability space and let ξ := (ξT , ξI , ξF ) be a neutrosophic random set on a BCK-algebra X. Then the neutrosophic falling shadow H̃ := (H̃T , H̃I , H̃F ) and of ξ := (ξT , ξI , ξF ) is called a commutative falling neutrosophic ideal of X if ξT (ωT ), ξI (ωI ) and ξF (ωF ) are commutative ideals of X for all ωT , ωI , ωF ∈ Ω.   1 0.3 H̃I (x) =  0.25 if x ∈ {0, 1, 2}, if x = 3, if x = 4,   0 0.5 H̃F (x) =  0.3 if x = 0, if x ∈ {1, 2, 4}, if x = 3. Example 4.2. Consider a set X = {0, 1, 2, 3, 4} with the binary operation ∗ which is given in Table 5 Given a probability space (Ω, A, P ), let H̃ := (H̃T , H̃I , H̃F ) Then (X; ∗, 0) is a BCK-algebra (see [9]). Consider (Ω, A, P ) = ([0, 1], A, m) and let ξ := (ξT , ξI , ξF ) be a neu- be a neutrosophic falling shadow of a neutrosophic random set Y.B. Jun, F. Smarandache, M.A. Öztürk, Commutative falling neutrosophic ideals in BCK-algebras. Neutrosophic Sets and Systems, Vol. 20, 2018 50 ξ := (ξT , ξI , ξF ). For x ∈ X, let for all x, y, z ∈ X. Then Ω(x; ξT ) := {ωT ∈ Ω | x ∈ ξT (ωT )}, Ω(x; ξI ) := {ωI ∈ Ω | x ∈ ξI (ωI )}, Ω(x; ξF ) := {ωF ∈ Ω | x ∈ ξF (ωF )}. x ∗ (y ∗ (y ∗ x)) ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ). Note that ((x ∗ y) ∗ z) ∗ (x ∗ (y ∗ (y ∗ x))) = ((x ∗ y) ∗ (x ∗ (y ∗ (y ∗ x)))) ∗ z ≤ ((y ∗ (y ∗ x)) ∗ y) ∗ z = ((y ∗ y) ∗ (y ∗ x)) ∗ z = (0 ∗ (y ∗ x)) ∗ z = 0 ∗ z = 0, Then Ω(x; ξT ), Ω(x; ξI ), Ω(x; ξF ) ∈ A (see [8]). Proposition 4.3. Let H̃ := (H̃T , H̃I , H̃F ) be a neutrosophic falling shadow of the neutrosophic random set ξ := (ξT , ξI , ξF ) on a BCK-algebra X. If H̃ := (H̃T , H̃I , H̃F ) is a commutative which yields falling neutrosophic ideal of X, then Ω((x ∗ y) ∗ z; ξT ) ∩ Ω(z; ξT ) ⊆ Ω(x ∗ (y ∗ (y ∗ x)); ξT ) Ω((x ∗ y) ∗ z; ξI ) ∩ Ω(z; ξI ) ⊆ Ω(x ∗ (y ∗ (y ∗ x)); ξI ) Ω((x ∗ y) ∗ z; ξF ) ∩ Ω(z; ξF ) ⊆ Ω(x ∗ (y ∗ (y ∗ x)); ξF ) ((x ∗ y) ∗ z) ∗ (x ∗ (y ∗ (y ∗ x))) = 0 ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ). (4.1) Since ξT (ωT ), ξI (ωI ) and ξF (ωF ) are commutative ideals and hence ideals of X, it follows that (x ∗ y) ∗ z ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ). and Hence Ω(x ∗ (y ∗ (y ∗ x)); ξT ) ⊆ Ω((x ∗ y) ∗ z; ξT ) Ω(x ∗ (y ∗ (y ∗ x)); ξI ) ⊆ Ω((x ∗ y) ∗ z; ξI ) Ω(x ∗ (y ∗ (y ∗ x)); ξF ) ⊆ Ω((x ∗ y) ∗ z; ξF ) ωT ∈ Ω((x ∗ y) ∗ z; ξT ), ωI ∈ Ω((x ∗ y) ∗ z; ξI ), ωF ∈ Ω((x ∗ y) ∗ z; ξF ). (4.2) for all x, y, z ∈ X. Therefore (4.2) is valid. Given a probability space (Ω, A, P ), let Proof. Let ωT ∈ Ω((x ∗ y) ∗ z; ξT ) ∩ Ω(z; ξT ), ωI ∈ Ω((x ∗ y) ∗ z; ξI ) ∩ Ω(z; ξI ), ωF ∈ Ω((x ∗ y) ∗ z; ξF ) ∩ Ω(z; ξF ) F(X) := {f | f : Ω → X is a mapping}. Define a binary operation ⊛ on F(X) as follows: (∀ω ∈ Ω) ((f ⊛ g)(ω) = f (ω) ∗ g(ω)) for all x, y, z ∈ X. Then (x ∗ y) ∗ z ∈ ξT (ωT ) and z ∈ ξT (ωT ), (x ∗ y) ∗ z ∈ ξI (ωI ) and z ∈ ξI (ωI ), (x ∗ y) ∗ z ∈ ξF (ωF ) and z ∈ ξF (ωF ). Since ξT (ωT ), ξI (ωI ) and ξF (ωF ) are commutative ideals of X, it follows from (2.7) that θ : Ω → X, ω 7→ 0. For any subset A of X and gT , gI , gF ∈ F(X), consider the followings: AgT := {ωT ∈ Ω | gT (ωT ) ∈ A}, AgI := {ωI ∈ Ω | gI (ωI ) ∈ A}, AgF := {ωF ∈ Ω | gF (ωF ) ∈ A} and so that Hence (4.1) is valid. Now let ωT ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξT ), ωI ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξI ), ωF ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξF ) (4.4) for all f, g ∈ F(X). Then (F(X); ⊛, θ) is a BCK/BCIalgebra (see [7]) where θ is given as follows: x ∗ (y ∗ (y ∗ x)) ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ) ωT ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξT ), ωI ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξI ), ωF ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξF ). (4.3) and ξT : Ω → P(F(X)), ωT 7→ {gT ∈ F(X) | gT (ωT ) ∈ A}, ξI : Ω → P(F(X)), ωI 7→ {gI ∈ F(X) | gI (ωI ) ∈ A}, ξF : Ω → P(F(X)), ωF 7→ {gF ∈ F(X) | gF (ωF ) ∈ A}. Then AgT , AgI , AgF ∈ A (see [8]). Y.B. Jun, F. Smarandache, M.A. Ozturk, Commutative falling neutrosophic ideals in BCK-algebras. 51 Neutrosophic Sets and Systems, Vol. 20, 2018 Theorem 4.4. If K is a commutative ideal of a BCK-algebra of X. X, then The converse of Theorem 4.5 is not true as seen in the following example. ξT (ωT ) = {gT ∈ F(X) | gT (ωT ) ∈ K}, ξI (ωI ) = {gI ∈ F(X) | gI (ωI ) ∈ K}, Example 4.6. Consider a set X = {0, 1, 2, 3, 4} with the binary ξF (ωF ) = {gF ∈ F(X) | gF (ωF ) ∈ K} operation ∗ which is given in Table 6 are commutative ideals of F(X). Proof. Assume that K is a commutative ideal of a BCK-algebra X. Since θ(ωT ) = 0 ∈ K, θ(ωI ) = 0 ∈ K and θ(ωF ) = 0 ∈ K for all ωT , ωI , ωF ∈ Ω, we have θ ∈ ξT (ωT ), θ ∈ ξI (ωI ) and θ ∈ ξF (ωF ). Let fT , gT , hT ∈ F(X) be such that (fT ⊛ gT ) ⊛ hT ∈ ξT (ωT ) and hT ∈ ξT (ωT ). Then Table 6: Cayley table for the binary operation “∗” ∗ 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 2 4 2 0 1 0 1 4 3 0 0 0 0 4 (fT (ωT ) ∗ gT (ωT )) ∗ hT (ωT ) = ((fT ⊛ gT ) ⊛ hT )(ωT ) ∈ K 4 0 1 2 3 0 Then (X; ∗, 0) is a BCK-algebra (see [9]). Consider (Ω, A, P ) = ([0, 1], A, m) and let ξ := (ξT , ξI , ξF ) be a neutrosophic random set on X which is given as follows:  {0, 1} if t ∈ [0, 0.2),  (fT ⊛ (gT ⊛ (gT ⊛ fT )))(ωT )   {0, 2} if t ∈ [0.2, 0.55), = fT (ωT ) ∗ (gT (ωT ) ∗ (gT (ωT ) ∗ fT (ωT ))) ∈ K, ξT : [0, 1] → P(X), x 7→ {0, 2, 4} if t ∈ [0.55, 0.75),    {0, 1, 2, 3} if t ∈ [0.75, 1], that is, fT ⊛ (gT ⊛ (gT ⊛ fT )) ∈ ξT (ωT ). Hence ξT (ωT ) is a commutative ideal of F(X). Similarly, we can verify that ξI (ωI )  is a commutative ideal of F(X). Now, let fF , gF , hF ∈ F(X) {0, 1} if t ∈ [0, 0.34),   be such that (fF ⊛ gF ) ⊛ hF ∈ ξF (ωF ) and hF ∈ ξF (ωF ). Then  {0, 4} if t ∈ [0.34, 0.66), ξI : [0, 1] → P(X), x 7→ {0, 1, 4} if t ∈ [0.66, 0.78),   (fF (ωF ) ∗ gF (ωF )) ∗ hF (ωF )  X if t ∈ [0.78, 1], = ((fF ⊛ gF ) ⊛ hF )(ωF ) ∈ K and and hF (ωF ) ∈ K. Then  {0} if t ∈ (0.87, 1],     (fF ⊛ (gF ⊛ (gF ⊛ fF )))(ωF ) {0, 2} if t ∈ (0.76, 0.87],  {0, 4} if t ∈ (0.58, 0.76], ξ : [0, 1] → P(X), x → 7 = fF (ωF ) ∗ (gF (ωF ) ∗ (gF (ωF ) ∗ fF (ωF ))) ∈ K, F   {0, 2, 4} if t ∈ (0.33, 0.58],    and so fF ⊛ (gF ⊛ (gF ⊛ fF )) ∈ ξF (ωF ). Hence ξF (ωF ) is a X if t ∈ [0, 0.33]. commutative ideal of F(X). This completes the proof. Then ξT (t), ξI (t) and ξF (t) are commutative ideals of X for Theorem 4.5. If we consider a probability space (Ω, A, P ) = all t ∈ [0, 1]. Hence the neutrosophic falling shadow H̃ := ([0, 1], A, m), then every commutative (∈, ∈)-neutrosophic ideal (H̃ , H̃ , H̃ ) of ξ := (ξ , ξ , ξ ) is a commutative falling neuT I F T I F of a BCK-algebra is a commutative falling neutrosophic ideal. trosophic ideal of X, and it is given as follows:  Proof. Let H̃ := (H̃T , H̃I , H̃F ) be a commutative (∈, ∈ 1 if x = 0,   )-neutrosophic ideal of X. Then T∈ (H̃; α), I∈ (H̃; β) and   0.45 if x = 1,  F∈ (H̃; γ) are commutative ideals of X for all α, β ∈ (0, 1] and 0.8 if x = 2, H̃T (x) = γ ∈ [0, 1). Hence a triple ξ := (ξT , ξI , ξF ) in which   0.25 if x = 3,    0.2 if x = 4, ξT : [0, 1] → P(X), α 7→ T∈ (H̃; α), and hT (ωT ) ∈ K. Since K is a commutative ideal of X, it follows from (2.7) that ξI : [0, 1] → P(X), β 7→ I∈ (H̃; β), ξF : [0, 1] → P(X), γ 7→ F∈ (H̃; γ) is a neutrosophic cut-cloud of H̃ := (H̃T , H̃I , H̃F ). Therefore H̃ := (H̃T , H̃I , H̃F ) is a commutative falling neutrosophic ideal  1    0.68 H̃I (x) =  0.22   0.66 Y.B. Jun, F. Smarandache, M.A. Öztürk, Commutative falling neutrosophic ideals in BCK-algebras. if x = 0, if x = 1, if x ∈ {2, 3}, if x = 4, Neutrosophic Sets and Systems, Vol. 20, 2018 52 and and  0    0.67 H̃F (x) = 0.31    0.24  {0}    {0, 3} ξF : [0, 1] → P(X), x 7→ {0, 1, 2, 4}    X if x = 0, if x ∈ {1, 3}, if x = 2, if x = 4. if t ∈ (0.84, 1], if t ∈ (0.76, 0.84], if t ∈ (0.58, 0.76], if t ∈ [0, 0.58]. But H̃ := (H̃T , H̃I , H̃F ) is not a commutative (∈, ∈)- Then ξT (t), ξI (t) and ξF (t) are ideals of X for all t ∈ [0, 1]. Hence the neutrosophic falling shadow H̃ := (H̃T , H̃I , H̃F ) of neutrosophic ideal of X since ξ := (ξT , ξI , ξF ) is a falling neutrosophic ideal of X. But it (3 ∗ 4) ∗ 2 ∈ T∈ (H̃; 0.4) and 2 ∈ T∈ (H̃; 0.6), is not a commutative falling neutrosophic ideal of X because if α ∈ [0, 0.27), β ∈ [0, 0.35) and γ ∈ (0.76, 0.84], then ξT (α) = but 3 ∗ (4 ∗ (4 ∗ 3)) = 3 ∈ / T∈ (H̃; 0.4). {0, 3}, ξI (β) = {0, 3} and ξF (γ) = {0, 3} are not commutative ideals of X respectively. We provide relations between a falling neutrosophic ideal and Since every ideal is commutative in a commutative BCKa commutative falling neutrosophic ideal . algebra, we have the following theorem. Theorem 4.7. Let (Ω, A, P ) be a probability space and let H̃ := (H̃T , H̃I , H̃F ) be a neutrosophic falling shadow of a neu- Theorem 4.9. Let (Ω, A, P ) be a probability space and let trosophic random set ξ := (ξT , ξI , ξF ) on a BCK-algebra. If H̃ := (H̃T , H̃I , H̃F ) be a neutrosophic falling shadow of a neuH̃ := (H̃T , H̃I , H̃F ) is a commutative falling neutrosophic ideal trosophic random set ξ := (ξT , ξI , ξF ) on a commutative BCKalgebra. If H̃ := (H̃T , H̃I , H̃F ) is a falling neutrosophic ideal of X, then it is a falling neutrosophic ideal of X. of X, then it is a commutative falling neutrosophic ideal of X. Proof. Let H̃ := (H̃T , H̃I , H̃F ) be a commutative falling neutrosophic ideal of a BCK-algebra X. Then ξT (ωT ), ξI (ωI ) and Corollary 4.10. Let (Ω, A, P ) be a probability space. For any ξF (ωF ) are commutative ideals of X for all ωT , ωI , ωF ∈ Ω. BCK-algebra X which satisfies one of the following assertions Thus ξT (ωT ), ξI (ωI ) and ξF (ωF ) are ideals of X for all ωT , ωI , (∀x, y ∈ X)(x ≤ y ⇒ x ≤ y ∗ (y ∗ x)), (4.5) ωF ∈ Ω. Therefore H̃ := (H̃T , H̃I , H̃F ) is a falling neutro(∀x, y ∈ X)(x ≤ y ⇒ x = y ∗ (y ∗ x)), (4.6) sophic ideal of X. (∀x, y ∈ X)(x ∗ (x ∗ y) = y ∗ (y ∗ (x ∗ (x ∗ y)))), (4.7) The following example shows that the converse of Theorem (∀x, y, z ∈ X)(x, y ≤ z, z ∗ y ≤ z ∗ x ⇒ x ≤ y), (4.8) 4.7 is not true in general. (∀x, y, z ∈ X)(x ≤ z, z ∗ y ≤ z ∗ x ⇒ x ≤ y), (4.9) Example 4.8. Consider a set X = {0, 1, 2, 3, 4} with the binary operation ∗ which is given in Table 7 let H̃ := (H̃ , H̃ , H̃ ) be a neutrosophic falling shadow of T Table 7: Cayley table for the binary operation “∗” ∗ 0 1 2 3 4 0 0 1 2 3 4 1 0 0 1 3 4 2 0 0 0 3 4 3 0 1 2 0 4 4 0 0 0 3 0 Then (X; ∗, 0) is a BCK-algebra (see [9]). Consider (Ω, A, P ) = ([0, 1], A, m) and let ξ := (ξT , ξI , ξF ) be a neutrosophic random set on X which is given as follows:  if t ∈ [0, 0.27),  {0, 3} {0, 1, 2, 3} if t ∈ [0.27, 0.66), ξT : [0, 1] → P(X), x 7→  {0, 1, 2, 4} if t ∈ [0.67, 1], ξI : [0, 1] → P(X), x 7→  {0, 3} {0, 1, 2, 4} if t ∈ [0, 0.35), if t ∈ [0.35, 1], I F a neutrosophic random set ξ := (ξT , ξI , ξF ) on X. If H̃ := (H̃T , H̃I , H̃F ) is a falling neutrosophic ideal of X, then it is a commutative falling neutrosophic ideal of X. References [1] A. Borumand Saeid and Y.B. Jun, Neutrosophic subalgebras of BCK/BCI-algebras based on neutrosophic points, Ann. Fuzzy Math. Inform. 14 (2017), no. 1, 87–97. [2] I.R. Goodman, Fuzzy sets as equivalence classes of random sets, in “Recent Developments in Fuzzy Sets and Possibility Theory”(R. Yager, Ed.), Pergamon, New York 1982, pp. 327–343. [3] K. Iséki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125–130. [4] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. [5] Y. Huang, BCI-algebra, Science Press, Beijing, 2006. Y.B. Jun, F. Smarandache, M.A. Ozturk, Commutative falling neutrosophic ideals in BCK-algebras. 53 Neutrosophic Sets and Systems, Vol. 20, 2018 [6] Y.B. Jun, Neutrosophic subalgebras of several types in BCK/BCI-algebras, Ann. Fuzzy Math. Inform. 14 (2017), no. 1, 75–86. [7] Y.B. Jun and C.H. Park, Falling shadows applied to subalgebras and ideals of BCK/BCI-algebras, Honam Math. J. 34 (2012), no. 2, 135–144. [8] Y.B. Jun, F. Smarandache and H. Bordbar, Neutrosophic falling shadows applied to subalgebras and ideals in BCK/BCI-algebras, Ann. 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[24] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116. Received : April 5, 2018. Accepted : April 20, 2018. [14] S.K. Tan, P.Z. Wang and E.S. Lee, Fuzzy set operations based on the theory of falling shadows, J. Math. Anal. Appl. 174 (1993), 242–255. [15] S.K. Tan, P.Z. Wang and X.Z. Zhang, Fuzzy inference relation based on the theory of falling shadows, Fuzzy Sets and Systems 53 (1993), 179–188. [16] P.Z. Wang and E. Sanchez, Treating a fuzzy subset as a projectable random set, in: “Fuzzy Information and Decision” (M. M. Gupta, E. Sanchez, Eds.), Pergamon, New York, 1982, pp. 212–219. [17] P.Z. Wang, Fuzzy Sets and Falling Shadows of Random Sets, Beijing Normal Univ. Press, People’s Republic of China, 1985. [In Chinese] Y.B. Jun, F. Smarandache, M.A. Öztürk, Commutative falling neutrosophic ideals in BCK-algebras. 54 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico On Neutrosophic Soft Prime Ideal Tuhin Bera1 and Nirmal Kumar Mahapatra2 1 2 Department of Mathematics, Boror S. S. High School, Bagnan, Howrah-711312, WB, India, E-mail : tuhin78bera@gmail.com Department of Mathematics, Panskura Banamali College, Panskura RS-721152, WB, India, E-mail : nirmal hridoy@yahoo.co.in Abstract The motivation of the present paper is to extend the concept of neutrosophic soft prime ideal over a ring. In this paper the concept of neutrosophic soft completely prime ideals, neutrosophic soft completely semi-prime ideals and neutrosophic soft prime k - ideals have been introduced. These are illustrated with suitable examples also. Several related properties, theorems and structural characteristics of each are studied here. Keywords Neutrosophic soft completely prime ideals; Neutrosophic soft completely semi-prime ideals; Neutrosophic soft prime k - ideals. 1 Introduction Because of the insufficiency in the available information situation, evaluation of membership values and nonmembership values are not always possible to handle the uncertainties appearing in daily life situations. So there exists an indeterministic part upon which hesitation survives. The neutrosophic set theory by Smarandache [1,2] which is a generalisation of fuzzy set and intuitionistic fuzzy set theory, makes description of the objective world more realistic, practical and very promising in nature. The neutrosophic logic includes the information about the percentage of truth, indeterminacy and falsity grade in several real world problems in law, medicine, engineering, management, industrial, IT sector etc which are not available in intuitionistic fuzzy set theory. But each of the theories suffers from inherent difficulties because of the inadequacy of parametrization tools. Molodtsov [3] introduced a nice concept of soft set theory which is free from the parametrization inadequacy syndrome of different theories dealing with uncertainty. The parametrization tool of soft set theory makes it very convenient and easy to apply in practice. The classical algebraic structures were extended over fuzzy set, intuitionistic fuzzy set, soft set, fuzzy soft set and intuitionistic fuzzy soft set by so many authors, for instance, Rosenfeld [4], Malik and Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 Mordeson [5,6], Lavanya and Kumar [8], Bakhadach et al. [9], Dutta et al. [10-12], Maji et al. [13], Aktas and Cagman [14], Augunoglu and Aygun [15], Zhang [16], Maheswari and Meera [17] and others. The notion of neutrosophic soft set theory (NSS) has been innovated by Maji [18]. Later, it has been modified by Deli and Broumi [19]. Cetkin et al. [20,21], Bera and Mahapatra [22-26] and others have produced their research works on fundamental algebraic structures on the NSS theory context. This paper presents the notion of neutrosophic soft completely prime ideals, neutrosophic soft completely semi-prime ideals and neutrosophic soft prime k-ideals along with investigation of some related properties and theorems. The content of the present paper is designed as following : Section 2 gives some preliminary useful definitions related to it. In Section 3, neutrosophic soft completely prime ideals is defined and illustrated by suitable examples along with investigation of its structural characteristics. Section 4 deals with the notion of neutrosophic soft completely semi-prime ideals with development of related theorems. The concept of neutrosophic soft prime k-ideals along with some properties has been introduced in Section 6. Finally, the conclusion of our work has been stated in Section 7. 2 Preliminaries We recall some basic definitions related to fuzzy set, soft set, neutrosophic soft set for the sake of completeness. 2.1 Definition [24] 1. A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is said to be continuous t - norm if ∗ satisfies the following conditions : (i) ∗ is commutative and associative. (ii) ∗ is continuous. (iii) a ∗ 1 = 1 ∗ a = a, ∀a ∈ [0, 1]. (iv) a ∗ b ≤ c ∗ d if a ≤ c, b ≤ d with a, b, c, d ∈ [0, 1]. A few examples of continuous t-norm are a ∗ b = ab, a ∗ b = min{a, b}, a ∗ b = max{a + b − 1, 0}. 2. A binary operation ⋄ : [0, 1] × [0, 1] → [0, 1] is said to be continuous t - conorm (s - norm) if ⋄ satisfies the following conditions : (i) ⋄ is commutative and associative. (ii) ⋄ is continuous. (iii) a ⋄ 0 = 0 ⋄ a = a, ∀a ∈ [0, 1]. (iv) a ⋄ b ≤ c ⋄ d if a ≤ c, b ≤ d with a, b, c, d ∈ [0, 1]. A few examples of continuous s-norm are a ⋄ b = a + b − ab, a ⋄ b = max{a, b}, a ⋄ b = min{a + b, 1}. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 55 Neutrosophic Sets and Systems, Vol. 20, 2018 56 2.2 Definition [1] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA , an indeterminacy-membership function IA and a falsity-membership function FA . TA (x), IA (x) and FA (x) are real standard or non-standard subsets of ]− 0, 1+ [. That is TA , IA , FA : X →]− 0, 1+ [. There is no restriction on the sum of TA (x), IA (x), FA (x) and so, − 0 ≤ sup TA (x) + sup IA (x) + sup FA (x) ≤ 3+ . 2.3 Definition [3] Let U be an initial universe set and E be a set of parameters. Let P (U ) denote the power set of U . Then for A ⊆ E, a pair (F, A) is called a soft set over U , where F : A → P (U ) is a mapping. 2.4 Definition [18] Let U be an initial universe set and E be a set of parameters. Let N S(U ) denote the set of all NSs of U . Then for A ⊆ E, a pair (F, A) is called an NSS over U , where F : A → N S(U ) is a mapping. This concept has been redefined by Deli and Broumi [19] as given below. 2.5 Definition [19] 1. Let U be an initial universe set and E be a set of parameters. Let N S(U ) denote the set of all NSs of U . Then, a neutrosophic soft set N over U is a set defined by a set valued function fN representing a mapping fN : E → N S(U ) where fN is called approximate function of the neutrosophic soft set N . In other words, the neutrosophic soft set is a parameterized family of some elements of the set N S(U ) and therefore it can be written as a set of ordered pairs, N = {(e, fN (e)) : e ∈ E} = {(e, {< x, TfN (e) (x), IfN (e) (x), FfN (e) (x) >: x ∈ U }) : e ∈ E} where TfN (e) (x), IfN (e) (x), FfN (e) (x) ∈ [0, 1], respectively called the truth-membership, indeterminacy-membership, falsity-membership function of fN (e). Since supremum of each T, I, F is 1 so the inequality 0 ≤ TfN (e) (x) + IfN (e) (x) + FfN (e) (x) ≤ 3 is obvious. 2. Let N1 and N2 be two NSSs over the common universe (U, E). Then N1 is said to be the neutrosophic soft subset of N2 if TfN1 (e) (x) ≤ TfN2 (e) (x), IfN1 (e) (x) ≥ IfN2 (e) (x), FfN1 (e) (x) ≥ FfN2 (e) (x), ∀e ∈ E and ∀x ∈ U . We write N1 ⊆ N2 and then N2 is the neutrosophic soft superset of N1 . Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 2.6 Proposition [22] An NSS N over the group (G, o) is called a neutrosophic soft group iff followings hold on the assumption that a ∗ b = min{a, b} and a ⋄ b = max{a, b}. TfN (e) (xoy −1 ) ≥ TfN (e) (x) ∗ TfN (e) (y), IfN (e) (xoy −1 ) ≤ IfN (e) (x) ⋄ IfN (e) (y), FfN (e) (xoy −1 ) ≤ FfN (e) (x) ⋄ FfN (e) (y)); ∀x, y ∈ G, ∀e ∈ E. 2.7 Definition [24] 1. A neutrosophic soft ring N over the ring (R, +, ·) is called a neutrosophic soft left ideal over R if fN (e) is a neutrosophic left ideal of R for each e ∈ E i.e., (i) fN (e) is a neutrosophic subgroup of (R, +) for each e ∈ E and   TfN (e) (x · y) ≥ TfN (e) (y) (ii) IfN (e) (x · y) ≤ IfN (e) (y)  FfN (e) (x · y) ≤ FfN (e) (y); for x, y ∈ R. 2. A neutrosophic soft ring N over the ring (R, +, ·) is called a neutrosophic soft right ideal over R if fN (e) is a neutrosophic right ideal of R for each e ∈ E i.e., (i) fN (e) is a neutrosophic subgroup of (R, +) for each e ∈ E and   TfN (e) (x · y) ≥ TfN (e) (x) (ii) IfN (e) (x · y) ≤ IfN (e) (x)  FfN (e) (x · y) ≤ FfN (e) (x); for x, y ∈ R. 3. A neutrosophic soft ring N over the ring (R, +, ·) is called a neutrosophic soft ideal over R if fN (e) is a both neutrosophic left and right ideal of R for each e ∈ E. 2.8 Definition [25] 1. Let ϕ : U → V and ψ : E → E be two functions where E is the parameter set for each of the crisp sets U and V . Then the pair (ϕ, ψ) is called an NSS function from (U, E) to (V, E). We write, (ϕ, ψ) : (U, E) → (V, E). If M is an NSS over U via parametric set E, we shall write (M, E) an NSS over U . 2. Let (M, E), (N, E) be two NSSs defined over U, V respectively and (ϕ, ψ) be an NSS function from (U, E) to (V, E). Then, (i) The image of (M, E) under (ϕ, ψ), denoted by (ϕ, ψ)(M, E), is an NSS over V and is defined by : (ϕ, ψ)(M, E) = (ϕ(M ), ψ(E)) = {< ψ(a), fϕ(M ) >: a ∈ E} where ∀b ∈ ψ(E), ∀y ∈ V , Tfϕ(M ) (b) (y) = Ifϕ(M ) (b) (y) = Ffϕ(M ) (b) (y) =    maxϕ(x)=y maxψ(a)=b [TfM (a) (x)], if x ∈ ϕ−1 (y) 0 , otherwise. minϕ(x)=y minψ(a)=b [IfM (a) (x)], if x ∈ ϕ−1 (y) 1 , otherwise. minϕ(x)=y minψ(a)=b [FfM (a) (x)], if x ∈ ϕ−1 (y) 1 , otherwise. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 57 Neutrosophic Sets and Systems, Vol. 20, 2018 58 (ii) The pre-image of (N, E) under (ϕ, ψ), denoted by (ϕ, ψ)−1 (N, E), is an NSS over U and is defined by : (ϕ, ψ)−1 (N, E) = (ϕ−1 (N ), ψ −1 (E)) where ∀a ∈ ψ −1 (E), ∀x ∈ U , Tfϕ−1 (N ) (a) (x) = TfN [ψ(a)] (ϕ(x)) Ifϕ−1 (N ) (a) (x) = IfN [ψ(a)] (ϕ(x)) Ffϕ−1 (N ) (a) (x) = FfN [ψ(a)] (ϕ(x)) If ψ and ϕ is injective (surjective), then (ϕ, ψ) is injective (surjective). 2.9 Definition [26] 1. An NSS M over (R, E) is said to be constant if each fM (e) is constant for e ∈ E i.e., (TfM (e) (x), IfM (e) (x), FfM (e) (x)) is same ∀e ∈ E, ∀x ∈ R. For M to be nonconstant, if for each e ∈ E the triplet (TfM (e) (x), IfM (e) (x), FfM (e) (x)) is atleast of two different kinds ∀x ∈ R. 2. Let R be a ring and M, N be two NSSs over (R, E). Then M oN = L (say) is also an NSS over (R, E) and is defined as following, for e ∈ E and x ∈ R,  maxx=yz [TfM (e) (y) ∗ TfN (e) (z)] TfL (e) (x) = 0 if x is not expressible as x = yz.  minx=yz [IfM (e) (y) ⋄ IfN (e) (z)] IfL (e) (x) = 1 if x is not expressible as x = yz.  minx=yz [FfM (e) (y) ⋄ FfN (e) (z)] FfL (e) (x) = 1 if x is not expressible as x = yz. 3. A neutrosophic soft ideal P over (R, E) is said to be a neutrosophic soft prime ideal if (i) P is not constant neutrosophic soft ideal, (ii) for any two neutrosophic soft ideals M, N over (R, E), M oN ⊆ P ⇒ either M ⊆ P or N ⊆ P . 2.10 Theorem [26] 1. Let P be an NSS over (R, E) such that cardinality of fP (e) is 2 i.e., |fP (e)| = 2 and [fP (e)](0r ) = (1, 0, 0) for each e ∈ E. If P0 = {x ∈ R : [fP (e)](x) = [fP (e)](0r )} is a prime ideal over R, then P is a neutrosophic soft prime ideal over (R, E). 2. Let P be an NSS over (R, E). Then P is a neutrosophic soft left (right) ideal over (R, E) iff Pb = {x ∈ R : [fP (e)](x) = (1, 0, 0)} with 0r ∈ Pb is a left (right) ideal of R. 3. S(6= φ) ⊂ R is an ideal of R iff there exists a neutrosophic soft ideal M over (R, E) where fM : E −→ N S(R) is defined as, ∀e ∈ E,  (r1 , r2 , r3 ) if x ∈ S [fM (e)](x) = (t1 , t2 , t3 ) if x ∈ / S. with r1 > t1 , r2 < t2 , r3 < t3 and r1 , r2 , r3 , t1 , t2 , t3 ∈ [0, 1]. In particular, S(6= φ) ⊂ R is an ideal of R iff the characteristic function χS is a Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 59 Neutrosophic Sets and Systems, Vol. 20, 2018 neutrosophic soft ideal over (R, E) where χS : E −→ N S(R) is defined as, ∀e ∈ E,  (1, 0, 0) if x ∈ S [χS (e)](x) = (0, 1, 1) if x ∈ / S. 4. An NSS M over (R, E) is a neutrosophic soft left (right) ideal iff each nonempty level set [fM (e)](α,β,γ) of the neutrosophic set fM (e) is a left (right) ideal of R where α ∈ Im TfM (e) , β ∈ Im IfM (e) , γ ∈ Im FfM (e) . 5. Let P be a neutrosophic soft left (right) ideal over (R, E). Then P0 = {x ∈ R : [fP (e)](x) = [fP (e)](0r )} is a left (right) ideal of R. 6. Let P be a neutrosophic soft prime ideal over (R, E). Then P0 = {x ∈ R : [fP (e)](x) = [fP (e)](0r )} is a prime ideal of R. 2.11 Definition [7] A left k-ideal I of a semiring S is a left ideal such that if a ∈ I and x ∈ S and if either a + x ∈ I or x + a ∈ I, then x ∈ I. Right k-ideal of a semiring is defined dually. A non-empty subset I of a semiring S is called a k-ideal if it is both a left k-ideal and a right k-ideal. 3 Neutrosophic soft completely prime ideal Here first we have defined a completely prime ideal of a ring and then defined a neutrosophic soft completely prime ideal. These are illustrated with suitable examples. Along with several related properties and theorems have been developed. Through out this paper, unless otherwise stated, E is treated as the parametric set and e ∈ E, an arbitrary parameter. Moreover the standard t-norm and s-norm are taken into consideration wherever needed through out this paper i.e., a∗b = min{a, b} and a ⋄ b = max{a, b}. 3.1 Definition An ideal S of a ring R is called a completely prime ideal of R if for x, y ∈ R, xy ∈ S ⇒ either x ∈ S or y ∈ S. 3.1.1 Example 1. For the ring (Z, +, ·) (Z being the set of integers), an ideal (2Z, +, ·) is a completely prime ideal. 2. We assume a ring R = {0, x, y, z}. The two binary operations addition and multiplication on R are given by the following tables : + 0 Table 1 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 · 0 Table 2 x y z Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 0 0 0 0 0 x 0 0 0 0 y 0 0 y y z 0 0 y y Neutrosophic Sets and Systems, Vol. 20, 2018 60 It is an abelian ring. With respect to these two tables, {0, x} and {0, y} are two ideals of R. From 2nd table, it is evident that {0, x} is a completely prime ideal of R but {0, y} is not so because z · z = y though z ∈ / {0, y}. 3. Consider the another ring R = {0, x, y, z} with two binary operations addition and multiplication on R are given by the following tables : Table 3 + 0 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 Table 4 · 0 x y z 0 0 0 0 0 x 0 0 0 x y 0 0 0 y z 0 0 0 x It is not an abelian ring. With respect to these two tables, {0, x} is an ideal of R but not completely prime ideal. Because y · z = 0, z · z = x, y · y = 0 but y, z ∈ / {0, x}. 3.2 Proposition If S is a completely prime ideal of a ring R then S is a prime ideal of R. Proof. Let S be a completely prime ideal of a ring R and A, B be two ideals of R such that AB ⊆ S. Suppose A 6⊆ S and B 6⊆ S. Then there exists x ∈ A and y ∈ B such that x, y ∈ / S. But xy ∈ S as AB ⊆ S. Since S is a completely prime ideal of R, so either x ∈ S or y ∈ S and this leads a contradiction to the fact x, y ∈ / S. Hence S is a prime ideal of R. 3.3 Definition A neutrosophic soft ideal N over (R, E) is called a neutrosophic soft completely prime ideal if ∀x, y ∈ R and ∀e ∈ E,   TfN (e) (x · y) ≤ max{TfN (e) (x), TfN (e) (y)} If (e) (x · y) ≥ min{IfN (e) (x), IfN (e) (y)}  N FfN (e) (x · y) ≥ min{FfN (e) (x), FfN (e) (y)}. 3.3.1 Example Consider the Example [3.1.1](2). We define an NSS M over (R, E) as following, ∀r ∈ R and ∀e ∈ E,  (1, 0.3, 0.1) if r ∈ {0, x} [fM (e)](r) = (0.8, 0.6, 0.4) if r ∈ / {0, x}. Then M is a neutrosophic soft completely prime ideal over (R, E). 3.4 Theorem An NSS N is a neutrosophic soft completely prime ideal over (R, E) iff for e ∈ b = {x ∈ R : [fN (e)](x) = (1, 0, 0)} is a E, |fN (e)| = 2, [fN (e)](0r ) = (1, 0, 0) and N Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 completely prime ideal of R. Proof. Let N be a neutrosophic soft completely prime ideal over (R, E). Then N b is an ideal over R by Theorem is a neutrosophic soft ideal over (R, E) and so N b is a complete prime ideal, let xy ∈ N b for x, y ∈ R. Then [2.11](2). To prove N [fN (e)](xy) = (1, 0, 0) for e ∈ E. But, 1 = TfN (e) (xy) ≤ max{TfN (e) (x), TfN (e) (y)}, 0 = IfN (e) (xy) ≥ min{IfN (e) (x), IfN (e) (y)}, 0 = FfN (e) (xy) ≥ min{FfN (e) (x), FfN (e) (y)}; This implies that TfN (e) (0r ) = 1 ≤ max{TfN (e) (x), TfN (e) (y)}, IfN (e) (0r ) = 0 ≥ min{IfN (e) (x), IfN (e) (y)}, FfN (e) (0r ) = 0 ≥ min{FfN (e) (x), FfN (e) (y)}; This shows that, either TfN (e) (0r ) ≤ TfN (e) (x) or TfN (e) (0r ) ≤ TfN (e) (y), either IfN (e) (0r ) ≥ IfN (e) (x) or IfN (e) (0r ) ≥ IfN (e) (y), either FfN (e) (0r ) ≥ FfN (e) (x) or FfN (e) (0r ) ≥ FfN (e) (y); But TfN (e) (0r ) ≥ TfN (e) (x), IfN (e) (0r ) ≤ IfN (e) (x), FfN (e) (0r ) ≤ FfN (e) (x), ∀x ∈ R. Hence TfN (e) (x) = TfN (e) (0r ), IfN (e) (x) = IfN (e) (0r ), FfN (e) (x) = FfN (e) (0r ), ∀x ∈ R b . Thus N b is a complete prime ideal. i.e., x, y ∈ N b is a completely prime ideal with the given conditions. As N b Conversely suppose N is an ideal of R, so N is a neutrosophic soft ideal over (R, E) by Theorem [2.11](2). For contrary, suppose N is not neutrosophic soft completely prime ideal. Then, TfN (e) (xy) > max{TfN (e) (x), TfN (e) (y)}, IfN (e) (xy) < min{IfN (e) (x), IfN (e) (y)}, FfN (e) (xy) < min{FfN (e) (x), FfN (e) (y)}; Since |fN (e)| = 2 and [fN (e)](0r ) = (1, 0, 0) then there exists x, y ∈ R so that [fN (e)](x) = [fN (e)](y) = (r1 , r2 , r3 ) 6= (1, 0, 0) (say) for 0 ≤ r1 < 1 and 0 < r2 , r3 ≤ 1. Then, TfN (e) (xy) > r1 , IfN (e) (xy) < r2 , FfN (e) (xy) < r3 ⇒ TfN (e) (xy) = 1, IfN (e) (xy) = FfN (e) (xy) = 0 ⇒ [fN (e)](xy) = (1, 0, 0) b ⇒ xy ∈ N b is completely prime ideal, so either x ∈ N b or y ∈ N b i.e., [fN (e)](x) = Since N [fN (e)](y) = (1, 0, 0). A contradiction arises to the fact that [fN (e)](x) = [fN (e)](y) = (r1 , r2 , r3 ) 6= (1, 0, 0). Thus, TfN (e) (xy) ≤ max{TfN (e) (x), TfN (e) (y)}, IfN (e) (xy) ≥ min{IfN (e) (x), IfN (e) (y)}, FfN (e) (xy) ≥ min{FfN (e) (x), FfN (e) (y)}; and so N is a neutrosophic soft completely prime ideal over (R, E). Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 61 Neutrosophic Sets and Systems, Vol. 20, 2018 62 3.5 Theorem Let N be a neutrosophic soft completely prime ideal over (R, E) with |fN (e)| = 2, [fN (e)](0r ) = (1, 0, 0) for each e ∈ E. Then N is a neutrosophic soft prime ideal over (R, E). b = {x ∈ R : [fN (e)](x) = (1, 0, 0)} Proof. Let the condition hold. By Theorem [3.4], N b is a prime ideal of R. is a completely prime ideal of R. Then by Proposition [3.2], N Hence N is a neutrosophic soft prime ideal over (R, E) by Theorem [2.11](1). 3.6 Theorem Let R be a ring. Then S(6= φ) ⊂ R be a completely prime ideal of R iff an NSS N over (R, E) is a neutrosophic soft completely prime ideal where fN : E −→ N S(R) is defined as :  (r1 , r2 , r3 ) if x ∈ S [fN (e)](x) = (t1 , t2 , t3 ) if x ∈ / S. with r1 > t1 , r2 < t2 , r3 < t3 and r1 , r2 , r3 , t1 , t2 , t3 ∈ [0, 1]. Proof. First let S(6= φ) ⊂ R be a completely prime ideal of R. Then S is an ideal of R and so by Theorem [2.11](3), N is a neutrosophic soft ideal over (R, E). To end the theorem, we shall just show that N is completely prime. For contrary, suppose TfN (e) (xy) > max{TfN (e) (x), TfN (e) (y)}, IfN (e) (xy) < min{IfN (e) (x), IfN (e) (y)}, FfN (e) (xy) < min{FfN (e) (x), FfN (e) (y)}; Then by definition of fN (e), we have [fN (e)](xy) = (r1 , r2 , r3 ) and [fN (e)](x) = [fN (e)](y) = (t1 , t2 , t3 ). This implies xy ∈ S but x, y ∈ / S which is a contradiction to the fact that S is a completely prime ideal of R. Hence N is a neutrosophic soft completely prime ideal over (R, E). Conversely, let N in given form be a neutrosophic soft completely prime ideal over (R, E). Then N is a neutrosophic soft ideal over (R, E) and so by Theorem [2.11](3), S is an ideal of R. To show S is a completely prime ideal of R, let xy ∈ S. Then, ⇒ ⇒ ⇒ ⇒ [fN (e)](xy) = (r1 , r2 , r3 ) TfN (e) (xy) = r1 , IfN (e) (xy) = r2 , FfN (e) (xy) = r3 max{TfN (e) (x), TfN (e) (y)} ≥ r1 , min{IfN (e) (x), IfN (e) (y)} ≤ r2 , min{FfN (e) (x), FfN (e) (y)} ≤ r3 either TfN (e) (x) ≥ r1 , IfN (e) (x) ≤ r2 , FfN (e) (x) ≤ r3 or TfN (e) (y) ≥ r1 , IfN (e) (y) ≤ r2 , FfN (e) (y) ≤ r3 either x ∈ S or y ∈ S Thus S is a completely prime ideal of R. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 3.6.1 Corollary A non empty subset S of a ring R is a completely prime ideal iff the characteristic function χS is a neutrosophic soft completely prime ideal over (R, E) where χS : E −→ N S(R) is defined by :  (1, 0, 0) if x ∈ S [χS (e)](x) = (0, 1, 1) if x ∈ / S. Proof. It is the particular case of Theorem [3.6]. 3.7 Theorem An NSS M over (R, E) is a neutrosophic soft completely prime ideal means each nonempty level set [fM (e)](α,β,γ) of the neutrosophic set fM (e), e ∈ E is a completely prime ideal of R where α ∈ Im TfM (e) , β ∈ Im IfM (e) , γ ∈ Im FfM (e) . Proof. Here M is a neutrosophic soft completely prime ideal over (R, E). Then M is a neutrosophic soft ideal over (R, E) and so by Theorem [2.11](4), [fM (e)](α,β,γ) is an ideal of R. To complete the theorem, let xy ∈ [fM (e)](α,β,γ) . Then, TfM (e) (xy) ≥ α, IfM (e) (xy) ≤ β, FfM (e) (xy) ≤ γ ⇒ max{TfM (e) (x), TfM (e) (y)} ≥ α, min{IfM (e) (x), IfM (e) (y)} ≤ β, min{FfM (e) (x), FfM (e) (y)} ≤ γ ⇒ either TfM (e) (x) ≥ α, IfM (e) (x) ≤ β, FfM (e) (x) ≤ γ or TfM (e) (y) ≥ α, IfM (e) (y) ≤ β, FfM (e) (y) ≤ γ ⇒ either x ∈ [fM (e)](α,β,γ) or y ∈ [fM (e)](α,β,γ) Thus [fM (e)](α,β,γ) is a completely prime ideal of R. 3.8 Proposition Let S be a completely prime ideal of a ring R. Then there exists a neutrosophic soft completely prime ideal M over (R, E) such that [fM (e)](α,β,γ) = S for e ∈ E and α, β, γ ∈ (0, 1). Proof. As S is a completely prime ideal of a ring R, so S is an ideal of R. For α, β, γ ∈ (0, 1) define an NSS M over (R, E) as following :  (α, β, γ) if x ∈ S [fM (e)](x) = (0, 1, 1) if x ∈ / S. Then by Theorem [2.11](3), M is a neutrosophic soft ideal over (R, E). If possible let M is not a neutrosophic soft completely prime ideal over (R, E). Then, TfM (e) (xy) > max{TfM (e) (x), TfM (e) (y)}, IfM (e) (xy) < min{IfM (e) (x), IfM (e) (y)}, FfM (e) (xy) < min{FfM (e) (x), FfM (e) (y)}; Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 63 Neutrosophic Sets and Systems, Vol. 20, 2018 64 Then by definition of fM (e), we have [fM (e)](xy) = (α, β, γ) and [fM (e)](x) = [fM (e)](y) = (0, 1, 1). This implies xy ∈ S but x, y ∈ / S which is a contradiction to the fact that S is a completely prime ideal of R. Hence M is a neutrosophic soft completely prime ideal over (R, E). Obviously [fM (e)](α,β,γ) = S for each e ∈ E. 3.9 Theorem Let (ϕ, ψ) : (R1 , E) −→ (R2 , E) be a neutrosophic soft homomorphism where R1 , R2 be two rings. Suppose (M, E) and (N, E) be two neutrosophic soft left (right) ideals over R1 and R2 , respectively. Then, 1. (ϕ, ψ)(M, E) is a neutrosophic soft left (right) ideal over R2 if (ϕ, ψ) is epimorphism. 2. (ϕ, ψ)−1 (N, E) is a neutrosophic soft left (right) ideal over R1 . Proof. 1. Let b ∈ ψ(E) and y1 , y2 , s ∈ R2 . For ϕ−1 (y1 ) = φ or ϕ−1 (y2 ) = φ, the proof is straight forward. So, we assume that there exists x1 , x2 , r ∈ R1 such that ϕ(x1 ) = y1 , ϕ(x2 ) = y2 , ϕ(r) = s. Then, Tfϕ(M ) (b) (y1 − y2 ) = ≥ ≥ = Tfϕ(M ) (b) (sy1 ) = ≥ ≥ max max [TfM (a) (x)] ϕ(x)=y1 −y2 ψ(a)=b max [TfM (a) (x1 − x2 )] ψ(a)=b max [TfM (a) (x1 ) ∗ TfM (a) (x2 )] ψ(a)=b max [TfM (a) (x1 )] ∗ max [TfM (a) (x2 )] ψ(a)=b max ψ(a)=b max [TfM (a) (x)] ϕ(x)=sy1 ψ(a)=b max [TfM (a) (rx1 )] ψ(a)=b max [TfM (a) (x1 )] ψ(a)=b Since, this inequality is satisfied for each x1 , x2 ∈ R1 satisfying ϕ(x1 ) = y1 , ϕ(x2 ) = y2 so we have, Tfϕ(M ) (b) (y1 − y2 ) ≥ ( max max [TfM (a) (x1 )]) ∗ ( max max [TfM (a) (x2 )]) ϕ(x1 )=y1 ψ(a)=b ϕ(x2 )=y2 ψ(a)=b = Tfϕ(M ) (b) (y1 ) ∗ Tfϕ(M ) (b) (y2 ) Also, Tfϕ(M ) (b) (sy1 ) ≥ maxϕ(x1 )=y1 maxψ(a)=b [TfM (a) (x1 )] = Tfϕ(M ) (b) (y1 ) Next, Ifϕ(M ) (b) (y1 − y2 ) = ≤ ≤ = min min [IfM (a) (x)] ϕ(x)=y1 −y2 ψ(a)=b min [IfM (a) (x1 − x2 )] ψ(a)=b min [IfM (a) (x1 ) ⋄ IfM (a) (x2 )] ψ(a)=b min [IfM (a) (x1 )] ⋄ min [IfM (a) (x2 )] ψ(a)=b ψ(a)=b Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 65 Neutrosophic Sets and Systems, Vol. 20, 2018 Ifϕ(M ) (b) (sy1 ) = ≤ ≤ min min [IfM (a) (x)] ϕ(x)=sy1 ψ(a)=b min [IfM (a) (rx1 )] ψ(a)=b min [IfM (a) (x1 )] ψ(a)=b Since, this inequality is satisfied for each x1 , x2 ∈ R1 satisfying ϕ(x1 ) = y1 , ϕ(x2 ) = y2 so we have, Ifϕ(M ) (b) (y1 − y2 ) ≤ ( min min [IfM (a) (x1 )]) ⋄ ( min ϕ(x1 )=y1 ψ(a)=b min [IfM (a) (x2 )]) ϕ(x2 )=y2 ψ(a)=b = Ifϕ(M ) (b) (y1 ) ⋄ Ifϕ(M ) (b) (y2 ) Also, Ifϕ(M ) (b) (sy1 ) ≤ minϕ(x1 )=y1 minψ(a)=b [IfM (a) (x1 )] = Ifϕ(M ) (b) (y1 ). Similarly, we can show that Ffϕ(M ) (b) (y1 − y2 ) ≤ Ffϕ(M ) (b) (y1 ) ⋄ Ffϕ(M ) (b) (y2 ), Ffϕ(M ) (b) (sy1 ) ≥ Ffϕ(M ) (b) (y1 ); This completes the proof. 2. For a ∈ ψ −1 (E) and x1 , x2 ∈ R1 , we have, Tfϕ−1 (N ) (a) (x1 − x2 ) = TfN [ψ(a)] (ϕ(x1 − x2 )) = TfN [ψ(a)] (ϕ(x1 ) − ϕ(x2 )) ≥ TfN [ψ(a)] (ϕ(x1 )) ∗ TfN [ψ(a)] (ϕ(x2 )) = Tfϕ−1 (N ) (a) (x1 ) ∗ Tfϕ−1 (N ) (a) (x2 ) Tfϕ−1 (N ) (a) (rx1 ) = TfN [ψ(a)] (ϕ(rx1 )) = ≥ ≥ = TfN [ψ(a)] (ϕ(r)ϕ(x1 )) TfN [ψ(a)] (sϕ(x1 )) TfN [ψ(a)] (ϕ(x1 )) Tfϕ−1 (N ) (a) (x1 ) Next, Ifϕ−1 (N ) (a) (x1 − x2 ) = IfN [ψ(a)] (ϕ(x1 − x2 )) = IfN [ψ(a)] (ϕ(x1 ) − ϕ(x2 )) ≤ IfN [ψ(a)] (ϕ(x1 )) ⋄ IfN [ψ(a)] (ϕ(x2 )) = Ifϕ−1 (N ) (a) (x1 ) ⋄ Ifϕ−1 (N ) (a) (x2 ) Ifϕ−1 (N ) (a) (rx1 ) = IfN [ψ(a)] (ϕ(rx1 )) = ≤ ≤ = IfN [ψ(a)] (ϕ(r)ϕ(x1 )) IfN [ψ(a)] (sϕ(x1 )) IfN [ψ(a)] (ϕ(x1 )) Ifϕ−1 (N ) (a) (x1 ) Similarly, Ffϕ−1 (N ) (a) (x1 − x2 ) ≤ Ffϕ−1 (N ) (a) (x1 ) ⋄ Ffϕ−1 (N ) (a) (x2 ) and Ffϕ−1 (N ) (a) (rx1 ) ≤ Ffϕ−1 (N ) (a) (x1 ); This proves the 2nd part. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 66 3.10 Theorem Let (ϕ, ψ) be a neutrosophic soft homomorphism from a ring R1 to a ring R2 . Suppose (M, E) and (N, E) are neutrosophic soft completely prime ideals over R1 and R2 , respectively. Then, 1. (ϕ, ψ)(M, E) is a neutrosophic soft completely prime ideal over R2 . 2. (ϕ, ψ)−1 (N, E) is a neutrosophic soft completely prime ideal over R1 . Proof. 1. If possible, let (M, E) be a neutrosophic soft completely prime ideal over R1 but (ϕ, ψ)(M, E) is not so over R2 . Then for b ∈ ψ(E) and y1 , y2 ∈ R2 , Tfϕ(M ) (b) (y1 y2 ) > max{Tfϕ(M ) (b) (y1 ), Tfϕ(M ) (b) (y2 )} ⇒ max max [TfM (a) (x)] > max{( max max [TfM (a) (x)]), ϕ(x)=y1 y2 ψ(a)=b ϕ(x)=y1 ψ(a)=b ( max max [TfM (a) (x)])} ϕ(x)=y2 ψ(a)=b ⇒ ⇒ max [TfM (a) (x)] > max{( max [TfM (a) (x)]), ( max [TfM (a) (x)])} ϕ(x)=y1 ϕ(x)=y1 y2 ϕ(x)=y2 max [TfM (a) (x)] ≥ max{TfM (a) (x1 ), TfM (a) (x2 )} ϕ(x)=y1 y2 Since the inequality holds for each x1 , x2 ∈ R1 satisfying ϕ(x1 ) = y1 , ϕ(x2 ) = y2 so we have TfM (a) (x1 x2 ) > max{TfM (a) (x1 ), TfM (a) (x2 )} which is a contradiction to the truth that (M, E) is a neutrosophic soft completely prime ideal over R1 . We can reach to the same conclusion taking the indeterminacy membership function (I) and falsity membership function (F ) also. Hence we get the first result. 2. For a ∈ ψ −1 (E) and x1 , x2 ∈ R1 , we have, Tfϕ−1 (N ) (a) (x1 x2 ) = TfN [ψ(a)] (ϕ(x1 x2 )) = TfN [ψ(a)] (ϕ(x1 )ϕ(x2 )) ≤ max{TfN [ψ(a)] (ϕ(x1 )), TfN [ψ(a)] (ϕ(x2 ))} = max{Tfϕ−1 (N ) (a) (x1 ), Tfϕ−1 (N ) (a) (x2 )} Ifϕ−1 (N ) (a) (x1 x2 ) = IfN [ψ(a)] (ϕ(x1 x2 )) = IfN [ψ(a)] (ϕ(x1 )ϕ(x2 )) ≥ min{IfN [ψ(a)] (ϕ(x1 )), IfN [ψ(a)] (ϕ(x2 ))} = min{Ifϕ−1 (N ) (a) (x1 ), Ifϕ−1 (N ) (a) (x2 )} Ffϕ−1 (N ) (a) (x1 x2 ) = FfN [ψ(a)] (ϕ(x1 x2 )) = FfN [ψ(a)] (ϕ(x1 )ϕ(x2 )) ≥ min{FfN [ψ(a)] (ϕ(x1 )), FfN [ψ(a)] (ϕ(x2 ))} = min{Ffϕ−1 (N ) (a) (x1 ), Ffϕ−1 (N ) (a) (x2 )} This shows the 2nd result. 4 Neutrosophic Soft Completely Semi-Prime Ideal In this section the concept of semi-prime ideal, completely semi-prime ideal of a ring R and neutrosophic soft completely semi-prime ideal are focussed. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 67 Neutrosophic Sets and Systems, Vol. 20, 2018 4.1 Definition 1. An ideal I of a ring R is called a semi-prime ideal if there is another ideal J of R such that JJ ⊆ I ⇒ J ⊆ I. 2. An ideal J of a ring R is called a completely semi-prime ideal if for x ∈ R, xx ∈ J ⇒ x ∈ J. xx is denoted by x2 . 4.1.1 Example 1. Let R = {0, x, y, z} be a ring. The two binary operations addition and multiplication on R are given by the following tables : + 0 Table 5 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 · 0 Table 6 x y z 0 0 0 0 0 x 0 x x 0 y 0 x y z z 0 0 z z Then {0, x} is a completely semi-prime ideal of R as 0·0 = 0, x·x = x, y·y = y, z·z = z. 2. Consider the Example [3.1.1](3). Then {0, x} is not a completely semi-prime ideal, because z · z = x, y · y = 0 but y, z ∈ / {0, x}. 4.2 Proposition Every completely prime ideal of a ring R is a completely semi-prime ideal of R. Proof. By taking y = x, the proof follows directly from Definition [3.1]. 4.3 Definition Let R be a ring and E be a parametric set. A neutrosophic soft ideal N over (R, E) is called a neutrosophic soft completely semi-prime ideal if ∀x, y ∈ R and ∀e ∈ E, TfN (e) (x2 ) ≤ TfN (e) (x), IfN (e) (x2 ) ≥ IfN (e) (x), FfN (e) (x2 ) ≥ FfN (e) (x). 4.3.1 Example Consider the Example [4.1.1](1). We define an NSS M over (R, E) as following, ∀r ∈ R and ∀e ∈ E, [fM (e)](r) =  (0.4, 0.1, 0.5) (0.2, 0.5, 0.8) if r ∈ {0, x} if r ∈ / {0, x}. Then M is a neutrosophic soft completely semi-prime ideal over (R, E). Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 68 4.4 Lemma A neutrosophic soft ideal N over (R, E) is a neutrosophic soft completely semi-prime ideal iff [fN (e)](x2 ) = [fN (e)](x), for every e ∈ E, x ∈ R. Proof. Let N be a neutrosophic soft ideal over (R, E) with [fN (e)](x2 ) = [fN (e)](x), ∀e ∈ E and ∀x ∈ R. Then by Definition [4.3], N is a neutrosophic soft completely semi-prime ideal over (R, E). Conversely, if N is a neutrosophic soft completely semi-prime ideal by Definition [4.3], TfN (e) (x2 ) ≤ TfN (e) (x), IfN (e) (x2 ) ≥ IfN (e) (x), FfN (e) (x2 ) ≥ FfN (e) (x) and as N is a neutrosophic soft ideal over (R, E), then TfN (e) (x2 ) ≥ TfN (e) (x), IfN (e) (x2 ) ≤ IfN (e) (x), FfN (e) (x2 ) ≤ FfN (e) (x). Hence [fN (e)](x2 ) = [fN (e)](x) for every e ∈ E, x ∈ R. 4.5 Theorem An NSS N over (R, E) is a neutrosophic soft completely semi-prime ideal iff for e ∈ E, S = {x ∈ R : [fN (e)](x) = [fN (e)](0r )}, 0r being the additive identity of ring R, is a completely semi-prime ideal of R. Proof. Let N be a neutrosophic soft completely semi-prime ideal over (R, E). Then [fN (e)](x2 ) = [fN (e)](x) for every e ∈ E, x ∈ R. Now let x2 ∈ S. Then [fN (e)](x2 ) = [fN (e)](0r ) ⇒ [fN (e)](x) = [fN (e)](0r ) ⇒ x ∈ S. Hence S is a completely semi-prime ideal of R. Conversely, if S is a completely semi-prime ideal of R. Then x2 ∈ S ⇒ x ∈ S. Since 2 x ∈ S, then [fN (e)](x2 ) = [fN (e)](0r ) and [fN (e)](x) = [fN (e)](0r ) ⇒ [fN (e)](x2 ) = [fN (e)](x). Hence by Lemma [4.4], N is a neutrosophic soft completely semi-prime ideal over (R, E). 4.6 Theorem An NSS N is a neutrosophic soft completely semi-prime ideal over (R, E) iff [fN (e)](α,β,γ) is a completely semi-prime ideal of R where α ∈ Im TfN (e) , β ∈ Im IfN (e) , γ ∈ Im FfN (e) . Proof. Let N be a neutrosophic soft completely semi-prime ideal over (R, E). Then [fN (e)](x2 ) = [fN (e)](x). Now, x2 ∈ [fN (e)](α,β,γ) ⇒ TfN (e) (x2 ) ≥ α, IfN (e) (x2 ) ≤ β, FfN (e) (x2 ) ≤ γ ⇒ TfN (e) (x) ≥ α, IfN (e) (x) ≤ β, FfN (e) (x) ≤ γ ⇒ x ∈ [fN (e)](α,β,γ) Hence, [fN (e)](α,β,γ) is a completely semi-prime ideal of R. Conversely, let [fN (e)](α,β,γ) be a completely semi-prime ideal of R. Then x2 ∈ [fN (e)](α,β,γ) ⇒ x ∈ ([fN (e)](α,β,γ) i.e., TfN (e) (x2 ) ≥ α, IfN (e) (x2 ) ≤ β, FfN (e) (x2 ) ≤ γ ⇒ TfN (e) (x) ≥ α, IfN (e) (x) ≤ β, FfN (e) (x) ≤ γ Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 69 Neutrosophic Sets and Systems, Vol. 20, 2018 Now, suppose [fN (e)](x2 ) 6= [fN (e)](x). Let [fN (e)](x) = (t1 , t2 , t3 ). Then x2 ∈ / [fN (e)](t1 ,t2 ,t3 ) but x ∈ [fN (e)](t1 ,t2 ,t3 ) which is a contradiction as [fN (e)](α,β,γ) is a completely semi-prime ideal of R. Hence [fN (e)](x2 ) = [fN (e)](x) and so N is a neutrosophic soft completely semi-prime ideal over (R, E) by Lemma [4.4]. 4.7 Theorem Let (ϕ, ψ) be a neutrosophic soft homomorphism from a ring R1 to a ring R2 . Suppose (M, E) and (N, E) are neutrosophic soft completely semi-prime ideals over R1 and R2 , respectively. Then, 1. (ϕ, ψ)(M, E) is a neutrosophic soft completely semi-prime ideal over R2 . 2. (ϕ, ψ)−1 (N, E) is a neutrosophic soft completely semi-prime ideal over R1 . Proof. 1. If possible, let (M, E) be a neutrosophic soft completely semi-prime ideal over R1 but (ϕ, ψ)(M, E) is not so over R2 . Then for b ∈ ψ(E) and y ∈ R2 , Tfϕ(M ) (b) (y 2 ) > Tfϕ(M ) (b) (y) ⇒ max 2 max [TfM (a) (x)] > max max [TfM (a) (x)] ϕ(x)=y ⇒ ⇒ ψ(a)=b ϕ(x)=y ψ(a)=b max [TfM (a) (x)] > max [TfM (a) (x)] ϕ(x)=y 2 ϕ(x)=y max [TfM (a) (x)] ≥ TfM (a) (x) ϕ(x)=y 2 Since the inequality holds for each x ∈ R1 satisfying ϕ(x) = y, so we have TfM (a) (x2 ) > TfM (a) (x) which is a contradiction to the fact that (M, E) is a neutrosophic soft completely semi-prime ideal over R1 . We can reach to the same conclusion taking the indeterminacy membership function (I) and falsity membership function (F ) also. Hence we get the first result. 2. For a ∈ ψ −1 (E) and x ∈ R1 , we have, Tfϕ−1 (N ) (a) (x2 ) = TfN [ψ(a)] (ϕ(x2 )) = TfN [ψ(a)] (ϕ(x))2 ≤ TfN [ψ(a)] (ϕ(x)) = Tfϕ−1 (N ) (a) (x), Ifϕ−1 (N ) (a) (x2 ) = IfN [ψ(a)] (ϕ(x2 )) = IfN [ψ(a)] (ϕ(x))2 ≥ IfN [ψ(a)] (ϕ(x)) = Ifϕ−1 (N ) (a) (x), Ffϕ−1 (N ) (a) (x2 ) = FfN [ψ(a)] (ϕ(x2 )) = FfN [ψ(a)] (ϕ(x))2 ≥ FfN [ψ(a)] (ϕ(x)) = Ffϕ−1 (N ) (a) (x); This proves the 2nd result. 5 5.1 Neutrosophic soft prime k-ideal Definition A neutrosophic soft ideal N over (R, E) is said to be a neutrosophic soft k-ideal over (R, E) if ∀x, y ∈ R and ∀e ∈ E,   TfN (e) (x) ≥ min{TfN (e) (x + y), TfN (e) (y)} If (e) (x) ≤ max{IfN (e) (x + y), IfN (e) (y)}  N FfN (e) (x) ≤ max{FfN (e) (x + y), FfN (e) (y)}. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 70 5.1.1 Example 1. Let Z be the set of all integers and E = {e1 , e2 , e3 } be a parametric set. We consider an NSS N over (Z, E) given by the following table : Z1 Z2 Z3 fN (e1 ) (0.3, 0.8, 0.5) (0.4, 0.6, 0.3) (0.6, 0.2, 0.1) Table 7 fN (e2 ) (0.4, 0.5, 0.7) (0.6, 0.2, 0.4) (1, 0, 0) fN (e3 ) (0.7, 0.6, 0.4) (0.7, 0.4, 0.2) (0.9, 0.1, 0.1) where Z1 = {±1, ±3, ±5, · · · }, Z2 = {±2, ±4, ±6, · · · }, Z3 = {0}. Then N is a neutrosophic soft k-ideal over (Z, E). To verify it, we shall show (i) fN (e) is neutrosophic subgroup of (Z, +) for each e ∈ E. (ii) fN (e) is both neutrosophic left and right ideal of Z for each e ∈ E. (iii) fN (e) is neutrosophic k-ideal of Z for each e ∈ E. If x ∈ Z1 , y ∈ Z2 then x − y ∈ Z1 . We then write Z1 − Z2 = Z1 and so on. Here Z1 − Z1 = Z2 or Z3 , Z1 − Z2 = Z1 , Z1 − Z3 = Z3 , Z2 − Z2 = Z2 or Z3 , Z2 − Z3 = Z2 , Z3 − Z3 = Z3 . Then Table 7 shows the result (i) obviously. Next Z1 .Z1 = Z1 , Z2 .Z2 = Z2 , Z3 .Z3 = Z3 , Z2 .Z1 = Z1 .Z2 = Z2 , Z1 .Z3 = Z3 .Z1 = Z3 , Z2 .Z3 = Z3 .Z2 = Z3 . Then the result (ii) also holds by Table 7. Finally Z1 + Z1 = Z2 or Z3 , Z1 + Z2 = Z1 , Z1 + Z3 = Z3 , Z2 + Z2 = Z2 or Z3 , Z2 + Z3 = Z2 , Z3 + Z3 = Z3 . The Table 7 then meets the result (iii) clearly. 2. Let R be the set of real numbers and E = {e1 , e2 , e3 } be a parametric set. Consider an NSS M over (R, E) given by the following table : Q Qc fM (e1 ) (0.6, 0.1, 0.3) (0.5, 0.4, 0.7) Table 8 fM (e2 ) (0.8, 0.2, 0.4) (0.4, 0.5, 0.6) fM (e3 ) (0.5, 0.6, 0.7) (0.3, 0.7, 1) where Q and Qc are the set of rational and irrational numbers, respectively. If x ∈ Q, y ∈ Qc then x − y ∈ Qc . We write Q − Qc = Qc and so on. Then Q − Q = Q, Q − Qc = Qc , Qc − Qc = Q or Qc . Clearly fM (e) is neutrosophic subgroup of (R, +) for each e ∈ E by Table 8. Next, Q.Q = Q, Q.Qc = Qc , Qc .Qc = Q or Qc . Then Table 8 shows that fM (e) is neutrosophic ideal of R for each e ∈ E. Finally Q + Q = Q, Q + Qc = Qc , Qc + Qc = Q or Qc . Then fM (e) is neutrosophic k-ideal of R for each e ∈ E by Table 8. Hence M is a neutrosophic soft k-ideal over (R, E). 5.2 Definition A neutrosophic soft k-ideal P over (R, E) is said to be a neutrosophic soft prime k-ideal if (i) P is not constant over (R, E), (ii) for any two neutrosophic soft ideals M, N over (R, E), M oN ⊆ P ⇒ either M ⊆ P or N ⊆ P . Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 5.3 Theorem Let P be a neutrosophic soft prime k-ideal over (R, E). Then P0 = {x ∈ R : [fP (e)](x) = [fP (e)](0r ), ∀e ∈ E} is a prime k-ideal of R. Proof. Let x, x + y ∈ P0 for x, y ∈ R. Then [fP (e)](x) = [fP (e)](x + y) = [fP (e)](0r ). Since P is a neutrosophic soft k-ideal over (R, E), so ∀e ∈ E, TfP (e) (y) ≥ min{TfP (e) (x + y), TfP (e) (x)} = TfP (e) (0r ), IfP (e) (y) ≤ max{IfP (e) (x + y), IfP (e) (x)} = IfP (e) (0r ), FfP (e) (y) ≤ max{FfP (e) (x + y), FfP (e) (x)} = FfP (e) (0r ); But TfP (e) (0r ) ≥ TfP (e) (y), IfP (e) (0r ) ≤ IfP (e) (y), FfP (e) (0r ) ≤ FfP (e) (y), ∀e ∈ E. Thus TfP (e) (y) = TfP (e) (0r ), IfP (e) (y) = IfP (e) (0r ), FfP (e) (y) ≤ FfP (e) (0r ), ∀e ∈ E i.e., [fP (e)](y) = [fP (e)](0r ) and so y ∈ P0 . Hence P0 is a k-ideal of R. Also by Theorem [2.11](6), P0 is a prime ideal of R. This completes the proof. 5.4 Theorem Let P be a neutrosophic soft prime k-ideal over (Z, E), Z being the set of integers with P0 = {x ∈ R : [fP (e)](x) = [fP (e)](0), ∀e ∈ E} = nZ, n being a natural number. Then |fP (e)| ≤ r, where r is the number of distinct positive divisor of n. Proof. Let a(6= 0) be an integer and d = gcd(a, n). Then there exists r, s ∈ Z − {0} such that ns = ar + d or ar = ns + d. We shall now estimate following two cases : Case 1 : When ns = ar + d, then ∀e ∈ E and as n ∈ P0 = nZ, TfP (e) (ar + d) = TfP (e) (ns) ≥ TfP (e) (n) = TfP (e) (0) ≥ TfP (e) (ar), IfP (e) (ar + d) = IfP (e) (ns) ≤ IfP (e) (n) = IfP (e) (0) ≤ IfP (e) (ar), FfP (e) (ar + d) = FfP (e) (ns) ≤ FfP (e) (n) = FfP (e) (0) ≤ FfP (e) (ar); Again P is a neutrosophic soft k-ideal over (Z, E). So, TfP (e) (d) ≥ min{TfP (e) (ar + d), TfP (e) (ar)} = TfP (e) (ar) ≥ TfP (e) (a), IfP (e) (d) ≤ max{IfP (e) (ar + d), IfP (e) (ar)} = IfP (e) (ar) ≤ IfP (e) (a), FfP (e) (d) ≤ max{FfP (e) (ar + d), FfP (e) (ar)} = FfP (e) (ar) ≤ FfP (e) (a); Case 2 : When ar = ns + d, then ∀e ∈ E and as n ∈ P0 = nZ, TfP (e) (ns + d) = TfP (e) (ar) ≥ TfP (e) (a), IfP (e) (ns + d) = IfP (e) (ar) ≤ IfP (e) (a), FfP (e) (ns + d) = FfP (e) (ar) ≤ FfP (e) (a); Again, TfP (e) (ns) ≥ TfP (e) (n) = TfP (e) (0) ≥ TfP (e) (a), IfP (e) (ns) ≤ IfP (e) (n) = IfP (e) (0) ≤ IfP (e) (a), FfP (e) (ns) ≤ FfP (e) (n) = FfP (e) (0) ≤ FfP (e) (a); Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 71 Neutrosophic Sets and Systems, Vol. 20, 2018 72 Now as P is a neutrosophic soft k-ideal over (Z, E) so, TfP (e) (d) ≥ min{TfP (e) (ns + d), TfP (e) (ns)} ≥ TfP (e) (a), IfP (e) (d) ≤ max{IfP (e) (ns + d), IfP (e) (ns)} ≤ IfP (e) (a), FfP (e) (d) ≤ max{FfP (e) (ns + d), FfP (e) (ns)} ≤ FfP (e) (a); Thus in either case ∀e ∈ E, TfP (e) (d) ≥ TfP (e) (a), IfP (e) (d) ≤ IfP (e) (a), FfP (e) (d) ≤ FfP (e) (a); Further since d is a divisor of a, there exists t ∈ Z − {0} such that a = dt. So ∀e ∈ E, TfP (e) (a) = TfP (e) (dt) ≥ TfP (e) (d), IfP (e) (a) = IfP (e) (dt) ≤ IfP (e) (d), FfP (e) (a) = FfP (e) (dt) ≤ FfP (e) (d); Hence TfP (e) (d) = TfP (e) (a), IfP (e) (d) = IfP (e) (a), FfP (e) (d) = FfP (e) (a), ∀e ∈ E. Thus for any integer a(6= 0) there exists a divisor d of n such that [fP (e)](d) = [fP (e)](a), ∀e ∈ E. If a = 0 then TfP (e) (a) = TfP (e) (0) = TfP (e) (n), IfP (e) (a) = IfP (e) (0) = IfP (e) (n), FfP (e) (a) = FfP (e) (0) = FfP (e) (n), ∀e ∈ E. This follows the theorem. 5.5 Lemma For a neutrosophic soft prime k-ideal N over (Z, E)(Z being the set of integers), N0 = pZ is a prime k-ideal of Z iff p is either zero or prime. This result is similar to the matter incase of prime ideal in the ring of integers in classical sense. So the proof is omitted. 5.6 Theorem Let N be a neutrosophic soft prime k-ideal over (Z, E), Z being the set of integers. Then |fN (e)| = 2 for each e ∈ E. Conversely, if N is an NSS over (Z, E) such that for each e ∈ E, [fN (e)](x) = (1, 0, 0) when p|x and [fN (e)](x) = (α, β, γ) when p 6 |x, p being a fixed prime and β > 0, γ > 0, α < 1, then N be a neutrosophic soft prime k-ideal over (Z, E). Proof. Let N be a neutrosophic soft prime k-ideal over (Z, E) with N0 = pZ. By Theorem [5.3], N0 is a prime k-ideal of Z. Hence by Lemma [5.5], p is prime i.e., p has only two distinct divisors namely 1, p. So by Theorem [5.4], |fN (e)| ≤ 2. But N being a neutrosophic soft prime k-ideal can not be constant, so |fN (e)| = 2, ∀e ∈ E. Conversely, let N be an NSS over (Z, E) satisfying the given conditions. Let x, y ∈ Z. If TfN (e) (x) = α or TfN (e) (y) = α then TfN (e) (x + y) = 1 or α and so TfN (e) (x + y) ≥ min{TfN (e) (x), TfN (e) (y)}. If TfN (e) (x) = 1 and TfN (e) (y) = 1 then p|x and p|y. It implies p|(x + y) and TfN (e) (x + y) = 1 = min{TfN (e) (x), TfN (e) (y)}. Thus in either case TfN (e) (x + y) ≥ min{TfN (e) (x), TfN (e) (y)}, ∀x, y ∈ Z, ∀e ∈ E. Next, if IfN (e) (x) = β or IfN (e) (y) = β then IfN (e) (x + y) = 0 or β and so, IfN (e) (x + y) ≤ max{IfN (e) (x), IfN (e) (y)}. If IfN (e) (x) = 0 and TfN (e) (y) = 0 then p|x and p|y. It implies p|(x + y) and IfN (e) (x + y) = 0 = min{IfN (e) (x), IfN (e) (y)}. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal Neutrosophic Sets and Systems, Vol. 20, 2018 Thus in either case IfN (e) (x + y) ≤ max{IfN (e) (x), IfN (e) (y)}, ∀x, y ∈ Z, ∀e ∈ E. Finally, if FfN (e) (x) = β or FfN (e) (y) = β then FfN (e) (x + y) = 0 or β and so FfN (e) (x + y) ≤ max{FfN (e) (x), FfN (e) (y)}. If FfN (e) (x) = 0 and FfN (e) (y) = 0 then p|x and p|y. It implies p|(x + y) and FfN (e) (x + y) = 0 = min{FfN (e) (x), FfN (e) (y)}. Thus in either case FfN (e) (x + y) ≤ max{FfN (e) (x), FfN (e) (y)}, ∀x, y ∈ Z, ∀e ∈ E. Further if [fN (e)](x) = (α, β, γ) then either [fN (e)](xy) = (α, β, γ) or [fN (e)](xy) = (1, 0, 0) i.e., TfN (e) (xy) ≥ TfN (e) (x), IfN (e) (xy) ≤ IfN (e) (x), FfN (e) (xy) ≤ FfN (e) (x). If [fN (e)](x) = (1, 0, 0) then p|x and so p|xy. Then [fN (e)](x) = [fN (e)](xy) = (1, 0, 0). Thus in either case we have ∀x, y ∈ Z and ∀e ∈ E, TfN (e) (xy) ≥ TfN (e) (x), IfN (e) (xy) ≤ IfN (e) (x), FfN (e) (xy) ≤ FfN (e) (x). So N is a neutrosophic soft ideal over (Z, E). We shall now prove that N is a neutrosophic soft k-ideal over (Z, E). If [fN (e)](x + y) = (α, β, γ) or [fN (e)](y) = (α, β, γ), then the inequalities in Definition [5.1] are obvious. If [fN (e)](x + y) = (1, 0, 0) or [fN (e)](y) = (1, 0, 0), then p|(x + y) and p|y. It implies p|x and so [fN (e)](x) = (1, 0, 0). Thus the inequalities in Definition [5.1] hold clearly. Therefore N is a neutrosophic soft k-ideal over (Z, E) and so N0 is a k-ideal over Z. Finally, we shall prove that N is a neutrosophic soft prime k-ideal over (Z, E). To prove it, we shall first show that N0 = pZ is a prime k-ideal of Z. Now, x ∈ N0 ⇔ [fN (e)](x) = [fN (e)](0) = (1, 0, 0) ⇔ p|x ⇔ x = pm, m ∈ Z ⇔ x ∈ pZ. Thus N0 = pZ, p being a prime and so N0 is a prime k-ideal of Z by Lemma [5.5]. Further, |fN (e)| = 2, ∀e ∈ E namely (1, 0, 0) and (α, β, γ). So N is not constant over (Z, E). Now assume two neutrosophic soft ideals S, Q over (Z, E) such that SoQ ⊆ N and S 6⊆ N, Q 6⊆ N . Then there exists x, y ∈ Z such that TfS (e) (x) > TfN (e) (x), IfS (e) (x) < IfN (e) (x), FfS (e) (x) < FfN (e) (x) and TfQ (e) (y) > TfN (e) (y), IfQ (e) (y) < IfN (e) (y), FfQ (e) (y) < FfN (e) (y), ∀e ∈ E. Then [fN (e)](x) = [fN (e)](y) = (α, β, γ) obviously and so x, y ∈ / N0 . It implies xy ∈ / N0 as it is a prime k-ideal of an abelian ring Z. So [fN (e)](xy) = (α, β, γ). Thus TfSoQ (e) (xy) ≤ TfN (e) (xy) = α, IfSoQ (e) (xy) ≥ IfN (e) (xy) = β, FfSoQ (e) (xy) ≥ FfN (e) (xy) = γ. But, TfSoQ (e) (xy) ≥ TfS (e) (x) ∗ TfQ (e) (y) > α, IfSoQ (e) (xy) ≤ IfS (e) (x) ⋄ IfQ (e) (y) < β, FfSoQ (e) (xy) ≤ FfS (e) (x) ⋄ FfQ (e) (y) < γ; It opposes the fact. This ends the theorem. 6 Conclusion The aim of this paper is to put forward the study of the concept neutrosophic soft prime ideal introduced in [26]. Here we have studied about neutrosophic soft completely prime ideal, neutrosophic soft completely semi-prime ideal and neutrosophic soft prime k-ideal. They are defined and illustrated by suitable examples. Their related properties and structural characteristics have been investigated also. Moreover a number of theorems have been developed in virtue of these notions. The concepts Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 73 Neutrosophic Sets and Systems, Vol. 20, 2018 74 will bring a new opportunity in research and development of algebraic structures over NSS theory context, we expect. References [1] F. Smarandache, Neutrosophy, neutrosophic probability, set and logic, Amer. Res. Press, Rehoboth, USA., (1998), p. 105, http://fs.gallup.unm.edu/eBookneutrosophics4.pdf (fourth version). [2] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Inter. J. Pure Appl. Math., 24, (2005), 287-297. [3] D. Molodtsov, Soft set theory- First results, Computer and Mathematics with Applications, 37(4-5), (1999), 19-31. [4] A. 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Dutta and A. Ghosh, Intuitionistic fuzzy prime ideals of a semiring (I), International Journal of Fuzzy Mathematics, 16(1), (2008). [12] T. K. Dutta and A. Ghosh, Intuitionistic fuzzy prime ideals of a semiring (II), International Journal of Fuzzy Mathematics, (2008). [13] P. K. Maji, R. Biswas and A. R. Roy, On intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics, 12(3), (2004), 669-683. [14] H. Aktas and N. Cagman, Soft sets and soft groups, Information Sciences, 177, (2007), 2726-2735. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 75 Neutrosophic Sets and Systems, Vol. 20, 2018 [15] A. Aygunoglu and H. Aygun, Introduction to fuzzy soft groups, Computer and Mathematics with Applications, 58, (2009), 1279-1286. [16] Z. Zhang, Intuitionistic fuzzy soft rings , International Journal of Fuzzy Systems, 14(3), (2012), 420-431. [17] A. R. Maheswari and C. Meera, Fuzzy soft prime ideals over right tenary nearrings, International Journal of Pure and Applied Mathematics, 85(3), (2013), 507-529. [18] P. K. Maji, Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics, 5(1), (2013), 157-168. [19] I. Deli and S. Broumi, Neutrosophic soft matrices and NSM-decision making, Journal of Intelligent and Fuzzy Systems, 28(5), (2015), 2233-2241. [20] V. Cetkin and H. Aygun, An approach to neutrosophic subgroup and its fundamental properties, Journal of Intelligent and Fuzzy Systems 29, (2015), 1941-1947. [21] V. Cetkin and H. Aygun, A note on neutrosophic subrings of a ring, 5th International Eurasian Conference on Mathematical Sciences and Applications, 16-19 August 2016, Belgrad-Serbia. [22] T. Bera and N. K. Mahapatra, Introduction to neutrosophic soft groups, Neutrosophic Sets and Systems, 13, (2016), 118-127, doi.org/10.5281/zenodo.570845 [23] T. Bera and N. K. Mahapatra, (α, β, γ)-cut of neutrosophic soft set and it’s application to neutrosophic soft groups, Asian Journal of Math. and Compt. Research, 12(3), (2016), 160-178. [24] T. Bera and N. K. Mahapatra, On neutrosophic soft rings, OPSEARCH, (2016), 1-25, DOI 10.1007/ s12597-016-0273-6. [25] T. Bera and N. K. Mahapatra, On neutrosophic normal soft groups, Int. J. Appl. Comput. Math., 2(4), (2016), DOI 10.1007/s40819-016-0284-2. [26] T. Bera and N. K. Mahapatra, A note on neutrosophic soft prime ideal, Communicated paper for possible publication. Received : April 9, 2018. Accepted : April 23, 2018. Tuhin Bera, Nirmal Kumar Mahapatra. On Neutrosophic Soft Prime Ideal 76 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico Single Valued Neutrosophic Soft Approach to Rough Sets, Theory and Application Emad Marei Department of Mathematics, Faculty of Science and Art, Sager, Shaqra University, Saudi Arabia. E-mail: via_marei@yahoo.com Abstract. This paper aims to introduce a single valued neutrosophic soft approach to rough sets based on neutrosophic right minimal structure. Some of its properties are deduced and proved. A comparison between traditional rough model and suggested model, by using their properties is concluded to show that Pawlak’s approach to rough sets can be viewed as a special case of single valued neutrosophic soft approach to rough sets. Some of rough concepts are redefined and then some properties of these concepts are deduced, proved and illustrated by several examples. Finally, suggested model is applied in a decision making problem, supported with an algorithm. Keywords: Neutrosophic set, soft set, rough set approximations, neutrosophic soft set, single valued neutrosophic soft set. a collected data. This model has been successfully used in the decision making problems and it has been modified in Set theory is a basic branch of a classical mathematics, many papers such as [13-17]. In 2011, F. Feng et al.[18] which requires that all input data must be precise, but introduced a soft rough set model and proved its properties. almost, real life problems in biology, engineering, E.A. Marei generalized this model in [19]. In 2013, P.K. economics, environmental science, social science, medical Maji [20] introduced neutrosophic soft set, which can be science and many other fields, involve imprecise data. In viewed as a new path of thinking to engineers, 1965, L.A. Zadeh [1] introduced the concept of fuzzy logic mathematicians, computer scientists and many others in which extends classical logic by assigning a membership various tests. In 2014, Broumi et al. [21] introuduced the function ranging in degree between 0 and 1 to variables. concept of rough neutrosophic sets. It is generalized and As a generalization of fuzzy logic, F. Smarandache in 1995, applied in many papers such as [22-31]. In 2015, E.A. initiated a neutrosophic logic which introduces a new Marei [32] introduced the notion of neutrosophic soft component called indeterminacy and carries more rough sets and its modification. information than fuzzy logic. In it, each proposition is estimated to have three components: the percentage of This paper aims to introduce a new approach to soft truth (t %), the percentage of indeterminacy (i %) and the rough sets based on the neutrosophic logic, named single percentage of falsity (f %), his work was published in [2]. valued neutrosophic soft (VNS in short) rough set From scientific or engineering point of view, neutrosophic approximations. Properties of VNS-lower and VNS-upper set’s operators need to be specified. Otherwise, it will be approximations are included along with supported proofs difficult to apply in the real applications. Therefore, Wang and illustrated examples. A comparison between traditional et al.[3] defined a single valued neutrosophic set and rough and single valued neutrosophic soft rough various properties of it. This thinking is further extended to approaches is concluded to show that Pawlak’s approach to many applications in decision making problems such as [4, rough sets can be viewed as a special case of single valued 5]. neutrosophic soft approach to rough sets. This paper delves Rough set theory, proposed by Z. Pawlak [6], is an into single valued neutrosophic soft rough set by defining effective tool in solving many real life problems, based on some concepts on it as a generalization of rough concepts. imprecise data, as it does not need any additional data to Single valued neutrosophic soft rough concepts (NRdiscover a knowledge hidden in uncertain data. Recently, concepts in short) include NR-definability, NRmany papers have been appeared to development rough set membership function, NR-membership relations, NRmodel and then apply it in many real life applications such inclusion relations and NR-equality relations. Properties of as [7-11]. In 1999, D. Molodtsov [12], suggested a soft set these concepts are deduced, proved and illustrated by model. By using it, he created an information system from 1 Introduction Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application 77 Neutrosophic Sets and Systems, Vol. 20, 2018 several examples. Finally, suggested model is applied in a decision making problem, supported with an algorithm. 2 Preliminaries In this section, we recall some definitions and properties regarding rough set approximations, neutrosophic set, soft set and neutrosophic soft set required in this paper. Definition 2.1 [6] Lower, upper and boundary approximations of a subset X  U , with respect to an equivalence relation, are defined as E ( X )  {[ x] E : [ x] E , X }, E ( X )  {[ x]E : [ x]E  X  }, BNDE ( X )  E( X )  E( X ), where [ x]E  {x,  U : E ( x)  E ( x, )}. Definition 2.2 [6] Pawlak determined the degree of crispness of any subset X  U by a mathematical tool, named the accuracy measure of it, which is defined as  ( X )  E ( X ) / E ( X ), E ( X )   . E Obviously, 0   E ( X )  1 . If E( X )  E ( X ) , then X is crisp (exact) set, with respect to E , otherwise X is rough set. Properties of Pawlak’s approximations are listed in the following proposition. Proposition 2.1 [6] Let (U , E ) be a Pawlak proximation space and let X , Y  U . Then, (a) E( X )  X  E( X ) . (b) E( ) =  = E( ) and E(U ) = U = E(U ) . (c) E( X  Y ) = E( X )  E(Y ) . (d) E( X  Y ) = E( X )  E(Y ) . (e) X  Y , then E ( X )  E (Y ) and E ( X )  E(Y ) . (f) E ( X  Y )  E ( X )  E (Y ) . (g) E( X  Y )  E( X )  E(Y ) . (h) E ( X c ) = [ E ( X )]c , X is the complement of X . (i) E ( X c ) = [ E ( X )]c . (j) E( E( X )) = E( E( X )) = E( X ) . (k) E( E( X )) = E( E( X )) = E( X ) . C Definition 2.3 [33] An information system is a quadruple IS = (U , A, V , f ) , where U is a non-empty finite set of objects, A is a non-empty finite set of attributes, V = {V , e  A} , V is the value set of attribute e , e e f : U  A  V is called an information (knowledge) function. Definition 2.4 [12] Let U be an initial universe set, E be a set of parameters, A  E and let P (U ) denotes the power set of U . Then, a pair S = ( F , A) is called a soft set over U , where F is a mapping given by F : A  P (U ) . In other words, a soft set over U is a parameterized family of subsets of U . For e  A, F (e) may be considered as the set of e -approximate elements of S . Definition 2.5 [2] A neutrosophic set A on the universe of discourse U is defined as A = { x, T ( x), I ( x), F ( x) : x  U }, where A A A  0  T ( x)  I ( x)  F ( x)  3 , andT , I , F    0,1    A A A Definition 2.6 [20] Let U be an initial universe set and E be a set of parameters. Consider A  E , and let P (U ) denotes the set of all neutrosophic sets of U . The collection ( F , A) is termed to be the neutrosophic soft set over U , where F is a mapping given by F : A  P (U ). Definition 2.7 [3] Let X be a space of points (objects), with a generic element in X denoted by x . A single valued neutrosophic set A in X is characterized by truth-embership function T A , indeterminacy-membership function I A and falsity-membership function FA . For each point x in X , TA(X),I A(X),F A(X)  0,1 . When X is continuous, a single valued neutrosophic set A can be written as A   X (T(x),I(x), F(x)) /x,x  X . When X is discrete, A can be written as A  in1 (T(xi ),I(x i ),F(x i )) /xi ,xi  X . 3 Single valued neutrosophic soft rough set approximations In this section, we give a definition of a single valued neutrosophic soft (VNS in short) set. VNS-lower and VNS-upper approximations are introduced and their properties are deduced, proved and illustrated by many counter examples. Definition 3.1 Let U be an initial universe set and E be a set of parameters. Consider A  E , and let P (U ) denotes the set of all single valued neutrosophic sets of U . The collection (G,A) is termed to be VNS set over U , where G is a mapping given by G : A  P (U ) . For more illustration the meaning of VNS set, we consider the following example Example 3.1 Let U be a set of cars under consideration and E is the set of parameters (or qualities). Each parameter is a neutrosophic word. Consider E = {elegant, trustworthy, sporty, comfortable, modern}. In this case, to define a VNS means to point out elegant cars, trustworthy cars and so on. Suppose that, there are five cars in the universe U , given by U  {h1 , h2 , h3 , h4 , h5} and the set of parameters A  {e1 , e2 , e3 , e4 } , where A  E and each ei is a specific criterion for cars: e1 stands for elegant, e 2 stands Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application Neutrosophic Sets and Systems, Vol. 20, 2018 78 for trustworthy, e 3 stands for sporty and e 4 stands for comfortable. A VNS set can be represented in a tabular form as shown in Table 1. In this table, the entries are c ij corresponding to the car hi and the parameter e j , where Cij = (true membership value of hi , indeterminacy-membership value of hi , falsity membership value of hi ) in G ( e i ) . e1 e2 e3 e4 h1 (.6, .6, .2) (.8, .4, .3) (.7, .4, .3) (.8, .6, .4) h3 Proof Let  h , T (h ), I (h ), F (h ) ,  h , T (h ), I (h ), F (h ) 2 e 2 e 2 and  h , T (h 1), e I 1(h e), 1F (eh 1)  G (2A) e. Then, 3 h5 (.4, .6, .6) (.6, .2, .4) (.6, .4, .3) (.7, .6, .6) (.6, .4, .2) (.8, .1, .3) (.7, .2, .5) (.7, .6, .4) (.6, .3, .3) (.8, .2, .2) (.5, .2, .6) (.7, .5, .6) (.8, .2, .3) (.8, .3, .2) (.7, .3, .4) (.9, .5, .7) Table1: Tabular representation of (G, A) of Example 3.1. Definition 3.2 Let (G , A) be a VNS set on a universe U . For any element h  U , a neutrosophic right neighborhood, with respect to e  A is defined as follows he = {hi  U : Te (hi )  Te (h), I e (hi )  I e (h), Fe (hi )  Fe (h)}. Definition 3.3 Let (G,A) be a VNS set on U. Neutrosophic right minimal structure is defined as follows   {U ,  , h e : h  U , e  A} Illustration of Definitions 3.2 and 3.3 is introduced in the following example Example 3.2 According Example 3.1, we can deduce the following results: h1e  h1e  h1e  h1e  {h } , h2 e  h2e  1 2 3 1 4 1 3 {h , h } , h2e  {h , h , h , h } , h2e  {h , h , h } , h3 e  h3e  {h , h } , 1 2 1 2 4 5 1 2 3 1 3 2 4 1 3 e 3 e 1 e 1 e 1 e 1 e 1 e = F (h ) . For every e  A , h1  h1e . Then h1 R e h1 and 1 e (b) Let h R h and h R h , then h  h and h  1 4 h3e2  {h , h , h , h } , h3e3  {h , h , h } , h4e1  {h , h3 , h } , h4e2  {h , h } , 1 3 4 5 1 3 5 4 5 1 4 2 e h . Hence, T ( h ) 2e  e 3 1  3 e 1 e   2 2 3 e T (h ) , I (h ) 1 e 2 e T (h ) , I (h ) e 2 e 3 1e   3 I (h ) , F (h ) 1 e e 2 e 3  I ( h ) and F ( h ) 1 e e 3 2 I ( h ) and 2 e F ( h ) . Consequently, we have T ( h ) T (h ) , I (h ) e 2 e F (h ) , T (h ) F (h ) e h4 3 e (a) Obviously, T (h ) = T (h ) , I (h ) = I (h ) and F (h ) then R e is reflexive relation. U h2 (c) R e may be not symmetric relation.  3 e  F (h ) . It e 1 follows that, h  h . Then h R h and then R is 3 1e 1 e 3 e transitive relation. The following example proves (c) of Proposition 3.1. Example 3.3 From Example 3.2, we have, h1e  {h1} and 1 h3 e1  {h , h } . Hence, (h2 , h1 )  Re1 but ( h , h )  Re1 . 1 3 1 3 Then, R e isn’t symmetric relation. Definition 3.4 Let (G,A) be a VNS set on U , and let  be a neutrosophic right minimal structure on it. Then, VNSlower and VNS-upper approximations of any subset X based on  , respectively, are S  X  {Y   : Y  X },  S X  {Y   : Y  X }. Remark 3.1 For any considered set X in a VNS set (G,A), the sets  c PNR X  S  X , N NR X  [ S X ] ,  b NR X  S X  P NR X are called single valued neutrosophic positive, single valued neutrosophic negative and single valued neutrosophic boundary regions of a considered set X , It follows that, respectively. The real meaning of single valued   {{h1 }, {h5 }, {h1 , h2 }, {h1 , h3 }, {h1 , h5 }, {h4 , h5 }, neutrosophic positive of X is the set of all elements which are surely belonging to X, single valued neutrosophic {h1 , h2 , h3 }, {h1 , h3 , h4 }, {h1 , h3 , h5 }, {h1 , h2 , h3 , h4 } negative of X is the set of all elements which are surely not , {h1 , h2 , h4 , h5 }, {h1 , h3 , h4 , h5 }, U , } belonging to X and single valued neutrosophic boundary of Proposition 3.1 Let (G , A) be a VNS set on a universe U , X is the elements of X which are not determined by (G,A).  is the family of all neutrosophic right neighborhoods on Consequently, the single valued neutrosophic boundary region of any considered set is the initial problem of any it, and let real life application. Re : U   , Re (h) = he VNS rough set approximations properties are introduced in Then, the following proposition. (a) R e is reflexive relation. Proposition 3.2 Let (G,A) be a VNS set on U, and let (b) R e is transitive relation. X , Z  U . Then the following properties hold h4 e  U , h4e  {h , h , h , h } , h5 e  h5e  h5e  {h } , h5e  {h , h } . 1 2 3 4 5 1 5 3 4 1 2 4 3 Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application 79 Neutrosophic Sets and Systems, Vol. 20, 2018 (a) S  X  X  S  X .   S( X  Z )  S X  S Z . (b) S   = S  =  . (i) Let h  S  ( X  Z ) . But S  ( X  Z )   {Y   : Y  X  Z } . Then, there exists Y such that Y  X  Z (c) S U = S U = U . (d) X  Z  S  X  S  Z . (e) X  Z  S X  S Z .   (f) S ( X  Z )  S X  SZ . (g) S ( X  Z )  S X  SZ . (h) S  ( X  Z )  S  X  S Z (i) S  ( X  Z )  S  X  S  Z . Proof (a) From Definition 3.3, obviously, we can deduce that, S X  X  S  X . (b) From Definition 3.4, we can deduce that S   and S   {Y   : Y  }   . (c) From Property (a), we have U  S U but U is the universe set, then S U  U . Also, from Definition 3.4, we have SU  {Y   : Y  U } , but U   . Then, SU  U (d) Let X  Z and h  S X , then there exists Y   such that h  Y  X . But X  Z , then h  Y  Z . Hence, h  SZ . Consequently S  X  S  Z . (e) Let X  Z and h  S  Z . But S  Z   {Y   : Y  . h  Y and Y  Z such that U   there exists Then. Z } But X  Z , then Y  X and h  Y . Hence h  S  Z . Thus S  X  S Z . (f) Let h  S ( X  Z )  {Y   : Y  X  Z} . So, there exists Y   such that, h  Y  X  Z , then h  Y  X and h  Y  Z . Consequently, h  S X and h  S  Z , then h  S X  SZ . Thus S ( X  Z )  S X  S Z . (g) Let h  S ( X  Z )  {Y   : Y  X  Z } . So, for all Y   , h  Y , we have Y  X  Z , then Y  X and Y  Z . Consequently, h  S  X and h  S  Z . So and h  Y . Then, Y  X , h  Y and Y  Z , h  Y . It follows that, h  S  X  S  Z . Thus S  ( X  Z )  S  X  S Z . The following example illustrates that the converse of Property (a) doesn’t hold Example 3.4 From Example 3.1, if X  {h3 } , then S X  . X  S  X and S X  X Hence. S  X  {h1 , h3 } and  The following example illustrates that the converse of Property (d) doesn’t hold Example 3.5 From Example 3.1, if X  {h2 } and Z  {h1, h2} , then S X   , S  Z  {h1 , h2 } . Thus S X  SZ . The following example illustrates that the converse of Property (e) doesn’t hold Example 3.6 From Example 3.1, if X  {h5 } and Z  {h2 , h5 } , then, S  X  {h5 } and S  Z  {h1 , h2 ,   h4 , h5 } . Hence, S X  S Z . The following example illustrates that the converse of Property (f) doesn’t hold Example 3.7 From Example 3.1, If X  {h1 , h3 , h4 } and Z  {h1 , h4 , h5 } , then S  X  {h1 , h3 , h4 } , S Z  {h1 , S  ( X  Z )  S  X Hence. S ( X  Z )  {h1} and h4 , h5 } .  S Z The following example illustrates that the converse of Property (g) doesn’t hold Example 3.8 From Example 3.1, if X  {h1 } and Z  {h2 } then S  X  {h1} , S  Z   and S ( X  Z )  {h1, h2} . Hence S  ( X  Z )  S X  S Z . The following example illustrates that the converse of Property (h) doesn’t hold Example 3.9 From Example 3.1, if X  {h1 , h2 , h4 } and Z  {h1 , h2 , h5 } then S  X  {h1 , h2 , h4 } , S  Z  {h1 , h2 , h4 , h5 } and S  ( X  Z )  {h1 , h2 } . Hence h  S  X  S  Z . Thus S ( X  Z )  S X  S Z . S  ( X  Z )  S  X  S Z (h) Let h  S  X  S  Z . Then, h  S  X or h  S  Z and then there exists Y   such that Y  X , h  Y or Y  X , The following example illustrates that the converse of Property (i) doesn’t hold h  Y . Consequently Example 3.10 From Example 3.1, if X  {h2 , h3 } and h  S  ( X  Z ) . Thus Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application Neutrosophic Sets and Systems, Vol. 20, 2018 80 Z  {h5 } then S  X  {h1 , h2 , h3 } , S  Z  {h5 } and S  ( X  Z )  U . Hence S  ( X  Z )  S  X  S  Z . Proposition 3.3 Let (G , A) be a neutrosophic soft set on a unverse U , and let X , Z  U . Then the following properties hold. (a) S S X  S X (b) S  S  X  S  X and h  Y  Z . Consequently, h S  X and h  S  Z , then h S X  S Z . Therefore S ( X  Z )  S X  S Z . The following example illustrates that the converse of Proposition 3.4 doesn’t hold. Example 3.13 From Example 3.1, if X  {h1 , h3 , h5 } and Z  {h1, h5} , then S  X  {h1 , h3 , h5 } , S  Z  {h1 , h5 } , S ( X  Z )   and S  X  S  Z  {h3 } . Hence, S  ( X  Z )  S X  S Z (d) S  S  X  S  X Proposition 3.5 Let (G , A) be a VNS set on U and let X , Z  U . Then the following properties don’t hold (a) S  X c  [ S  X ]c Proof (b) S  X c  [ S  X ]c (c) S  S  X  S  X (a) Let W = S  X and some h W   {Y   : Y  X } . Then, for e  A , we have h  Y  W . So h  SW . Hence W (c) S  ( X  Z )  S  X  S  Z  SW . Thus, SW  S  S W . Also, from Property (a) of Proposition 3.2, we have S X  X and by using Property (d) of Proposition 3.2, we get S S X  S X . Consequently. S X = S S X The following example proves Properties (a) and (b) of Proposition 3.5. Example 3.14 From Example 3.1, if X  {h1} . Then,  c S  X  S  X  {h1 } , S X c  {h4 , h5 }and S X  U . Thus c  c  c c S  X  [ S X ] and S X  [ S  X ] (b) Let W  S  X and h  W , from Definition 3.4, we have W  {Y   : Y  X }. Then there exists Y   , such The following example proves Property (c) of Proposition 3.5. Example 3.15 From Example 3.1, if X  {h1 , h2 } and Z  {h1 } . Then S  X  {h1 , h2 } , S  Z  {h1} , S  ( X  Z )  {h1 , h2 } . Hence S  ( X  Z )  S  X  S  Z . that Y  X and h  Y . Hence, there exists Y   , such that Y  W and h  Y , it follows that h  S W . Consequently W  S W . Also, by using Property (a) of Proposition 3.2, we have W  S W . Thus S  S W  S W Properties (c) and (d) can be proved directly from Proposition 3.2. The following example illustrates that the converse of Property (c) doesn’t hold. Example 3.11 From Example 3.1, if X  {h4 } . Then S  X  {h4 } and S S  X   . Hence, S S  X  S  X . The following example illustrates that the converse of Property (c) doesn’t hold. Example 3.12 From Example 3.1, if X  {h1 , h2 , h5 } , then S  X  {h1 , h2 , h5 } and S  S X  {h1 , h2 , h4 , h5 } . Hence  S S X  S X Proposition 3.4 Let (G , A) be a VNS set on X , Z  U . Then U and let S ( X  Z )  S X  S Z Proof Let h  S ( X  Z )  {Y   : Y  ( X  Z )} . So, there exists Y   such that h  Y  ( X  Z ) , then h  Y  X Remark 3.2 A comparison between traditional rough and single valued neutrosophic soft rough approaches, by using their properties, is concluded in Table 2, as follows 4 Single valued neutrosophic soft rough concepts In this section, some of single valued neutrosophic soft rough concepts (NR-concepts in short) are defined as a generalization of traditional rough concepts. Definition 4.1 Let (G , A) be a VNS set on U . A subset X  U is called (a) NR-definable (NR-exact) set if S  X  S  X  X (b) Internally NR-definable set if S  X  X and S  X  X (c) Externally NR-definable set if S  X  X and S  X  X (d) NR-rough set if S  X  X and S  X  X The following example illustrates Definition 4.1. Example 4.1 From Example 3.1, we can deduce that {h1 } , {h5 }, {h1 , h2 }, {h1 , h3 }, {h1 , h5 }, {h4 , h5 } , {h1 , h2 , h3},{h1 , h3 , h4 },{h1 , h3 , h5},{h1 , h4 , h5},{h1 , h2 , h3 , h4 } , {h1 , h2 , h4 , h5 }, {h1 , h3 , h4 , h5 } are NR-definable sets, {h1 , h2 , h5 }, {h1 , h2 , h3 , h5 } are internally NR-definable sets, {h4 }, {h1 , h4 }, {h1 , h2 , h4 } are externally NR-definable sets and the rest of proper subsets of U are Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application 81 Neutrosophic Sets and Systems, Vol. 20, 2018 NR-rough sets. We can determine the degree of single valued neutrosophic soft-crispness (exactness) of any subset X  U by using NR-accuracy measure, denoted by C X , which is defined as follows Definition 4.2 Let (G,A) be a VNS on U , and let X  U . Then C X  S  X S  X , X   Remark 4.1 Let (G,A) be a VNS on U . A subset X  U is NR-definable (NR-exact) if and only if C X  1 . Definition 4.3 Let (G,A) be a VNS on U and let X  U , x  X . NR-membership function of an element x to a set X denoted by  X x is defined as follows:  X x | xA  X | / | xA |, where x A  {xe : e  A} and xe is a neutrosophic right neighborhood, defined in Definition 3.2. Proposition 4.1 Let (G,A) be a VNS on U , X  U and let  X x be the membership function defined in Definition 4.3. Then  X x  [0,1] Proof Where   x A  X  x A then 0  x A  X  x A and then 0   X x  1. Proposition 4.2 Let (G,A) be a VNS on U and let X  U , then X x  1  x  X Proof Let  X x  1, then x A  X  x A . Consequantly x A  X . From Proposition 3.1, we have Re is a reflexive relation for all e  A . Hence x  xe e  A . It follows that x  x A . Thus x  X The following example illustrates that the converse of Proposition 4.2 doesn’t hold. Example 4.2 From Example 3.2, we get h3 A  {h1 , h3 } . If X  {h , h , h } , then  X h3  1 2 . Although h3  X 2 3 5 Proposition 4.3 Let (G,A) be a VNS on U and let X , Z  U . If X  Z , then the following properties hold (a)  X x  Z x (b)  S X x   S Z x   (c)  S  X x   S Z x Proof (a) Where X  U , for any x  U we can deduce that  X x   Z x . Thus x A  X  x A  Z then  xA  Z , x A  X We get the proof of Properties (b) and (c) of Proposition 4.3, directly from property (a) of Proposition 4.3 and properties (d) and (e) of Proposition 3.2. Traditional rough properties VNS rough properties S  ( X  Z )  S  X  S Z E( X  Z )  E X  EZ E ( X Y ) = E ( X ) E (Y ) S ( X  Z )  S X  S Z E ( E ( X )) = E ( X ) E ( E ( X )) = E ( X ) E ( X c ) = [ E ( X )]c c E ( X c ) = [ E ( X )] S S  X  S  X S  S X  S X S  X c  [ S  X ]c S  X c  [ S  X ]c Table 2: Comparison between traditional, VNS rough Proposition 4.4 Let (G,A) be a VNS on U and let XU, then the following properties hold (a) S X x   X x  (b)  X x   x S X  (c) S X x   x S X   Proof can be obtained directly from Propositions 3.2 and property (a) of Proposition 4.3. Definition 4.4 Let (G,A) be a VNS set on U , and let x  U , X  U . NR-membership relations, denoted by  and  are defined as follows   x  X if x  S  X and x  X if x  S X Proposition 4.5 Let (G,A) be a VNS set on U , and let x  U , X  U . Then (a) x  X  x  X (b) x  X  x  X Proof (a) Let x  X , hence by using Definition 4.4, we get x  S X . But from Proposition 3.2, we have S X  X , then x X . (b) Let x  X , according to Proposition 3.2, we have  X  S  X , then x  S X , by using Definition 4.4,  we can deduce that x  X . Consequently x  X  x  X . The following example illustrates that the converse of Proposition 4.5 doesn’t hold. Example 4.3 From Example 3.1, if X  {h2 , h5 } , then  S  X  {h5 } and S X  {h1 , h2 , h4 , h5 } . Hence, h2  X , although h2  X and h4  X , although h4  X . Proposition 4.6 Let (G,A) be a VNS on Then the following properties hold Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application U and let X  U . Neutrosophic Sets and Systems, Vol. 20, 2018 82 (a) x  X   X x  1 (b)  X x  1  x  X Proposition 4.9 Let (G,A) be a VNS on  U . Then Proof can be obtained directly from Definition 4.4 and Propositions 4.2 and 4.5. Proof comes directly From Proposition 3.2. The following example illustrates that the converse of property (a) does not hold. Example 4.4 From Example 3.1, if X  {h1 , h4 } then S  X  {h1} and h4 A  {h4 } , it follows that  X h4  1 . Although h4  X The following example illustrates that, the converse of Proposition 4.9 doesn’t hold. Example 4.9 In Example 3.1, if X  {h1 , h4 } and Z  {h1 , h2 , h5 } , then S  X  {h1}, S  Z  {h1 , h2 , h5 } , S  X  {h1 , h4 } and  S  Z  {h1 , h2 , h4 , h5 } . Hence, X   Z and X  Z . Although X  Z U and let X , Z X  Z  X  Z  X  Z The following example illustrates that the converse of property (b) does not hold. Example 4.5 From Example 3.1, if X  {h2 } , then S  X  {h1 , h2 } and h2 A  {h1 , h2 } , it follows that h2  X , although  X h2  1 From Definition 4.5 and Proposition 4.3, the following remarks can be deduced Remark 4.2 Let (G,A) be a VNS on U and let X , Z  U . If X   Z , then the following properties hold (a)  S X x   S Z x  Proposition 4.7 Let (G,A) be a VNS on X  U . Then (a)  X x  0  x  X U and let (b)  X x  0  x  X Proof is straightforward and therefore is omitted. The following example illustrates that the converse of property (a), does not hold. Example 4.6 From Example 3.1, if X  {h1 , h3 , h4 } and from Example 3.2, we get h2 A  {h1 , h2 } , then  X h2  0 , although h2  X The following example illustrates that the converse of property (b), does not hold. Example 4.7 From Example 3.1, if X  {h1 , h4 , h5 } , then S  X  {h1 , h4 , h5 } , from Example 3.2, we get h2 A  {h1 , h2 } , it follows that  X h2  0 , although h2  X Proposition 4.8 Let (G,A) be a VNS on U and let XU. The following property does not hold  X x  0  x  X The following example proves Proposition 4.8. Example 4.8 From Example 3.1, if X  {h2 } then S  X  {h1, h2} , from Example 3.2, we get h1 A  {h1} , it follows that h1  X , although  X h1  0 Definition 4.5 Let (G,A) be a VNS on U and let X , Z  U . NR-inclusion relations, denoted by   and   which are defined as follows X   Z If S  X  S  Z X   Z If S  X  S Z  (b) S X x  Z x  (c) S X x   x S Z   Remark 4.3 Let (G,A) be a VNS on U and let X , Z  U . If X   Z , then the following properties hold (a)  x   x SX S Z (b)  X x   x S Z  (c) S X x   x S Z   Definition 4.6 Let (G,A) be a VNS on U and let X , Z  U . NR-equality relations are defined as follows X  Z If  X  Z If If X   Z S X  SZ S  X  S Z  X  Z  X  Z The following example illustrates Definition 4.6. Example 4.10 According to Example 3.1. Let A  {e1} , then   {U ,  ,{h1}, {h5 }, {h1 , h2 }, {h1 , h3}, {h1 , h3 , h4 }} . If X1  {h2}, X 2  {h3}, X 3  {h1 , h2 }, X 4  {h2 , h3} and X 5  {h2 , h4 } , then S X1  S X 2   , S  X 1  S  X 3  {h1 , h2 } , S X 4  S X 5   and S  X 4  S  X 5  U . Consequently X 1   X 2 , X 1   X 3 and X 4  X 5 Proposition 4.10 Let (G,A) be a VNS set on X , Z  U . Then (a) X   S  X U and let (b) X   S  X (c) X  Z  X   Z (d) X  Z , Z    X   Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application 83 Neutrosophic Sets and Systems, Vol. 20, 2018 (e) X  Z , X  U  Z  U (f) X  Z , Z     X   (g) X  Z , X   U  Z   U Proof. From Definition 4.6 and Propositions 3.2 and 3.3 we get the proof, directly. From Definition 4.6 and Proposition 4.3, the following remarks can be deduced Remark 4.4 Let (G,A) be a VNS on U and let X , Z  U . If X  Z , then the following properties hold (a) S X x  S Z x   parameters of the person X. To solve this problem, we need the following definitions Definition 5.1 Let (G,A) be a VNS set on U  {h1, h2 ,..., hn } as the objects and A  {e1 , e2 ,.., em } is the set of parameters. The value matrix is a matrix whose rows are labeled by the objects, its columns are labeled by the parameters and the entries Cij are calculated by Cij  (Tej (hi )  I ej (hi )  Fej (hi )), 1  i  n,1  j  m Definition 5.2 Let (G,A) be a VNS set on U  {h1, h2 ,..., hn } , where A  {e1 , e2 ,.., em } . The score of an object h j is defined as follows S ( hi )  m j 1 Cij (b) S X x  Z x  (c) S X x   x S Z   Remark 4.5 Let (G,A) be a VNS on U and let X , Z  U . If X   Z , then the following properties hold (a)  x   x SX U and (b)  m  S (hi )  2m, hi  U S Z (b)  X x   x S Z  (c) S X x   x S Z  Remark 5.1 Let (G, A) be a VNS set on A  {e1 , e2 , then is the set of parameters. .., em } (a)  1  Cij  2, 1  i  n,1  j  m  The real meaning of C A is the degree of crispness of A . Hence, if C A  1 , then A is NR-definable set. It means that the collected data are sufficient to determine the set A . Also, from the meaning of the neutrosophic right neighborhood, we can deduce the most suitable choice by using the following algorithm. The following remark is introduced to show that Pawlak’s approach to rough sets can be viewed as a special case of proposed model. Remark 4.6 Let (G,A) be a VNS on U and let X , Z  U . Algorithm If we consider the following case 1. Input VNS set (G,A) ( If Te (hi )  0.5 , then e(h)  1 , otherwise e(h)  0 ) 2. Compute the accuracy measures of all singleton sets and the neutrosophic right neighborhood of an element h is 3. Consider the objects of NR-definable singleton sets replaced by the following equivalence class 4. Compute the value matrix of the considered objects [h] e  {hi  U : e(hi )  e(h), e  A}. 5. Compute the score of all considered objects in a tabular form Then VNS-lower and VNS-upper approximations will be traditional Pawlak’s approximations. It follows that NR- 6. Find the maximum score of the considered objects concepts will be Pawlak’s concepts. Therefor Pawlak’s 7. If there are more than one object has the maximum approach to rough sets can be viewed as a special case of scare, then any object of them could be the suitable suggested single valued neutrosophic soft approach to choice rough sets. 8. If there is no NR-definable singleton set, then we consider the objects of all NR-definable sets consisting 5 A decision making problem two elements and then repeat steps (4-7), else, consider In this section, suggested single valued neutrosophic the objects of all NR-definable sets consisting three soft rough model is applied in a decision making problem. elements and then repeat steps (4-7),and so on... We consider the problem to select the most suitable car which a person X is going to choose from n cars (h1, h2 ,..., For illustration the previous technique, the following hn ) by using m parameters ( e1 , e2 ,.., em ). Since these data are not crisp but neutrosophic, the example is introduced. selection is not straightforward. Hence our problem in this Example 5.1 According to Example 3.1, we can create section is to select the most suitable car with the choice Tables 3, as follows Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application Neutrosophic Sets and Systems, Vol. 20, 2018 84 Singleton sets {h1} {h2 } {h3 } {h4 } {h5 } 1 0 0 0 1 C X Table 3: Accuracy measures of all singleton sets. Hence C {h1 }  C {h5 }  1 . It follows that h1 and h5 are the NR-definable singleton sets. Consequently h1 and h5 are concidered objects. Therefore Table 4 can be created as follows Object h1 e1 e3 e2 e4 (.6,.6,.2) (.8,.4,.3) (.7,.4,.3) (.8,.6,.4) (.8,.2,.3) (.8,.3,.2) (.7,.3,.4) (.9,.5,.7) h5 Table 4: Tabular representation of considered objects. The value matrix of considered objects can be viewed as Table 5. Object h1 e1 e2 e3 e4 1 0.9 0.8 1 0.7 0.9 0.6 0.7 h5 Table 5: Value matrix of considered objects. Finally, the scores of considered objects are concluded in Table 6, as follows Object Score of the object 3.7 2.9 Table 6: The scores of considered objects. h1 h5 Clearly, the maximum score is 3.7, which is scored by the car h1 . Hence, our decision in this case study is that a car h1 is the most suitable car for a person X , under his choice parameters. Also, the second suitable car for him is a car h5 . Obviously, the selection is dependent on the choice parameters of the buyer. Consequently, the most suitable car for a person X need not be suitable car for another person Y . Conclusion This paper introduces the notion of single valued neutrosophic soft rough set approximations by using a new neighborhood named neutrosophic right neighborhood. Suggested model is more realistic than the other traditional models, as each proposition is estimated to have three components: the percentage of truth, the percentage of indeterminacy and the percentage of falsity. 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An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116. [28] H. L. Yang, C.L. Zhang, Z.L. Guo, Y.L. Liu, and X. Liao. A hybrid model of single valued neutrosophic sets and rough sets: single valued neutrosophic rough set model. Soft Computing (2016) 1-15. doi:10.1007/s00500-016-2356-y. Received : April 11, 2018. Accepted : April 25, 2018. [29] K. Mondal, S. Pramanik, and F. Smarandache. Several trigonometric Hamming similarity measures of rough neutrosophic sets and their applications in decision making. In New Trends in Neutrosophic Theory and Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application 86 Neutrosophic Sets and Systems, Vol. 20, 2018 University of New Mexico A novel approach to nano topology via neutrosophic sets 1 M. Lellis Thivagar, 2 Saeid Jafari 3 V. Sutha Devi 4 V. Antonysamy 1,3,4 School of Mathematics, Madurai Kamaraj University, Madurai -625021, Tamilnadu, India. E-mail1 : mlthivagar@yahoo.co.in , E-mail3 : vsdsutha@yahoo.co.in, E-mail 4 : tonysamsj@yahoo.com 2 Department of Mathematics, College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark. E-mail2 : jafaripersia@gmail.com Abstract: The main objective of this study is to introduce a new hybrid intelligent structure called Neutrosophic nano topology. Fuzzy nano topology and intuitionistic nano topology can also be deduced from the neutrosophic nano topology. Based on the neutrosophic nano approximations we have classified neutrosophic nano topology. Some properties like neutrosophic nano interior and neutrosophic nano closure are derived. Keywords and phrases: Neutrosophic sets, Fuzzy sets, Intuitionistic sets, Neutrosophic nano topology, Fuzzy nano topology, Intuitionistic nano topology 2010 AMS SUBJECT CLASSIFICATION: 54A05 1 INTRODUCTION Nano topology explored by Thivagar et.al can be described as a collection of nano approximations, a non-empty finite universe and empty set for which equivalence classes are buliding blocks. It is named as nano topology because whatever may be the size of the universe it has at most five open sets. After this, there has been many models built upon different aspect, i.e, universe, relations, object and operators. One of the interesting generalizations of the theories of fuzzy sets and intuitionistic fuzzy sets is the theory of neutrosophic sets introduced by F.Smarandache. Neutrosophic set is described by three functions : a membership function, indeterminacy function and a nonmembership function that are independently related. The theories of neutrosophic set have achieved greater success in various areas such as medical diagnosis, database, topology, image processing and decision making problem. While the neutrosophic set is a powerful tool to deal with indeterminate and inconsistent data, the theory of rough set is a powerful mathematical tool to deal with incompleteness. Neutrosophic sets and rough sets are two different topics, none conflicts the other. The main objective of this study is to introduce a new hybrid intelligent structure called neutrosophic nano topology. The significance of introducing hybrid structures is that the computational techniques, based on any one of these structures alone, will not always yield the best results but a fusion of two or more of them can often give better results. The rest of this paper is organized as follows. Some preliminary concepts required in our work are briefly recalled in section 2. In section 3 , the concept of neutrosophic nano topology is investigated. Section 4 concludes the paper with some properties on neutrosophic nano interior and neutrosophic nano closure. M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets Neutrosophic Sets and Systems, Vol. 20, 2018 2 87 Preliminaries The following recalls requisite ideas and preliminaries necessitated in the sequel of our work. Definition 2.1 [8]: Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U , R) is said to be the approximation space. Let X ⊆ U . (i) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by LR (X). ∪ That is, LR (X) = {R(x) : R(x) ⊆ X}, where R(x) denotes the equivalence x∈U class determined by x. (ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly ∪classified as X with respect to R and it is denoted by UR (X). That {R(x) : R(x) ∩ X ̸= φ}. is, UR (X) = x∈U (iii) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R and it is denoted by BR (X). That is, BR (X) = UR (X) − LR (X). Remark 2.2 [8]: If (U , R) is an approximation space and X, Y ⊆ U , then the following statements hold: (i) LR (X) ⊆ X ⊆ UR (X). (ii) LR (φ) = UR (φ) = φ and LR (U) = UR (U) = U . (iii) UR (X ∪ Y ) = UR (X) ∪ UR (Y ). (iv) UR (X ∩ Y ) ⊆ UR (X) ∩ UR (Y ) (v) LR (X ∪ Y ) ⊇ LR (X) ∪ LR (Y ). (vi) LR (X ∩ Y ) = LR (X) ∩ LR (Y ). (vii) LR (X) ⊆ LR (Y ) and UR (X) ⊆ UR (Y ), whenever X ⊆ Y . (viii) UR (X C ) = [LR (X)]C and LR (X C ) = [UR (X)]C . (ix) UR UR (X) = LR UR (X) = UR (X). (x) LR LR (X) = UR LR (X) = LR (X). Definition 2.3 [8]: Let U be an universe, R be an equivalence relation on U and τR (X) = {U, φ, LR (X), UR (X), BR (X)} where X ⊆ U . τR (X) satisfies the following axioms: (i) U and φ ∈ τR (X). (ii) The union of the elements of any sub-collection of τR (X) is in τR (X). (iii) The intersection of the elements of any finite sub-collection of τR (X) is in τR (X). M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets Neutrosophic Sets and Systems, Vol. 20, 2018 88 That is, τR (X) forms a topology on U called the nano topology on U with respect to X. We call (U , τR (X)) as the nano topological space. The elements of τR (X) are called nano-open sets. Proposition 2.4 [8]: Let U be a non-empty finite universe and X ⊆ U . Then the following statements hold: (i) If LR (X) = φ and UR (X) = U , then τR (X) = {U, φ}, is the indiscrete nano topology on U . (ii) If LR (X) = UR (X) = X, then the nano topology, τR (X) = {U, φ, LR (X)}. (iii) If LR (X) = φ and UR (X) ̸= U , then τR (X) = {U, φ, UR (X)}. (iv) If LR (X) ̸= φ and UR (X) = U , then τR (X) = {U, φ, LR (X), BR (X)}. (v) If LR (X) ̸= UR (X) where LR (X) ̸= φ and UR (X) ̸= U , then τR (X) = {U , φ, LR (X), UR (X), BR (X)} is the discrete nano topology on U . Definition 2.5 [3]: Let X be a non empty set. A fuzzy set A is an object having the form A = {< x : µA (x), x ∈ X}, where 0 ≤ µA (x) ≤ 1 represent the degree of membership of each x ∈ X to the set A. Definition 2.6 [2]: Let X be a non empty set. An intuitionstic set A is of the form A = {< x : µA (x), νA (x), x ∈ X}, where µA (x) and νA (x) represent the degree of membership function and the degree of non membership respectively of each x ∈ X to the set A and 0 ≤ µA (x) + νA (x) ≤ 1 for all x ∈ X. Definition 2.7 [6]: Let X be an universe of discourse with a generic element in X denoted by x, the neutrosophic set is an object having the form A = {< x : µA (x), σA (x), νA (x) >, x ∈ X}, where the functions µ, σ, ν : X → [0, 1] define respectively the degree of membership or truth , the degree of indeterminancy, and the degree of non-membership (or Falsehood) of the element x ∈ X to the set A with the condition. −0 ≤ µA (x) + σA (x) + νA (x) ≤ 3. 3 Neutrosophic Nano Topological Space In this section we introduce the notion of neutrosophic nano topology by means of nano neutrosophic nano approximations namely neutrosophic nano lower, neutrosophic nano upper and neutrosophic nano boundary. From Neutrosophic nano topology we have also defined and deduced intuitionistic nano topology and fuzzy nano topology. Definition 3.1 : Let U be a non-empty set and R be an equivalence relation on U . Let F be a neutrosophic set in U with the membership function µF , the indeterminancy function σF and the non-membership function νF . The neutrosophic nano lower, neutrosophic nano upper approximation and neutrosophic nano boundary of F in the approximation (U , R) denoted by N (F ), N (F )and BN (F ) are respectively defined as follows: (i) N (F ) = {< x, µR(A) (x), σR(A) (x), νR(A) (x) > /y ∈ [x]R , x ∈ U }. (ii) N (F ) = {< x, µR(A) (x), σR(A) (x), νR(A) (x) > /y ∈ [x]R , x ∈ U }. (iii) BN(F)= N (F ) − N (F ). M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets 89 Neutrosophic Sets and Systems, Vol. 20, 2018 where µR(A) (x) = µR(A) (x) = ∨ ∧ y∈[x]R y∈[x]R µA (y), σR(A) (x) = µA (y), σR(A) (x) = ∨ ∧ y∈[x]R y∈[x]R σA (y), νR(A) (x) = σA (y), νR(A) (x) = ∧ ∨ y∈[x]R y∈[x]R νA (y). νA (y). Definition 3.2 : Let U be an universe, R be an equivalence relation on U and F be a neutrosophic set in U and if the collection τN (F ) = {0N , 1N , N (F ), N (F ), BN (F )} forms a topology then it is said to be a neutrosophic nano topology. We call (U , τN (F )) as the neutrosophic nano topological space. The elements of τN (F ) are called neutrosophic nano open sets. Remark 3.3 : From Neutrosophic nano topology we can deduce and define the fuzzy nano topology and intuitionistic nano topology. Fuzzy nano topology is obtained by considering the membership values alone whereas in case of intuitionistic nano topology both membership and non member ship values are considered. Definition 3.4 : Let U be a non-empty set and R be an equivalence relation on U . Let F be an intuitionistic set in U with the membership function µF and the nonmembership function νF . The intuitionistic nano lower, intuitionistic nano upper approximation and intuitionistic nano boundary of F in the approximation (U , R) denoted by I(F ), I(F )and BI (F ) are respectively defined as follows: (i) I(F ) = {< x, µR(A) (x), νR(A) (x) > /y ∈ [x]R , x ∈ U }. (ii) I(F ) = {< x, µR(A) (x), νR(A) (x) > /y ∈ [x]R , x ∈ U }. (iii) BI (F )= I(F ) − I(F ). ∧ ∨ where µRI (A) (x) = y∈[x]R µA (y), νRI (A) (x) = y∈[x]R νA (y). µR(A) (x) = ∨ y∈[x]R µA (y), νRI (A) (x) = ∧ y∈[x]R νA (y). Definition 3.5 : Let U be an universe, R be an equivalence relation on U and F be an intuitionistic set in U and if the collection τI (F ) = {0N , 1N , I(F ), I(F ), BI (F )} forms a topology then it is said to be a intuitionistic nano topology. We call (U , τI (F )) as the intuitionistic nano topological space. The elements of τI (F ) are called intuitionistic nano open sets. Definition 3.6 : Let U be a non-empty set and R be an equivalence relation on U . Let F be a fuzzy set in U with the membership function µF . Then the fuzzy nano lower, fuzzy nano upper approximation of F and fuzzy nano boundary of F in the approximation (U, R) denoted by F(F ), F(F )and BF (F ) are respectively defined as follows: (i) F(F ) = {< x, µR(A) (x) > /y ∈ [x]R , x ∈ U }. (ii) F(F ) = {< x, µR(A) (x) > /y ∈ [x]R , x ∈ U }. (iii) BF (F )= F(F ) − F(F ). ∧ ∨ where µR(A) (x) = y∈[x]R µA (y), µR(A) (x) = y∈[x]R µA (y) Definition 3.7 : Let U be an universe, R be an equivalence relation on U and F be a fuzzy set in U and if the collection τF (F ) = {0N , 1N , F(F ), F(F ), BF (F )} forms a topology then it is said to be a fuzzy nano topology. We call (U , τF (F )) as the fuzzy nano topological space. The elements of τF (F ) are called fuzzy nano open sets. M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets Neutrosophic Sets and Systems, Vol. 20, 2018 90 Remark 3.8 : Thus from the above definitions of intuitionistic and fuzzy nano topologies we can assure that throughout this paper all the properties and examples also holds good when it is possible for neutrosophic nano topology. Remark 3.9 : Since our main purpose is to construct tools for developing neutrosophic nano topological spaces, we must introduce 0N , 1N and certain neutrosophic set operations in X as follows: Definition 3.10 : Let U be a nonempty set and the neutrosophic sets A and B in the form A = {< x : µA (x), σA (x), νA (x) >, x ∈ U }, B = {< x : µB (x), σB (x), νB (x) >, x ∈ U}. Then the following statements hold: (i) 0N = {< x, 0, 0, 1 >: x ∈ U} and 1N = {< x, 1, 1, 0 >: x ∈ U}. (ii) A ⊆ B iff µA (x) ≤ µB (x), σA (x) ≤ σB (x), νA (x) ≥ νB (x)f or all x ∈ U }. (iii) A = B iff A ⊆ Band B ⊆ A. (iv) AC = {< x, νA (x), 1 − σA (x), µA (x) >: x ∈ U}. (v) A ∩ B = {x, µA (x) ∧ µB (x), σA (x) ∧ σB (x), νA (x) ∨ νB (x)f or all x ∈ U }. (vi) A ∪ B = {x, µA (x) ∨ µB (x), σA (x) ∨ σB (x), νA (x) ∧ νB (x)f or all x ∈ U }. Theorem 3.11 [8]: Let U be a non-empty finite universe and X ⊆ U . Let τR (X) be the nano topology on U with respect to X. Then [τR (X)]C , whose elements are AC for A ∈ τR (X), is a topology on U . Remark 3.12 : [τN (F )]C is called the dual neutrosophic nano topology of τN (F ). Elements of [τN (F )]C are called neutrosophic nano closed sets. Thus, we note that a neutrosophic set N(G) of U is neutrosophic nano closed in τN (F ) if and only if U − N (G) is neutrosophic nano open in τN (F ). Example 3.13 : Let U = {p1 , p2 , p3 } be the universe of discourse. Let U /R = {{p1 , p2 }, {p3 }} be an equivalence relation on U and A = {< p1 , (0.7, 0.6, 0.5) >, < p2 , (0.3, 0.4, 0.5) >, < p3 , (0.1, 0.5, 0.1) >} be a neutrosophic set on U then N (A) = {< p1 , (0.3, 0.4, 0.5) >, < p2 , (0.3, 0.4, 0.5) >, < p3 , (0.1, 0.5, 0.1) >}, N (A) = {< p1 , (0.7, 0.6, 0.5) > , < p2 , (0.7, 0.6, 0.5) >, < p3 , (0.1, 0.5, 0.1) >} , B(A) = {< p1 , (0.5, 0.6, 0.5) >, < p2 , (0.5, 0.6, 0.5) > , < p3 , (0.1, 0.5, 0.1) >}. Then the collection τN (A) = {0N , 1N , {< p1 , (0.3, 0.4, 0.5) >, < p2 , (0.3, 0.4, 0.5) >, < p3 , (0.1, 0.5, 0.1) >}, {< p1 , (0.7, 0.6, 0.5) >, < p2 , (0.7, 0.6, 0.5) > , < p3 , (0.1, 0.5, 0.1) >}, {< p1 , (0.5, 0.6, 0.5) >, < p2 , (0.5, 0.6, 0.5) >, < p3 , (0.1, 0.5, 0.1) > }} is a neutrosophic nano topology on U and [τN (A)]C is also a neutrosophic nano topology on U . Thus τI (A) = {0N , 1N , {< p1 , (0.3, 0.5) >, < p2 , (0.3, 0.5) >, < p3 , (0.1, 0.1) > }, {< p1 , (0.7, 0.5) >, < p2 , (0.7, 0.5) >, < p3 , (0.1, 0.1) >}, {< p1 , (0.5, 0.5) >, < p2 , (0.5, 0.5) > , < p3 , (0.1, 0.1) >}} and τF (A) = {0N , 1N , {< p1 , (0.3) >, < p2 , (0.3) >, < p3 , (0.1) > }, {< p1 , (0.7) >, < p2 , (0.7) >, < p3 , (0.1) >}, {< p1 , (0.5) >, < p2 , (0.5) >, < p3 , (0.1) > }} are the intuitionistic nano topology and fuzzy nano topology. Remark 3.14 : In neutrosophic nano topological space, the neutrosophic nano boundary cannot be empty. Since the difference between neutrosophic nano upper and neutrosophic nano lower approximations is defined here as the maximum and minimum of the values in the neutrosophic sets. M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets Neutrosophic Sets and Systems, Vol. 20, 2018 91 Proposition 3.15 : Let U be a non-empty finite universe and F be a neutrosophic set on U . Then the following statements hold: (i) The collection τN (F ) = {0N , 1N }, is the indiscrete neutrosophic nano topology on U. (ii) If N (F ) = N (F ) = N (F ), then the neutrosophic nano topology, τN (F ) = {0N , 1N , N (F ), BN (F )}. (iii) If N (F ) = BN (F ), then τN (F ) = {0N , 1N , N (F ), N (F )} is a neutrosophic nano topology (iv) If N (F ) = BN (F ) then τN (F ) = {0N , 1N , N (F ), BN (F )}. (v) The collection τN (F ) = {0N , 1N , N (F ), N (F ), BN (F )} is the discrete neutrosophic nano topology on U . 4 Neutrosophic nano closure and interior In this section we have defined neutrosophic nano closure and neutrosophic nano interior on neutrosophic nano topological space. Based on this we also prove some properties. Definition 4.1 : If (U , τN (F )) is a neutrosophic nano topological space with respect to neutrosophic subset of U and if A be any neutrosophic subset of U , then the neutrosophic nano interior of A is defined as the union of all neutrosophic nano open subsets of A and it is denoted by NF int(A). That is, NF int(A) is the largest neutrosophic nano open subset of A. The neutrosophic nano closure of A is defined as the intersection of all neutrosophic nano closed sets containing A and it is denoted by NF cl(A). That is, NF cl(A) is the smallest neutrosophic nano closed set containing A. Remark 4.2 : Let (U , τN (F )) be a neutrosophic nano topological space with respect to F where F is a neutrosophic subset of U . The neutrosophic nano closed sets in U are 0N ,1N , (N (F ))C , (N (F ))C and (BN (F ))C . Theorem 4.3 [8]: Let (U , τR (X)) be a nano topological space with respect to X ⊆ U then N cl(X) = U . Remark 4.4 : The above theorem need not be true for all neutrosophic nano topological space (U , τN (F )) with respect to F where F is a neutrosophic subset of U . That is NF cl(A) need not be equal to U which can be shown by the following example. Example 4.5 : Let U = {p1 , p2 , p3 , p4 , p5 } be the universe of discourse. Let U /R = {{p1 , p4 }, {p2 , p3 }, {p5 }} be an equivalence relation on U and A = {< p1 , (0.2, 0.3, 0.4) > , < p4 , (0.2, 0.3, 0.4) >, < p5 , (0.4, 0.6, 0.2) >} be a neutrosophic set on U . Then N (A) = {< p1 , (0.2, 0.3, 0.4) >, < p4 , (0.2, 0.3, 0.4) >, < p5 , (0.4, 0.6, 0.2) >}, N (A) = {< p1 , (0.2, 0.3, 0.4) > , < p4 , (0.2, 0.3, 0.4) >, < p5 , (0.4, 0.6, 0.2) >} B(A) = {< p1 , (0.2, 0.3, 0.4) >, < p4 , (0.2, 0.3, 0.4) > , < p5 , (0.2, 0.4, 0.4) >}. Now we have τN (A) = {0N , 1N , {< p1 , (0.2, 0.3, 0.4) >, < p4 , (0.2, 0.3, 0.4) >, < p5 , (0.4, 0.6, 0.2) >}, {< p1 , (0.2, 0.3, 0.4) >, < p4 , (0.2, 0.3, 0.4) > , < p5 , (0.2, 0.4, 0.4) >}} which is a neutrosophic nano topology on U . [τN (A)]c = {0N , 1N , {< p1 , (0.2, 0.3, 0.4) >, < p4 , (0.2, 0.3, 0.4) >, < p5 , (0.4, 0.6, 0.2) >}, {< p1 , (0.2, 0.3, 0.4) > , < p4 , (0.2, 0.3, 0.4) >, < p5 , (0.2, 0.4, 0.4) >}. Here NF cl(A) ̸= U M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets Neutrosophic Sets and Systems, Vol. 20, 2018 92 Theorem 4.6 : Let (U , τN (F )) be a neutrosophic nano topological space with respect to F where F is a neutrosophic subset of U . Let A and B be neutrosophic subsets of U . Then the following statements hold: (i) A ⊆ NF cl(A). (ii) A is nano closed if and only if NF cl(A) = A. (iii) NF cl(0N ) = 0N and NF cl(1N ) = 1N . (iv) A ⊆ B ⇒ NF cl(A) ⊆ NF cl(B). (v) NF cl(A ∪ B) = NF cl(A) ∪ NF cl(B). (vi) NF cl(A ∩ B) ⊆ NF cl(A) ∩ NF cl(B). (vii) NF cl(NF cl(A)) = NF cl(A). Proof : (i) By definition of neutrosophic nano closure, A ⊆ NF cl(A). (ii) If A is neutrosophic nano closed, then A is the smallest neutrosophic nano closed set containing itself and hence NF cl(A) = A. Conversely, if NF cl(A) = A, then A is the smallest neutrosophic nano closed set containing itself and hence A is neutrosophic nano closed. (iii) Since 0N and 1N are neutrosophic nano closed in (U , τN (F )), NF cl(0N ) = 0N and NF cl(1N ) = 1N . (iv) If A ⊆ B, since B ⊆ NF cl(B), then A ⊆ NF cl(B). That is, NF cl(B) is a Neutrosophic nano closed set containing A. But NF cl(A) is the smallest Neutrosophic nano closed set containing A. Therefore, NF cl(A) ⊆ NF cl(B). (v) Since A ⊆ A ∪ B and B ⊆ A ∪ B, NF cl(A) ⊆ NF cl(A ∪ B) and NF cl(B) ⊆ NF cl(A ∪ B). Therefore, NF cl(A) ∪ NF cl(B) ⊆ NF cl(A ∪ B). By the fact that A ∪ B ⊆ NF cl(A) ∪ NF cl(B), and since NF cl(A ∪ B) is the smallest nano closed set containing A ∪ B, soNF cl(A ∪ B) ⊆ NF cl(A) ∪ NF cl(B). Thus, NF cl(A ∪ B) = NF cl(A) ∪ NF cl(B). (vi) Since A ∩ B ⊆ A and A ∩ B ⊆ B, NF cl(A ∩ B) ⊆ NF cl(A) ∩ NF cl(B). (vii) Since NF cl(A) is nano closed, NF cl(NF cl(A)) = NF cl(A). Theorem 4.7 : (U , τN (F )) be a neutrosophic nano topological space with respect to F where F is a neutrosophic subset of U . Let A be a neutrosophic subset of U . Then (i) 1N − NF Int(A) = NF cl(1N − A). (ii) 1N − NF cl(A) = NF Int(1N − A). Remark 4.8 : Taking complements on either side of(i) and (ii) Theorem 4.8, we get (NF Int(A)) = 1N − NF cl(1N − A)) and (NF cl(A)) = 1N − (NF Int(1N − A)). Example 4.9 : Let U = {a, b, c} and U /R = {{a, b}, {c}}. Let F = {< a, (0.4, 0.5, 0.5) > , < b, (0.4, 0.5, 0.5) >, < c, (0.5, 0.5, 0.5) >} be a neutrosophic set on U then the τN (A) = {0N , 1N , {< a, (0.4, 0.5, 0.5) >, < b, (0.4, 0.5, 0.5) >, < c, (0.5, 0.5, 0.5) >}} is a neutrosophic nano topology on U . [τN (A)]c = {0N , 1N , {< a, (0.5, 0.5, 0.4) >, < b, (0.5, 0.5, 0.4) > , < c, (0.5, 0.5, 0.5) >}}. If A = {< a, (0.7, 0.6, 0.5) >, < b, (0.3, 0.4, 0.5) >, < c, (0.7, 0.5, 0.5) > }, then (NF Int(A))C = 1N NF cl(1N − A) = 1N . That is, 1N − NF Int(A) = NF cl(1N − A) Also, 1N − NF cl(A) = NF Int(1N − A) = 0N M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets Neutrosophic Sets and Systems, Vol. 20, 2018 93 Theorem 4.10 : Let (U , τN (F )) be a neutrosophic nano topological space with respect to F where F is a neutrosophic subset of U . Let A and B be neutrosophic subsets of U , then the following statements hold: (i) A is neutrosophic nano open if and only if NF Int(A) = A. (iii) NF Int(0N ) = 0N and NF Int(1N ) = 1N . (iv) A ⊆ B ⇒ NF Int(A) ⊆ NF Int(B). (v) NF Int(A) ∪ NF Int(B) ⊆ NF Int(A ∪ B). (vi) NF Int(A ∩ B) = NF Int(A) ∩ NF Int(B). (vii) NF Int(NF Int(A)) = NF Int(A). Proof : (i) A is neutrosophic nano open if and only if 1N − A is neutrosophic nano closed, if and only if NF cl(1N − A) = 1N − A, if and only if 1N − NF cl(1N − A) = A if and only if NF Int(A) = A, by Remark 4.8. (ii) Since 0N and 1N are neutrosophic nano open, NF Int(0N ) = 0N and NF Int(1N ) = 1N . (iii) A ⊆ B ⇒ 1N − B ⊆ 1N − A. Therefore, NF cl(1N − B) ⊆ NF cl(1N − A). That is, 1N − NF cl(1N − A) ⊆ 1N − NF cl(1N − B). That is, NF IntA ⊆ NF IntB. Proof of (iv), (v) and (vi) follow similarly from Theorem 4.7 and Remark 4.8. Conclusion: Neutrosophic set is a general formal framework, which generalizes the concept of classic set, fuzzy set, interval valued fuzzy set, intuitionistic fuzzy set, and interval intuitionistic fuzzy set. Since the world is full of indeterminacy, the neutrosophic nano topology found its place into contemporary research world. This paper can be further developed into several possible such as Geographical Information Systems (GIS) field including remote sensing, object reconstruction from airborne laser scanner, real time tracking, routing applications and modeling cognitive agents. In GIS there is a need to model spatial regions with indeterminate boundary and under indeterminacy. Hence this neutrosophic nano topological spaces can also be extended to a neutrosophic spatial region. References [1] Albowi S.A, Salama.A, and Mohamed Eisa.,New concepts of neutrosophic sets, International Journal Of Mathematics and Computer Applications Research, Vol.3, Issue 3, Oct (2013),95-102. [2] Atanassov.K, Intuitionistic fuzzy sets, Fuzzy sets and systems, 20(1986), 87-96. [3] Chang.C.L., Fuzzy topological spaces, J.Math.Anal.Appl. 24(1968),182-190. [4] Hanfay.I, Salama.A and Mahfouz.K, Correlation of neutrosophic data, International Refreed Journal Of Engineering and Science, Vol.(1), Issue 2, (2012) 33-33. [5] Jensen .R.,and Shen.Q.,Fuzzy - rough sets assisted attribute selection, IEEE Transactions on Fuzzy system., (2007), 15, 73-89. M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets 94 Neutrosophic Sets and Systems, Vol. 20, 2018 [6] Salama.A and Alblowi.S.A., Generalized neutrosophic set and generalized neutrosophic topological spaces, Journal Computer Sci. Engineering, Vol.2 No.7, (2012), 129-132. [7] Smarandache.F, ”A unifying field in logics neutrosophy neutrosophic probability, set and Logic”, Rehoboth American Research Press 1999. [8] Lellis Thivagar.M and Carmel Richard.,On nano forms of weakly open sets, International Journal of Mathematics and Statistics Invention, Vol.1 No.1,(2013), 31-37. [9] Lellis Thivagar.M and Sutha Devi.V,On Multi-granular nano topology, South East Asian Bulletin of Mathematics, Springer Verlag., Vol.40, (2016), 875-885. [10] Lellis Thivagar.M and Sutha Devi.V,Computing technique for recruitment process via nano topology, Sohag J. Math.,(2016) 3, No. 1, 37-45. [11] M. Lellis Thivagar, Paul Manuel and V.Sutha Devi,A detection for patent infringement suit via nano topology induced by graph Cogent Mathematics, Taylor and Francis, (2016), Vol.3:1161129. [12] Xiao.Q.M., Zhang.Z.L.,Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform.Sciences., (2006), 725-733. Received : April 19, 2018. Accepted : May 7, 2018. M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy. A novel approach to nano topology via neutrosophic sets 95 Neutrosophic Sets and Systems, Vol. 19, 2018 University of New Mexico NC-VIKOR Based MAGDM Strategy under Neutrosophic Cubic Set Environment 1 2 3 4 Surapati Pramanik , Shyamal Dalapati , Shariful Alam , Tapan Kumar Roy , 1 Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District –North 24 Parganas, Pin code-743126, West Bengal, India. E-mail: sura_pati@yahoo.co.in 2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal, India. E-mail: dalapatishyamal30@gmail.com 3 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal, India. E-mail: salam50in@yahoo.co.in 4 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal, Abstract. Neutrosophic cubic set consists of interval neutrosophic set and single valued neutrosophic set simultaneously. Due to its unique structure, neutrosophic cubic set can express hybrid information consisting of single valued neutrosophic information and interval neutrosophic information simultaneously. VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) strategy is an important decision making strategy which selects the optimal alternative by utilizing maximum group utility and minimum of an individual regret. In this paper, we propose VIKOR strategy in neutrosophic cubic set environment, namely NC-VIKOR. We first define NC-VIKOR strategy in neutrosophic cubic set environment to handle multi-attribute group decision making (MAGDM) problems, which means we combine the VIKOR with neutrosophic cubic number to deal with multi-attribute group decision making problems. We have proposed a new strategy for solving MAGDM problems. Finally, we solve MAGDM problem using our newly proposed NC-VIKOR strategy to show the feasibility, applicability and effectiveness of the proposed strategy. Further, we present sensitivity analysis to show the impact of different values of the decision making mechanism coefficient on ranking order of the alternatives. Keywords: MAGDM, NCS, NC-VIKOR strategy. 1. Introduction Smarandache [1] introduced neutrosophic set (NS) by defining the truth membership function, indeterminacy function and falsity membership function as independent components by extending fuzzy set [2] and intuitionistic fuzzy set [3]. Each of three independent component of NS belons to [-0, 1+]. Wang et al. [4] introduced single valued neutrosophic set (SVNS) where each of truth, indeterminacy and falsity membership degree belongs to [0, 1]. Many researchers developed and applied the NS and SVNS in various areas of research such as conflict resolution [5], clustering analysis [6-9], decision making [10-39], educational problem [40, 41], image processing [42-45], medical diagnosis [46, 47], social problem [48, 49]. Wang et al. [50] proposed interval neutrosophic set (INS). Ye [51] defined similarity measure of two interval neutrosophic sets and applied it to solve multi criteria decision making (MCDM) problem. By combining SVNS and INS Jun et al. [52], and Ali et al. [53] proposed neutrosophic cubic set (NCS). Thereafter, Zhan et al. [54] presented two weighted average operators on NCSs and applied the operators for MADM problem. Banerjee et al. [55] introduced the grey relational analysis based MADM strategy in NCS environment. Lu and Ye [56] proposed three cosine measures between NCSs and presented MADM strategy in NCS environment. Pramanik et al. [57] defined similarity measure for NCSs and proved its basic properties and presented a new multi criteria group decision making strategy with linguistic variables in NCS environment. Pramanik et al. [58] proposed the score and accuracy functions for NCSs and prove their basic properties. In the same study, Pramanik et al. [59] developed a strategy for ranking of neutrosophic cubic numbers (NCNs) based on the score and accuracy functions. In the same study, Pramanik et al. [58] first developed a TODIM (Tomada de decisao interativa e multicritévio), called the NC-TODIM and presented new NC-TODIM [58] strategy for solving (MAGDM) in NCS environment. Shi and Ye [59] introduced Dombi aggregation operators of NCSs and applied them for MADM problem. Pramanik et al. [60] proposed an ex- Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 96 i. ii. tended technique for order preference by similarity to ideal solution (TOPSIS) strategy in NCS environment for solving MADM problem. Ye [61] present operations and aggregation method of neutrosophic cubic numbers for MADM. Pramanik et al. [62] presented some operations and properties of neutrosophic cubic soft set. Opricovic [63] proposed the VIKOR strategy for a MAGDM problem with conflicting attributes [64-65]. In 2015, Bausys and Zavadskas [66] extended the VIKOR strategy to INS environment and applied it to solve MCDM problem. Further, Hung et al. [67] proposed VIKOR method for interval neutrosophic MAGDM. Pouresmaeil et al. [68] proposed an MAGDM strategy based on TOPSIS and VIKOR in SVNS environment. Liu and Zhang [69] extended VIKOR method in neutrosophic hesitant fuzzy set environment. Hu et al. [70] proposed interval neutrosophic projection based VIKOR method and applied it for doctor selection. Selvakumari et al. [70] proposed VIKOR Method for decision making problem using octagonal neutrosophic soft matrix. VIKOR strategy in NCS environment is yet to appear in the literature. Research gap: MAGDM strategy based on NC-VIKOR. This study answers the following research questions: Is it possible to extend VIKOR strategy in NCS environment? Is it possible to develop a new MAGDM strategy based on the proposed NC-VIKOR method in NCS environment? Motivation: The above-mentioned analysis [64-69] describes the motivation behind proposing a novel NC-VIKOR method based MAGDM strategy under the NCS environment. This study develops a novel NC-VIKOR based MAGDM strategy that can deal with multiple decision-makers. The objectives of the paper are: i. To extend VIKOR strategy in NCS environment. ii. To define aggregation operator. iii. To develop a new MAGDM strategy based on proposed NC-VIKOR in NCS environment. To fill the research gap, we propose NC-VIKOR strategy, which is capable of dealing with MAGDM problem in NCS environment. The main contributions of this paper are summarized below: i. We developed a new NC-VIKOR strategy to deal with MAGDM problems in NCS environment. ii. We introduce a neutrosophic cubic number aggregation operator and prove its basic properties. iii. In this paper, we develop a new MAGDM strategy based on proposed NC-VIKOR method under NCS environment to solve MAGDM problems. iv. In this paper, we solve a MAGDM problem based on proposed NC-VIKOR method. The remainder of this paper is organized as follows: In the section 2, we review some basic concepts and operations related to NS, SVNS, NCS. In Section 3, we develop a novel MAGDM strategy based on NCVIKOR to solve the MADGM problems with NCS environment. In Section 4, we solve an illustrative numerical example using the proposed NC-VIKOR in NCS environment. Then in Section 5, we present the sensitivity analysis. The conclusions of the whole paper and further direction of research are given in Section 6. 2. Preliminaries Definition 1. Neutrosophic set Let X be a space of points (objects) with a generic element in X denoted by x, i.e. x  X. A neutrosophic set [1] A in X is characterized by truth-membership , indeterminacy-membership function t A (x) function iA ( x ) and falsity-membership function f A (x) , where t A ( x ) , iA ( x ) , f A ( x ) are the functions from X   to ] 0, 1  [ i.e. t A , iA , f A : X  ] 0, 1  [ that means t A (x) , iA ( x ) , f A ( x ) are the real standard or nonstandard subset of ] 0, 1  [. Neutrosophic set can be expressed as A = {<x , ( t A (x) , iA (x) , f A ( x ) )>:  x  X} and  0  tA (x)  iA (x)  f A (x)  3 . Example 1. Suppose that X = { x1 , x 2 , x 3 , ...,x n } be the universal set of n points. Let A1 be any neutrosophic set in X. Then A 1 expressed as A1 = {< x1 , (0.7, 0.4, 0.3)>: x 1  X}. Definition 2. Single valued neutrosophic set Let X be a space of points (objects) with a generic element in X denoted by x. A single valued neutrosophic set [4] B in X is expressed as: B = {< x: ( t B ( x ) , i B ( x) , f B ( x ) )>: x  X}, where t B ( x ) , i B (x) , f B ( x )  [0, 1]. For each x  X, t B ( x ) , i B ( x) , f B ( x )  [0, 1] and 0  t B (x) + i B (x ) + f B (x)  3. Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 97 Definition 3. Interval neutrosophic set ~ ~ A1 (h)  A 2 (h) = {< h, [max{ t A~1 (h), t A~ 2 (h)},max ~ An interval neutrosophic set [50] A of a non empty set H is expreesed by truth-membership function t A~ (h ) the indeterminacy membership function i A~ ( h ) and falsity membership function f A~ (h ) . For each h  H, ~ t A~ (h ) , i A~ (h ) , f A~ (h )  [0, 1] and A defined as follows:     ~ A = {< h, [ t A~ (h ) , t A~ (h ) ], [ i A~ ( h ) , i A~ ( h ) ],  [ f A~ (h ) , f A~ (h ) ]:  h  H}.   Here, t A~ (h ) , t A~ (h ) , iA~ (h ) , i A~ (h ) , f A~ (h) , f A~ (h ) : H  ]  0, 1  [ and  0  sup t A~ (h )  sup iA~ (h )  sup f A~ (h )  3 .     Here, we consider t A~ (h ) , t A~ (h ) , i A~ (h ) , i A~ ( h ) , f A~ (h ) , f A~ (h ) : H  [0, 1] for real applications.  Example 2. Assume that H = { h1, h2 , h3 ,... , hn} be a non-empty set. ~ be any interval neutrosophic set. Then Let A 1 ~ expressed as ~ = {< : [0.30, 0.70], [0.20, 0.45], h1 A1 A1 [0.18, 0.39]: h H}.  Definition 4. Neutrosophic cubic set A neutrosophic cubic set [52, 53] in a non-empty set H ~ is defined as N = {< h, A(h ) , A(h) >:  h  H}, where ~ A and A are the interval neutrosophic set and neutrosophic set in H respectively. Neutrosophic cubic ~ set can be presented as an order pair N = < A , A >, then we call it as neutrosophic cubic (NC) number. Example 3. Suppose that H = { h1, h2 , h3 ,... , hn} be a non-empty set. Let N1 be any NC-number. Then N1 can be expressed as N1 = {< h1 ; [0.35, 0.47], [0.20, 0.43], [0.18, 0.42], (0.7, 0.3, 0.5)>: h1 H}.  Some operations of NC-numbers: [52, 53] { t A~1 (h), t A~ 2 (h)}], [min { iA~1 (h), iA~2 (h)}, min { i A~1 (h),     i A~ 2 (h)}], [min { f A~1 (h), f A~2 (h)}, min { f A~1 (h),  f A~ 2 (h)}]>: h  H} and A1 (h)  A 2 (h) = {< h, max { t A1 (h), t A2 (h)}, min { iA1 (h), iA 2 (h)}, min { f A1 (h), f A 2 (h)}>:  h H}.  Example 4. Assume that N1 = < [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4)> and N 2 = < [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6)> be two NC-numbers. Then N1  N2 = < [0.56, 0.7], [0.17, 0.42], [0.15, 0.26], (0.7, 0.3, 0.4)>. ii. Intersection of any two NC-numbers Intersection of N1 and N2 denoted by N1  N2 is defined as follows: ~ ~ N1  N2 = < A1 (h)  A2 (h), A1 (h)  A2 (h) h H ~ ~ >, where A1 (h)  A 2 (h) = {< h, [min { t A~ 1 (h), t A~ 2 (h)}, min { t A~1 (h), t A~ 2 (h)}], [max { iA~1 (h), iA~ 2 (h)}, max { i A~1 (h), i A~ 2 (h)}], [max { f A~1 (h), f A~ 2 (h)}, max { f A~1 (h),  f A~ 2 (h)}]>: h  H} and A1 (h)  A 2 (h) = {< h, min { t A1 (h), t A2 (h)}, max { iA1 (h), iA 2 (h)}, max { f A1 (h),  f A 2 (h)}>:  h H}. Example 5. Assume that N1 = < [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5)> and N 2 = < [0.67, 0.75], [0.22, 0.44], [0.17, 0.21], (0.8, 0.4, 0.4)> be two NC numbers. Then N1  N2 = < [0.45, 0.57], [0.22, 0.33], [0.18, 0.46], (0.7, 0.3, 0.4)>. iii. Compliment of a NC-number ~ Let N1   A1 , A1  be a NCS in H. Then compliment ~ ~ of N1   A1 , A1  is denoted by N1c = {< h, A1c (h), A1c (h)>:  h  H}. ~ c Here, A1 = {< h, [ t A~ c (h), t A~ c (h)], [ iA~ c (h), iA~ c (h)], 1 i. Union of any two NC-numbers ~ ~ Let N1   A1 , A1  and N2   A 2 , A 2  be any two NC-numbers in a non-empty set H. Then the union of N1 and N 2 denoted by N1  N2 is defined as follows: [f  ~ A1c (h), f  ~ A1c 1 H}, where, t (h)]>:  h 1  ~ A1c 1 (h) = {1} -     t A~1 (h), t A~1c (h) = {1} - t A~1 (h), iA~ c (h) = {1} - iA~1 (h), 1      iA~ c (h) = {1} - i A~1 (h), f A~ c (h) = {1} - f A~1 (h), f A~ c (h) 1 1 ~ ~ N1  N2 = < A1 (h)  A 2 (h), A1 (h)  A 2 (h) h H >, where Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment 1 Neutrosophic Sets and Systems, Vol. 20, 2018 98  = {1} - f A~1 (h), and t A1c (h) = {1} - t A1 (h), iA´1c (g) = {1} - iA1 (h), f A1c (h) = = {1} - f A1 (h). Example 6. Assume that N1 be any NC-number in H in the form: N1 = < [.45, .57], [.27, .33], [.18, .46], (.7, .3, .5)>. Then compliment of N1 is obtained as N1c = < [0.18, 0.46], [0.67, 0.73], [0.45, 0.57], (0.5,0.7, 0.7) >. iv. Containment ~ Let N1   A1 , A1  = {< h, [ t A~ 1 ( h ) , t A~ (h ) ], [ iA~1 (h ) , 1  ~ A1 (h ) , f A~ (h) ], ( t A1 (h ), i A1 (h ), f A1 (h ) ) >: 1 ~ h H} and N2   A 2 , A 2  = {< h, [ t A~ 2 (h ) , t A~ (h ) ], iA~1 ( h ) ], [ f 2 ~ A2  ~ A2 [ i (h) , i ( h ) ], [ f ~ A2 (h ) , f  ~ A2 ( h ) ],  a ij if j G (2) a *ij    1 a ij if j C Where, aij is the performance rating of i th alternative for attribute  j and max aj is the maximum performance rating among alternatives for attribute  j . VIKOR strategy The VIKOR strategy is an MCDM or multi-criteria decision analysis strategy to deal with multi-criteria optimization problem. This strategy focuses on ranking and selecting the best alternatives from a set of feasible alternatives in the presence of conflicting criteria for a decision problem. The compromise solution [63, 64] H} ( t A 2 (h ), i A 2 (h ), f A 2 (h ) ) >: h reflects a feasible solution that is the closest to the ideal, be any two NC-numbers in a non-empty set H, then, (i) N 1  N 2 if and only if and a compromise means an agreement established by mutual concessions. The Lp -metric is used to develop  t (h)  t A~2 (h ) , t A~1 (h)  t A~2 (h ) , ~ A1 the stategy [65]. The VIKOR strategy is developed using the following form of L p –metric   iA~1 (h )  iA~2 (h) , iA~1 ( h )  iA~ 2 (h ) , and t A1 (h )  t A 2 (h ),  i A1 (h )  i A 2 (h ), f A1 (h )  f A 2 (h ) for all h H.  1  p  ;i 1,2,3,....,m. Definition 7. Let N1= < [a1, a2], [b1, b2], [c1, c2], (a, b, c) > and N2 = < [d1, d2], [e1, e2], [f1, f2], (d, e, f) > be any two NCnumbers, then distance [58] between them is defined by D (N1, N2) = 1 [ a 1  d 1  a 2  d 2  b1  e1  b 2  e 2  9 c1  f 1  c 2  f 2  a  d  b  e  c  f ] (1) Definition 2.14: Procedure of normalization In general, benefit type attributes and cost type attributes can exist simultaneously in MAGDM problem. Therefore the decision matrix must be normalized. Let  1 p p n  L pi      j   ij /  j   j     j 1     f (h )  f A~2 (h) , f A~1 (h )  f A~ 2 (h ) ~ A1 a ij be a NC-numbers to express the rating value of i-th alternative with respect to j-th attribute (  j). When attribute  j C or  j  G (where C and G be the set of cost type attribute and set of benefit type attributes respectively) The normalized values for cost type attribute and benefit type attribute are calculated by using the following expression (2). In the VIKOR strategy, L1i (as Si) and Li , i (as Ri ) are utilized to formulate ranking measure. The solution obtained by min Si reflects the maximum group utility (‘‘majority” rule), and the solution obtained by min Ri indicates the minimum individual regret of the “opponent”. Suppose that each alternative is evaluated by each criterion function, the compromise ranking is prepated by comparing the measure of closeness to the ideal alternative. The m alternatives are denoted as A1, A2, A3, ..., Am. For the alternative Ai, the rating of the j th aspect is denoted by  ij , i.e.  ij is the value of j th criterion function for the alternative Ai; n is the number of criteria. The compromise ranking algorithm of the VIKOR strategy is presented using the following steps: Step 1: Determine the best  j and the worst  j values of all criterion functions j =1, 2,..., n . If the Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment 99 Neutrosophic Sets and Systems, Vol. 20, 2018 j-th function represents a benefit then: j  max ij , j  min ij i i Step 2: Compute the values Si and Ri ; i = 1, 2,..., m, by these relations:     max w      /      , n Si   w j j  ij / j  j , j 1 Ri  j j j  j ij  j Here, wj is the weight of the criterion that expressss its relative importance. Step 3: Compute the values Qi: i = 1, 2,..., m, using the following relation: Qi  Si  S / S  S  1   R i  R  / R   R .         Here, S  max Si , S  min Si i i   R  max R i , R  min R i i i Here, v represents ‘‘the decision making mechanism coefficient” (or ‘‘the maximum group utility”). Here we consider v = 0.5 . Step 4: Preference ranikng order of the the alternatives is done by sorting the values of S, R and Q in decreasing order. 3. VIKOR strategy for solving MAGDM problem in NCS environment In this section, we propose a MAGDM strategy in NCS environment. Assume that  {1 ,  2 , 3 ,..., r } be a set of r alternatives and  {1 , 2 , 3 , ...,s } be a set of s attributes. Assume that W {w1 , w 2 , w 3 , ...,w s } be the weight vector of the attributes, where w k  0 s and  w k  1 . Assume that E {E1 , E2 , E3 ,..., EM } be k 1 the set of M decision makers and   { 1 ,  2 ,  3 , ..., M } be the set of weight vector of M decision makers, where  p  0 and   p 1 . p 1 The proposed MAGDM strategy consists of the following steps: Step: 1. Construction of the decision matrix Let DM p = (aijp) r  s (p = 1, 2, 3, …, t) be the p-th decision matrix, where information about the alternative  i provided by the decision maker or expert E p with respect to attribute  j (j = 1, 2, 3, …, s). The p-th decision matrix denoted by DM p (See Equation (3)) is constructed as follows:     1  2 ...  s    a p a p ... a p  1s  DM p   1 p11 p12 (3)  2sp   2 a 21 a22 .  . ... .  a p a p ... .a p  r2 rs r1  r  Here p = 1, 2, 3,…, M; i = 1, 2, 3,…, r; j = 1, 2, 3,…, s. Step: 2. Normalization of the decision matrix In decision making situation, cost type attributes and benefit type attributes play an important role to select the best alternative. Cost type attributes and benefit type attributes may exist simultaneously, so the decision matrices need to be normalized. We use Equation (2) for normalizing the cost type attributes and benefit type attributes. After normalization, the normalized decision matrix (Equation (3)) is represented as follows (see Equation 4):    1 DM p    2 .  r 1 2 ... s   p p p a *11 a *12 ... a *1s  p p p  a * 21 a * 22 a* 2s  . ... .   p p p a * r1 a * r 2 ... . a * rs  (4) Here, p = 1, 2, 3,…, M; i = 1, 2, 3,…, r; j = 1, 2, 3,…, s. Step: 3. Aggregated decision matrix For obtaining group decision, we aggregate all the individual decision matrices ( DM p , p 1, 2,..., M) to an aggregated decision matrix (DM) using the neutrosophic cubic numbers weighted aggregation (NCNWA) operator as follows: a ij  NCNWA  ( a 1ij , a ij2 ,... , a ijM )  (1a 1ij   2 a ij2   3a 3ij  ...  M a ijM ) = M M M  M   [   p t ij( p ) ,   p t ij( p ) ],[   p iij( p ) ,   p iij( p ) ], p 1 p 1 p 1  p 1 M M M M M  [   p f ij( p ) ,   p f ij ( p ) ], (   p t ij( p ) ,   p iij( p ),   p f ij( p ) ]   (5) p 1 p 1 p 1 p 1 p 1  The NCNWA operator satisfies the following properties: 1. Idempotency 2. Monotoncity 3. Boundedness Property: 1. Idempotency If all a 1ij , a ij2 ,... , a ijM  a are equal, then a ij  NCNWA (a1ij , a ij2 ,... ,a ijM )  a Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 100 Proof: Since a 1ij  a ij2  ...  a ijM  a , based on the Equation (5), we get a ij  NCNWA  ( a 1ij a ij2 ... a ijM )  (1a 1ij   2 a ij2   3a 3ij  ...  M a ijM ) = (1a   2 a   3a  ...  M a ) = M M M M    [ t    p , t    p ],[ i    p , i    p ], p 1 p 1 p 1  p 1  [ f    p , f    p ], ( t   p , i   p , f   p ]   p 1 p 1 p 1 p 1 p 1   M M M     M  M  =  [t , t ],[i , i ],[ f , f ], ( t, i, f ])   a. Property: 3. Monotonicity Assume that { a 1ij , a ij2 , .. , a ijM } and { a *ij1 , a *ij2 , ..., a *ijM } be any two set of collections of M NC-numbers with the condition a ijp  a *ijp (p = 1, 2, ..., M), then NCNWA  ( a 1ij , a ij2 ,..., a ijM )  NCNWA  ( a *ij1 , a *ij2 ,..., a *ijM ). Proof: From the given condition t ij( p)  t ij*( p) , we have  (p) p ij  t  *(p)   p t ij M M p 1 p 1  *( p )    p tij( p )    p tij . From the given condition tij ( p)  tij *( p) , we have  (p) p ij  t M  p t ij   t p 1 M M p 1 p 1  *( p )    p tij p 1 . *(p)  p t ij(p)   p t ij M M p 1 p 1 . From the given condition iij( p)  iij*( p) , we have *(p)  p i ij(p)   p i ij M M p 1 p 1 *( p )    p iij( p )    p iij . From the given condition tij( p)  tij*( p) , we have *(p)  p f ij(p)   p f ij M M p 1 p 1 *( p )    p f ij( p )    p f ij From the above relations, we obtain NCNWA  ( a 1ij , a ij2 ,..., a ijM )  NCNWA  ( a *ij1 , a *ij2 ,..., a *ijM ). Property: 2. Boundedness Let { a 1ij , a ij2 , ...,a ijM } be any collection of M NC-numbers. If p [min {f p ( p ) ij }, min {f p }],(max {t ijp}, min {iijp}, min {f ijp})  ( p ) ij p p p [max {f ij( p )}, max {f ij( p )}],(min {t ijp}, max {iijp}, max {f ijp})  . p M  *( p )    p iij p p p . Then, a -  NCNWA  ( a 1ij a ij2 ... a ijM )  a  . From the given condition iij( p)  iij *( p) , we have Proof: From Property 1 and Property 2, we obtain p 1 p (p) (p)  {iij ( p )}, max {iij ( p )}], a   [min {t ij },[min {t ij }],[max p p p  *(p)   i *( p )    p tij( p )    p tij i  p i ij (p)   p i ij ( p ) p ij . From the given condition tij( p)  tij*( p) , we have p M From the given condition iij( p)  iij*( p) , we have M  *( p )    p f ij( p )    p f ij ( p) (p)  {iij ( p )}, min {iij ( p )}], a   [max{t ij },[max{t ij }],[min p p  *(p) ( p ) p ij  *(p)  p f ij (p)   p f ij p 1 p  *(p)  p i ij (p)   p i ij M M p 1 p 1  *( p )    p iij( p )    p iij NCNWA  ( a 1ij , a ij2 ,..., a ijM )  NCNWA  ( a  , a  , ..., a  )  a  . From the given condition f ij( p)  f ij *( p) , we have p f  (p) ij  pf  *(p) ij M M p 1 p 1  *( p )    p f ij( p )    p f ij . and NCNWA  ( a 1ij , a ij2 ,..., a ijM )  NCNWA  ( a  , a  ,..., a  )  a  . So, we have a -  NCNWA  ( a 1ij , a ij2 ,..., a ijM )  a  . Therefore, the aggregated decision matrix is defined as follows: From the given condition f ij ( p)  f ij *( p) , we have Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment 101 Neutrosophic Sets and Systems, Vol. 20, 2018 1 2 ... .s      a  1 11 a 12 ... a 1s  DM    2 a 21 a 22 a 2s    ....  ..........    a a ... a  rs   r r1 r2 4. Illustrative example (6) Here, i = 1, 2, 3, …, r; j = 1, 2, 3, …, s; p =1, 2, …., M. Step: 4. Define the positive ideal solution and negative ideal solution a ij   [ max tij , max tij ], [ min i ij , min i ij ], i i i (7) i [ min f ij , min i ij ], ( max t ij , min f ij , min f ij )  i i i i i a ij   [min tij , min tij ],[max i ij , max i ij ], i i (8) i i [max f ij , max i ij ], (min t ij , max f ij , max f ij )  i i i i i Step: 5. Compute and  Zi i and represent the average and worst group Z i i scores for the alternative Ai respectively with the relations s w j  D (a ij , a *ij ) j 1 D (a ij , a ij ) i   To demonstrate the feasibility, applicability and effectiveness of the proposed strategy, we solve a MAGDM problem adapted from [51]. We assume that an investment company wants to invest a sum of money in the best option. The investment company forms a decision making board involving of three members (E1, E2, E3) who evaluate the four alternatives to invest money. The alternatives are Car company (  1 ), Food company (  2 ), Computer company (  3 ) and Arms company (  4 ). Decision makers take decision to evaluate alternatives based on the attributes namely, risk factor ( 1 ), growth factor ( 2 ), environment impact ( 3 ). We consider three criteria as benefit type based on Pramanik et al. [58]. Assume that the weight vector of attributes is W  (0.36, 0.37, 0.27)T and weight vector of decision makers or experts is   (0.26, 0.40, 0.34)T . Now, we apply the proposed MAGDM strategy using the following steps. (9)  *    w j  D (a ij , a ij )  Zi  max     j    D (a ij , a ij )  Here, wj is the weight of  j . (10) The smaller values of and correspond Z i to the i better average and worse group scores for alternative Ai , respectively. Step: 6. Calculate the values of i (i = 1, 2, 3, …, r) ( Zi  Z  ) (i    )    ( 1 ) i    (11) (Z  Z ) (    ) Here, i  min i , i  max i , i  i i  i Z  min Zi , Z  max Zi i i (12) and  depicts the decision making mechanism coefficient. If   0.5 , it is for “the maximum group utility”; If   0.5 , it is “ the minimum regret”; and it is both if   0.5 . Step: 7. Rank the priority of alternatives and  according Rank the alternatives by  i , Zi i to the rule of traditional VIKOR strategy. The smaller value reflects the better alternative. Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 102 Multi attribute group decision making problem Decision making analysis phase Construction of the decision matrix Normalization of the decision matrices Step-1 Step- 2 Aggregated decision matrix Step- 3 Define the positive ideal solution and negative ideal solution Step-4 Compute i and Z i Calculate the values of i Rank the priority of alternatives Step-5 Step- 6 Step- 7 Figure.1 Decision making procedure of proposed MAGDM method Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment 103 Neutrosophic Sets and Systems, Vol. 20, 2018 Step: 1. Construction of the decision matrix We construct the decision matrices as follows: …………………………………………………………………………………………………………………………….. Decision matrix for DM1 in NCN form    1   2  3  4 1 2 3   < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) >  < [.6, .8], [.2, .3], [.2, .4], (.8, .3, .4) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) >   < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.6, .8], [.2, .3], [.2, .4], (.8, .3, .4) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) >   < [.3, .4], [.5, .6], [.5, .7], (.4, .6, .7) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) >     1  2  3   4   1 2 3  < [.3, .4], [.5, .6], [.5, .7], (.4, .6, .7) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) >   < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) >  < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) >   < [.6, .8], [.2, .3], [.2, .4], (.8, .3, .4) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) >  (13) Decision matrix for DM2 in NCN form (14) Decision matrix for DM3 in NC-number form   1 2 3    1 < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) >      2 < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) >    3 < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.6, .8], [.2, .3], [.2, .4], (.8, .3, .4) > < [.6, .8], [.2, .3], [.2, .4], (.8, .3, .4) >    < [.7, .9], [.1, .2], [.1, .2], (.9, .2,.2) > < [.4, .5], [.4, .5], [.4, .5], (.5, .5, .5) > < [.3, .4], [.5, .6], [.5, .7], (.4, .6, .7) >   4  (15) Step: 2. Normalization of the decision matrix Since all the criteria are considered as benefit type, we do not need to normalize the decision matrices (DM 1, DM2, DM3). Step: 3. Aggregated decision matrix Using equation eq. (5), the aggregated decision matrix of (13, 14, 15) is presented below:    1   2  3   4 1 2 3   < [.44, .56], [.36, .46], [.36, .51], (.56, .46,.50) > < [.48, .60], [.32, .42], [.32, .42], (.60, .42,.42) > < [.62, .80], [.18, .28], [.18, .28], (.80, .28, .28) >  < [.45, .58], [.35, .45], [.35, .47], (.58, .45, .47) > < [.50, .64], [.30, .40], [.30, .40], (.64, .40, .40) > < [.60, .76], [.20, .30], [.20, .30], (.76, .30,.30) >   < [.62, .80], [.18, .28], [.18, .28], (.80, .28, .28) > < [.64, .84], [.16, .26], [.16, .32], (.84, .26, .32) > < [.47, .60], [.33, .43], [.33, .47], (.60, .43, .47) >   < [.56, .73], [.24, .34], [.24, .41], (.73, .34, .41) > < [.40, .50], [.40, .50], [.40, .50], (.50, .50, .50) > < [.56, .73], [.24, .34], [.24, .37], (.73, .34,.37) >  (16) Step: 4. Define the positive ideal solution and negative ideal solution The positive ideal solution a ij = 1 2 3 < [.62, .80],[.18, .28],[.18, .28], (.80, .28,.28) > < [.64, .84],[.16, .26],[.16, .32], (.84, .26,.32) > < [.62, .80],[.18, .28],[.18, .28], (.80, .28, .28) > and the negative ideal solution 1 a ij = 2 3 < [.44, .56], [.36, .46], [.36, .51], (.56, .46,.50) > < [.40, .50], [.40, .50], [.40, .50], (.50, .50,.50) > < [.47, .60], [.33, .43], [.33, .43], (.60, .43, .47) > ……………………………………………………………………………………………………………………………… And and  Step: 5. Compute Zi i  0.36  0.2   0.37  0.16   0.27  0  Using Equation (9) and Equation (10), we obtain Z  max  ,  ,    0.24,  0.36  0.2   0.37  0.16   0.27  0  1       0.43, 0.25   0.16   0.37    0.36  0.18   0.37  0.14   0.27  0.02  2       0.42,  0.37   0.25   0.16   0.36  0   0.37  0   0.27  0.19  3     0.32,    0.37   0.25   0.16   0.36  0.08   0.37  0.25   0.27  0.07  4       0.57. 0.16   0.37   0.25   1   0.37   0.25    0.16   0.36  0.18   0.37  0.14   0.27  0.02  Z 2  max    0.21, ,  ,   0.37   0.25   0.16   0.36  0   0.37  0   0.27  0.19  Z3  max  ,  ,    0.32,  0.37   0.25   0.16   0.36  0.08   0.37  0.25   0.27  0.07  Z 4  max    0.37. ,  ,  0.37   0.25   0.16  Step: 6. Calculate the values of i Using Equations (11), (12) and   0.5 , we obtain Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 104 (0.24  0.21 ) (0.43  0.32 )  0.5   0.31, 0.16 0.25 (0.42  0.32 ) (0.21  0.21)  0.2,  0.5  2  0.5 0.25 0.16 1  0.5   3  0.5   4  0.5 (0.32  0.32 ) (0.32  0.21)  0.5   0.34, 0.25 0.16 Step: 7. Rank the priority of alternatives The preference order of the alternatives based on the traditional rules of the VIKOR startegy is  2  1   3   4 . ………………………………………………………… (0.57  0.32 ) (0.37  0.21) .  0.5 1 0.25 0.16 ………………....................................................................................................................... ........................................... 5. The influence of parameter  Table 1 shows how the ranking order of alternatives ( i ) changes with the change of the value of  Values of Values of Preference order of alternatives i   = 0.1 1 = 0.22,  2 = 0.04,  3 = 0.62,  4 = 1  2  1   3   4  = 0.2 1 = 0.24,  2 = 0.08,  3 = 0.55,  4 = 1  2  1   3   4  = 0.3 1 = 0.26,  2 = 0.12,  3 = 0.48,  4 = 1  2  1   3   4  = 0.4 1 = 0.29,  2 = 0.16,  3 = 0.41,  4 = 1  2  1   3   4  = 0.5 1 = 0.31,  2 = 0.2,  3 = 0.34,  4 = 1  2  1   3   4  = 0.6 1 = 0.34,  2 = 0.24,  3 = 0.28,  4 = 1  2  3  1   4  = 0.7 1 = 0.36,  2 = 0.28,  3 = 0.21,  4 = 1 3   2  1   4  = 0.8 1 = 0.39,  2 = 0.32,  3 = 0.14,  4 = 1 3   2  1   4  = 0.9 1 = 0.42,  2 = 0.36,  3 = 0.07,  4 = 1 3   2  1   4 Table1. Values of  i (i = 1, 2, 3, 4) and ranking of alternatives for different values of  . …………………………………………………………………………………………………………………….. Figure 2 represents the graphical representation of alternatives ( A i ) versus  i (i = 1, 2, 3, 4) for different values of  . ……………………………………………………………………………………………………………………………. Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 105 Line of lowest values of  Line of greatest values of  0.95 0.57 0.38 values of  0.76 0.19 0.1 0.2 0.3 0.5 0.7 0.8 Va lue s 0.6 of  0.4 0.9 1 2 3 4 Fig 2. Graphical representation of ranking of alternatives for different values of  . ……………………………………………………………………………………………………………………………… [3] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets 6. Conclusions and Systems, 20 (1986), 87-96. In this paper, we have extended the traditional VIKOR [4] H. Wang, F. Smarandache, Y. Zhang, and R. strategy to NC-VIKOR. We introduced neutrosophic cubic Sunderraman. Single valued neutrosophic sets. 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[82] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22. [83] Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloud computing services. Future Generation Computer Systems, 86, 12-29. [84] Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066. [85] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116. Received : April 30, 2018. Accepted : May 10, 2018. Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, NC-VIKOR Based MAGDM under Neutrosophic Cubic Set Environment Neutrosophic Sets and Systems, Vol. 20, 2018 109 University of New Mexico Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 1 3 3 Surapati Pramanik , Rama Mallick , Anindita Dasgupta 1,3 Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District –North 24 Parganas, Pin code-743126, West Bengal, India. 1 E-mail: sura_pati@yahoo.co.in, 2Email: aninditadasgupta33@gmail.com 2 Umeschandra College, Department of Mathematics, Surya Sen Street ,Kolkata-700012, West Bengal, India, 1Email: ramamallick23@gmail.com Abstract Multi-attribute decision making (MADM) is a mathematical tool to solve decision problems involving conflicting attributes. With the increasing complexity, uncertainty of objective things and the neutrosophic nature of human thought, more and more attention has been paid to the investigation on multi attribute decision making in neutrosophic environment, and convincing research results have been reported in the literature. While modern algebra and number theory have well documented and established roots deep into India's ancient scholarly history, the understanding of the springing up of neutrosophics, specifically neutrosophic decision making, demands a closer inquiry. The objective of the study is to present a brief review of the pioneering contributions of personalities as diverse as those of P. P. Dey, K. Mondal, P. Biswas, D. Banerjee, S. Dalapati, P. K. Maji, A. Mukherjee, T. K. Roy, B. C. Giri, H. Garg, S. Bhattacharya. A survey of various concepts, issues, etc. related to neutrosophic decision making is discussed. New research direction of neutrosophic decision making is also provided. Keywords:Bipolar neutrosophic sets, VIKOR method, multi attribute group decision making. 1 Introduction Every human being has to make decision in every sphere of his/her life. So decision making should be pragmatic and elegant. Decision making involves multi attributes. Multi attribute decision making (MADM) refers to making selections among some courses of actions in the presence of multiple, usually conflicting attributes. MADM is the most well-known branch of decision making. To solve a MADM one needs to employ sorting and ranking (see Figure 1). It has been widely recognized that most real world decisions take place in uncertain environment where crisp values cannot capture the reflection of the complexity, indeterminacy, inconsistency and uncertainty of the problem. To deal with crisp MADM problem [1], classical set or crisp set [2] is employed. The classical MADM generally assumes that all the criteria and their respective weights are expressed in terms of crisp numbers and, thus, the rating and the ranking of the alternatives are determined. However, practical decision making problem involves imprecision or vagueness. Imprecision or vagueness may occur from different sources such as unquantifiable information, incomplete information, non-obtainable information, and partial ignorance. To tackle uncertainty, Zadeh [3] proposed the fuzzy set by introducing membership degree of an element. Different strategies [4-9] have been proposed for dealing with MADM in fuzzy environment. In fuzzy set, non-membership membership function is the complement of membership function. However, nonmembership function may be independent in real situation. Sensing this, Atanassov [10] proposed intuitionistic fuzzy set by incorporating nonmembership as an independent component. Many MADM strategies [11-14] in intuitionistic fuzzy environment have been studied in the literature. Deschrijver and Kerre [15] proved that intuitionistic fuzzy set is equivalent to interval valued fuzzy set [16], an extension of fuzzy set. In real world decision making often involves incomplete, indeterminate and inconsistent information. Fuzzy set and intuitionistic fuzzy set Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 cannot deal with the situation where indeterminacy component is independent of truth and falsity components. To deal with this situation, Smarandache [17] defined neutrosophic set. In 2005, Wang et al. [18] defined interval neutrosophic set. In 2010, Wang et al. [19] introduced the single valued neutrosophic set (SVNS) as a sub class of neutrosophic set. SVNS have caught much attention of the researchers. SVNS have been applied in many areas such as conflict resolution [20], decision making [21-30], image processing [31-33], medical diagnosis [34], social problem [35-36], and so on. In 2013, a new journal, “Neutrosophic Sets and Systems” came into being to propagate neutrosophic study, which can be seen in the journal website, namely, http://fs.gallup.unm.edu/nss. By hybridizing the concept of neutrosophic sets or SVNSs with the various established sets, several neutrosophic hybrid sets have been introduced in the literature such as neutrosophic soft sets [37], neutrosophic soft expert set [38], single valued neutrosophic hesitant fuzzy sets [39], interval neutrosophic hesitant sets [40], interval neutrosophic linguistic sets [41], rough neutrosophic set [42, 43], interval rough neutrosophic set [44], bipolar neutrosophic set [45], bipolar rough neutrosophic set [46], tri-complex rough neutrosophic set [47], hyper complex rough neutrosophic set [48], neutrosophic refined set [49], bipolar neutrosophic refined sets [50], neutrosophic cubic set [51], etc. So many new areas of decision making in neutrosophic hybrid environment began to emerge. Young researchers demonstrate great interest to conduct research on decision making in neutrosophic as well as neutrosophic hybrid environment. According to Pramanik [52], the concept of neutrosophic set was initially ignored, criticized by many [53, 54], while it was supported only by a very few, mostly young, unknown, and uninfluential researchers. As we see Smarandache [55, 55, 56, 57] leads from the front and makes the paths for research by publishing new books, journal articles, monographs, etc. In India, W. B. V. Kandasamy [58, 59] did many research works on neutrosophic algebra, neutrosophic cognitive maps, etc. She is a well-known researcher in neutrosophic study. Pramanik and Chackrabarti [36] and Pramanik [60, 61] did some work on neutrosophic related problems. Initially, publishing neutrosophic research paper in a recognized journal was a hard work. Pramanik and his colleagues were frustrated by the rejection of several neutrosophic research papers without any valid reasons. After the publication of the International Journal 110 namely, “Neutrosophic Sets and Systems” Pramanik and his colleagues explored the area of decision making in neutrosophic environment to establish their research work. In 2016, to present history of neutrosophic theory and applications, Smarandache [62] published an edited volume comprising of the short biography and research work of neutrosophic researchers. “The Encyclopedia of Neutrosophic Researchers” includes the researchers, who published neutrosophic papers, books, or defended neutrosophic master theses or Ph. D. dissertations. It encourages researchers to conduct study in neutrosophic environment. The fields of neutrosophics have been extended and applied in various fields, such as artificial intelligence, data mining, soft computing, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements, and decision making. Decision making in incomplete / indeterminate / inconsistent information systems has been deeply studied by the Indian researchers. New trends in neutrosophic theory and applications can be found in [62-67]. Considering the potentiality of SVNS and its various extensions and their importance of decision making, we feel a sense of commitment to survey the contribution of Indian mathematicians to multi attribute decision making. The venture is exclusively new and therefore it may be considered as an exploratory study. Research gap: Survey of new research in MADM conducted by the Indian researchers. Statement of the problem: Contributions of selected Indian researchers to multiattribute decision making in neutrosophic environment: An overview. Motivation: The above-mentioned analysis describes the motivation behind the present study. Objectives of the study The objective of the study is:  To present a brief review of the pioneering contributions of personalities as diverse as those Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 of Dr. Partha Pratim Dey, Dr. Pranab Biswas, Dr. Durga Banerjee, Mr. Kalyan Mondal, Shyamal Dalapati, Dr. P. K. Maji, Prof. T. K. Roy, Prof. B. C. Giri, Prof. Anjan Mukherjee, Dr. Harish Garg and Dr. Sukanto Bhattacharya. 111 Rest of the paper is organized as follows: In section 2, we review some basic concepts related to neutrosophic set. Section 3 presents the contribution of the selected Indian researchers. Section 4 presents conclusion and future scope of research. ......................................................................................................................................... For Single DecisionMa For Group king Decision Start Making Single decision maker Multiple decision makers Step1. Formulate the decision matrix Step1. Formulate the decisionmatrices Step2. Formulate weighted aggregated decision matrices Step3. Apply decision making method Step2. Apply decision making method Step4. Rank the priority Step3. Rank the priority Stop Figure 1. Decision making steps Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 112 2. Preliminaries In this section we recall some basic definitions related to this topic. ship degree of an element x  X to some implicit counter-property corresponding to a bipolar neutrosophic set P. Definition.2.1 Neutrosophic Set Definition 2.5: Neutrosophich hesitant fuzzy set Let 𝑋 be a fixed set, a neutrosophic hesitantfuzzy set [39] (NHFS) on X is defined as: M={<x,T(x),I(x),F(x)>|x ∈ 𝑋 },where T(x) ={ 𝛼|𝛼 ∈ 𝑇(𝑥)},I(x) ={𝛽|𝛽 ∈ 𝐼(𝑥)} and F(x) ={𝛾|𝛾 ∈ 𝐹(𝑥)} are the three sets of some different values in the interval [0, 1], which represent the possible truthmembership hesitant degree, indeterminacymembership hesitant degree, and falsity-membership hesitant degree of the element xϵX to the set M, and satisfies the following conditions: 𝛼𝜖[0,1], 𝛽𝜖[0,1], 𝛾𝜖[0,1] and 0 ≤ 𝑠𝑢𝑝 𝛼 + 𝑠𝑢𝑝𝛽 + 𝑠𝑢𝑝𝛾 ≤ 3 where 𝛼 = ⋃ ∈ ( ) 𝑚𝑎𝑥{𝛼} , 𝛽 = ⋃ ∈ ( ) 𝑚𝑎𝑥𝛽 and 𝛾 = ⋃ ∈ ( ) 𝑚𝑎𝑥{𝛾} for𝑥 ∈ 𝑋. The 𝑡𝑟𝑖𝑝𝑙𝑒𝑡 𝑚 = {𝑇(𝑥), 𝐼(𝑥), 𝐹(𝑥)} is called a neutrosophic hesitant fuzzy element (NHFE) which is the basic unit of the NHFS and is denoted by the symbol m={T, I, F}. Let X be the universe. A neutrosophic set (NS) [17] P in X is characterized by a truth membership function TP, an indeterminacy membership function IP and a falsity membership function FP whereTP, IP and FP are real standardor non-standard subset of ] -0,1+[. It can be defined as: P={<x,(TP(x),IP(x),FP(x))>:xϵX,TP,IP,FP ϵ]-0,1+[} There is no restriction on the sum ofTP(x),IP(x) and FP(x) and so 0-≤TP(x)+IP(x)+FP(x)≤3+. Definition 2.2 Single valued neutrosophic set Let X be a space of points (objects) with generic element in X denoted by x. A single valued neutrosophic set [19] P is characterized by a truth-membership functionTP(x), an indeterminacy-membership function IP(x), and a falsity-membership functionFP(x). For each point x in X, TP(x),IP(x),FP(x)  [0, 1]. A SVNS A can be written as: A = {<x:TP(x),IP(x),FP(x)>, x  X}. Definition 2.3 Interval valued neutrosophic set Let X be a space of points (objects) with generic elements in X denoted by x. An interval valued neutrosophic set [18] P is characterized by an interval truthmembership function TP(x)=[𝑇 , 𝑇 ], an interval indeterminacy-membership function IP(x)=[𝐼 , 𝐼 ], and an interval falsity-membership function FP(x)=[ 𝐹 , 𝐹 ]. For each point xϵX, TP(x), IP(x), FP(x)  [0, 1]. An IVNS P can be written as: P = {< x: TP(x),IP(x),FP(x)>x  X}. Definition 2.4: Bipolar neutrosophic set A bipolar neutrosophic set [45] P in X is defined as an object of the formP={<x, Tm (x),Im(x),Fm(x), n m m m T n ( x ) ,I (x), F n ( x ) >: x  X}, whereT , I ,F :X  [1, 0] and T n , I n , F n : X  [-1, 0] . The positive membership degree Tm (x), Im(x), Fm(x) denotes respectively the truth membership, indeterminate membership and false membership degree of an elecorresponding to a bipolar neutrosophment  X ic set P and the negative membership degree T n ( x ), In(x), F n ( x ) denotes respectively the truth membership, indeterminate membership and false member- Definition 2.6: Interval neutrosophic hesitant fuzzy set Let X be a nonempty fixed set, an Interval neutrosophic hesitant fuzzy set [67] onX is defined as : 𝑃 = {〈𝑥, 𝑇(𝑥), 𝐼(𝑥), 𝐹(𝑥)〉|𝑥 ∈ 𝑋}. Here𝑇(𝑥), 𝐼(𝑥) and 𝐹(𝑥) are sets of some different interval values in [0, 1], which denotes respectively the possible truth-membership hesitant degree, indeterminacy-membership hesitant degree, and falsity-membership hesitant degree of the element 𝑥 ∈ Ω to the set P. Then,T(x)={𝛼 |𝛼 ∈ 𝑇(𝑥)}, 𝑤here 𝛼 = [𝛼 , 𝛼 ] is an interval number; 𝛼 = 𝑖𝑛𝑓 𝛼 and 𝛼 = 𝑠𝑢𝑝𝛼 represents the lower and upper limits of 𝛼 , respectively; 𝐼(𝑥) = 𝛽 |𝛽 ∈ 𝐼(𝑥) , 𝑤here 𝛽 = [𝛽 , 𝛽 ] is an interval number; 𝛽 = inf 𝛽 and 𝛽 = sup 𝛽 represents the lower and upper limits of 𝛽 , respectively; F(x)= {𝛾|𝛾 ∈ 𝐹(𝑥) , where 𝛾 = [𝛾 , 𝛾 ]is an intervalnumber; 𝛾 = 𝑖𝑛𝑓𝛾 and, 𝛾 = 𝑠𝑢𝑝𝛾 represents the lower and upper limits of 𝛾, respectively and satisfied the condition 0 ≤ 𝑠𝑢𝑝𝛼 + 𝑠𝑢𝑝𝛽 + 𝑠𝑢𝑝𝛾 ≤ 3 where 𝛼 = ⋃ ∈ ( ) 𝑚𝑎𝑥{𝛼 } , 𝛽 = ⋃ ∈ ( ) 𝑚𝑎𝑥 𝛽 𝑎𝑛𝑑𝛾 = ⋃ ∈ ( ) 𝑚𝑎𝑥{𝛾} for𝑥 ∈ 𝑋. Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 The triplet 𝑝 = {𝑇(𝑥), 𝐼(𝑥), 𝐹(𝑥)} is called an interval neutrosophic hesitant fuzzy element or simply INHFE, which is denoted by the symbol 𝑝 = {𝑇, 𝐼, 𝐹}. Definition 2.7 Triangular fuzzy neutrosophic sets Let X be the finite universe and F [0, 1] be the set of all triangular fuzzy numbers on [0, 1]. A triangular fuzzy neutrosophicset (TFNS) [68] P with TP(x):X→ 𝐹[0,1],IP:X→ [0,1] and FP:X→ in X is defined as: P={<x:TP(x),IP(x),Fp(x)>,xϵX}, where TP(x):X → 𝐹[0,1] , IP:X → [0,1] and FP:X → [0,1] . The triangular fuzzy numbers TP(x) =(𝑇 , 𝑇 , 𝑇 ), IP(x)=(𝐼 , 𝐼 , 𝐼 ) and FP(x) =(𝐹 , 𝐹 , 𝐹 ), respectively, denotesrespectively the possible truthmembership, indeterminacy-membership and a falsity-membership degree of x in P and for every x  X 0≤ 𝑇 (𝑥) + 𝐼 (𝑥) + 𝐹 (𝑥) ≤ 3. The triangular fuzzy neutrosophic value (TFNV)P is symbolized by <(l,m,n),(p,q,r),(u,v,w)>where,(𝑇 (𝑥), 𝑇 (𝑥), 𝑇 (𝑥)) = (𝑙, 𝑚, 𝑛) , 𝐼 (𝑥), 𝐼 (𝑥), 𝐼 (𝑥) = (𝑝, 𝑞, 𝑟) and (𝐹 (𝑥), 𝐹 (𝑥), 𝐹 (𝑥)) = (u,v,w). Definition2.8Neutrosophic soft set Let V be an initial universe set and E be a set of parameters. Consider A ⊂ E. Let P( V ) denote the set of all neutrosophic sets of V. The collection ( F, A ) is termed to be the soft neutrosophic set [37] over V, where F is a mapping given by F : A → P(V). Definition 2.9 Neutrosophic cubic set Let U be the space of points with generic element in U denoted by u  U. A neutrosophic cubic set [51]in  = {< u, A (u),  (u) >: u  U} in U defined as N which A (u) is the interval valued neutrosophic set and  (u) is the neutrosophic set in U. A neutrosophic  = <A,  >. We use cubic set in U denoted by N  (U ) as a notation which implies that collection of CN all neutrosophic cubic sets in U. Definition 2.10 Rough Neutrosophic Sets Let X be a non empty set and R be an equivalence relation on X . Let P be a neutrosophic set in Y with the membership function TP, indeterminacy function IP and non-membership function FP. The lower and the upper approximations of P in the approximation (X, R) denoted 113 are respectively defined as by 𝐿(𝑃) 𝑎𝑛𝑑 𝐿(𝑃) follows: L( P )   x,T L( P )( x ), I L( P )( x ), F L( P )( x )  / y  [ x ] R ,x  X , L( P )   x,T L( P )( x ), I L( P )( x ), F L( P )( x )  / y  [ x ] R ,x  X , T L ( P ) ( x )   y  [ x ] R T P ( y ), I L (P ) (x)   y [x] R I P (y), F L( P) (x)   y [x] R F P (y), T L(P) ( x)   y [x]R T P ( y), IL(P)(x)  y[x]R I P(y), FL(P)(x)  y[x]R FP(y) So, 0  sup T L ( P ) ( x )  sup I L ( P ) ( x )  sup F L ( P ) ( x )  3. 0  sup T L ( P ) ( x )  sup I L ( P ) ( x )  sup F L ( P ) ( x )  3. Here  and  denotes “max” and “min’’ operators respectively. TP(y), IP(y) and FP(y) are the membership, indeterminacy and non-membership function of y with respect to P and also L (P ) and L ( P ) are two neutrosophic sets in X. Therefore, NS mapping L , L :L(X)  L(X) are, respectively, referred to as the lower and the upper rough NS approximation operators, and the pair ( L ( P ), L ( P )) is called the rough neutrosophic set [42] in (Y, R). Definition 2.11Refined Neutrosophic Sets LetX be a universe. A neutrosophic refined set (NRS) [49]A on X can be defined as follows:   x , (T A1 (x), T A2 (x ), ..., T Ap (x )), ( I 1A (x), I A2 (x ), ..., I Ap (x )),  A   p 1 2  ( F A (x ), F A (x), ..., F A (x))   Here, T A1 (x ), T A2 (x ), ..., T Ap (x) : X  [0,1], I 1A (x), I A2 (x), ..., I Ap (x) : X  [0,1], F A1 (x), T F F A2 (x), ..., 2 A F Ap (x) : X  [0,1] . For any 1 A (x ),T 1 A ( x ) , F A2 ( x ) , . . . , F Ap and x ϵ X  ,  I ( x ) , I ( x ) , . . . , I ( x )  and truth-membership ( x )  is the ( x ) , . . . , T Ap ( x ) 1 A 2 A p A sequence, indeterminacy-membership sequence and falsity-membership sequence of the element x, respectively. Section 3 The contribution of the selected Indian researchers Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 3.1 Dr. Partha Pratim Dey Dr. Partha Pratim Dey was born at Chak, P. O.Islampur, Murshidabad, West Bengal, India, PIN742304. Dr. Dey qualified CSIR-NET-Junior Research Fellowship (JRF) in 2008. His paper entitled“Fuzzy goal programming for multilevel linear fractional programming problem"coauthored with Surapati Pramanik was awarded as the best paper in West Bengal State Science and Technology Congress (2011) in mathematics. He obtained Ph. D. in Science from Jadavpur University, India in 2015.Title of his Ph. D. Thesis [70] is:“Some studies on linear and non-linear bi-level programming problems in fuzzy envieonment``. He continues his research in the feild of fuzzy multi-criteria decision making and extends them in neutrosophic environment. Curently, he is an assistant teacher of Mathematics in Patipukur Pallisree Vidyapith, Patipukur, Kolkata-48. His research interest includes decision making in neutrosophic environemnt and optimization. Contribution: In 2015, Dey, Pramanik, and Giri [71] proposed a novel MADM strategy based on extended grey relation analysis (GRA) in interval neutrosophic environment with unknown weight of the attributes. Maximizing deviation method is employed to determine the unknown weight information of the atributes. Dey et al. [71] also developed linguistic scale to transform linguistic variable into interval neutrosophic values. They employed the developed strategy for dealing with practical problem of selecting weaver for Khadi Institution. Partha Pratim Dey, coming from a weaver family, is very familiar with the parameters of weaving and criteria of selection of weavers. Several parameters are defined by Dey et al. [71] to conduct the study. Dey et al. [72] proposed a TOPSIS strategy at first in single valued neutrosophic soft expert set environmnet in 2015. Dey et al. [72] determined the weights of the parameters by employing maximizing 114 deviation method and demonstrated an illustrative example of teacher selection problem. According to Google Scholar Citation, this paper [72] has been cited by 15 studies so far. In 2015, Dey et al. [73] established TOPSIS startegy in generalized neutrosophic soft set environmnet and solved an illustrative MAGDM problem. In neutrosophic soft set environment, Dey et al. [74] grounded a new MADM strategy based on grey relational projection technique. In 2016, Dey et al. [75] developed two new strategies for solving MADM problems with interval-valued neutrosophic assessments. The empolyed measures [75] are namely, i) weighted projection measure and ii) angle cosine and projection measure. Dey et al. [76] defined Hamming distance function and Euclidean distance function between bipolar neutrosophic sets. In the same study, Dey et al. [76] defined bipolar neutrosophic relative positive ideal solution (BNRPIS) and neutrosophic relative negative ideal solution(BNRNIS) and developed an MADM strategy in bipolar neutrosophic environemnt. Deyet et al. [77] presented a GRA strategy for solving MAGDM problem under neutrosophic soft environment and solved an illustrative numerical example to show the effectiveness of the proposed strategy. In 2016, Dey et al. [78] discussed a solution strategy for MADM problems with interval neutrosophic uncertain linguistic information through extended GRA method. Dey et al. [78] also proposed Euclidean distance between two interval neutrosophic uncertain linguistic values. Pramanik, Dey, Giri, and Smarandache [79] defined projection, bidirectional projection and hybrid projection measures between bipolar neutrosophic sets in 2017 and proved their basic properties. In the same study [79], the same authors developed three new MADM strategies based on the proposed projection measures. They validated their result by solving a numerical example of MADM. In 2017, Pramanik, Dey, Giri, and Smarandache [80] defined some operation rules for neutrosophic cubic sets and introduced the Euclidean distance between them.nThe authors also defined neutrosophic cubic positive and negative ideal solutions and established a new MADM strategy. In 2018, Dey, Pramanik, Ye and Smarandache [81] introduced cross entropy and weighted cross entropy measures for bipolar neutro- Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 115 Neutrosophic Sets and Systems, Vol. 20, 2018 sophic sets and interval bipolar neutrosophic sets and proved their basic properties. The authors also developed two new multi-attribute decision-making strategies in bipolar and interval bipolar neutrosophic set environment. The authors solved two illustrative numerical examples and compared the obtained results with existing strategies to demonstrate the feasibility, applicability, and efficiency of their strategies. Pramanik, Dey and Giri [82] defined hybrid vector similarity measure between single valued refined neutrosophic sets (SVRNSs) and proved their basic properties and developed an MADM strategy and employed them to solve an illustrative example of MADM in SVRNS environment. Pramanik, Dey and Smarandache [83] defined the correlation coefficient measure Cor (L1, L2) between two interval bipolar neutrosophic sets (IBNSs) L1, L2 and proved the following properties: (1) Cor (L1, L2) = Cor (L2, L1) ; (2) 0  Cor (L1, L2)  1; (3) Cor (L1, L2) = 1, if L1= L2. In the same study, the authors defined weighted correlation coefficient measure Corw(L1, L2) between two IBNSs L1, L2 and established the following properties: (1) Corw(L1, L2) = Corw (L2, L1); (2) 0 Corw(L1, L2) 1; (3) Corw(L1, L2) = 1, if L1= L2. The authors [83] also developed a novel MADM straegy based on weighted correlation coefficient measure and empolyed to solve an investment problem and compared the solution with existing startegies. Pramanik, Dey, and Smarandache [84] defined Hamming and Euclidean distances measures, similarity measures based on maximum and minimum operators between two IBNSs and proved their basic properties. In the same research, Pramanik et al. [84] deveolped a novel MADM strategy in IBNS environment. In fuzzy environment, work of Dey and Pramanik [85] obtained the best paper award in Mathematics in 2011 at 18th West Bengal State Science & Technology Congress Tilte of the paper was:‘ Fuzzy goal programming for multilevel linear fractional programming problems’. In 2015, Dr. Dey obtained “Diploma Certificate” from Neutrosophic Science InternationalAssociation (NISA) for his outstanding performance in neutrosophic research. He was awarded the certificate of outstanding contribution in reviewing for the International Journal “Neutrosophic Sets and Systems“. His works in neutrosophics draw much attention of the researchers international level. According to “ResearchGate’’ a social networking site for scientists and researchers, citation of his research exceeds 200. He is an active member of ‘‘Indian society for neutrosophic study’’. Dr. Dey is very much intersted in neutrosophic study. He continues his research work with great mathematician like Prof. Florentin Smarandache and Prof. Jun Ye. 3.2 Kalyan Mondal Kalyan Mondal was born at Shantipur, Nadia, West Bengal, India, Pin-741404. He qualified CSIR-NETJunior Research Fellowship (JRF) in 2012. He is a research scholar in Mathematics of Jadavpur University, India since 2016. Title of his Ph. D. thesis is: “Some decision making models based on neutrosophic strategy”. His paper entiled “MAGDM based on contra-harmonic aggregation operator in neutrosophic number (NN) environment’’ coauthored with Surapsati Pramanik and Bibhas C. Giri was awarded outstanding paper in West Bengal State Science and Technology Congress (2018) in mathematics. He continues his research in the field neutrosophic multi-attribute decision making; aggregation operators; soft computing; pattern recognitions; neutrosophic hybrid systems, rough neutrosophic sets, neutrosophic numbers, neutrosophic game theory, neutrosophic algebraic structures. Presently, he is an assistant teacher of Mathematics in Birnagar High School (HS) Birnagar, Ranaghat, Nadia, Pin-741127, West Bengal, India. Contribution: In 2014, Mondal and Pramanik [86]initiated to study teacher selection problem using neutrosophic logic. Pramanik and Mondal [87] defined cosine similarity measure for rough neutrosophic sets as CRNS(A, B) between two rough neutrosophic sets A, B and established the following properties: (1) CRNS(A, B) = CRNS (B, A); (2) 0 CRNS(A, B) 1; (3) CRNS(A, B) = 1, iff A= B. In the same study, Pramanik and Mondal [87] applied cosine similarity measure for medical diagnosis. Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 Mondal et al. [88] proposed a rough cotangent similarity measure in 2015 and studied some of its basic properties. The authors demonstrated an application of cotangent similarity measure of rough neutrosophic sets for medical diagnosis. Pramanik and Mondal [89] introduced interval neutrosophic MADM strategy with completely unknown attribute weight information based on extended grey relational analysis. In 2015, Mondal and Pramanik [90] presents rough neutrosphic MADM strategy based on GRA. They also extended the neutrosophic GRA strategy to rough neutrosophic GRA strategy and applied it to MADM problem. The authors first defined accumulated geometric operator to transform rough neutrosophic number (neutrosophic pair) to single valued neutrosophic number. In 2015, Mondal and Pramanik [91] presented a neutrosophic MADM strategy for school choice problem. The authors used five criteria to modeling the school choice problem in neutrosophic environment. In 2015, Mondal and Prammanik [92] defined cotangent similarity measure for neutrosophic sets as COTNRS(N, P) between two refined neutrosophic sets N, P and established the following properties: (1) COTNRS(N, P) = COTNRS (P, N); (2) 0 COTNRS(N, P)  1; (3) COTNRS(P, N) = 1, if P = N. In the same study, Mondal and Pramanik [92] presented an application of cotangent similarity measure of neutrosophic single valued sets in a decision making problem for educational stream selection. Mondal and Pramanik [93] also defined rough accuracy score function and proved their basic properties. The authors also introduced entropy based weighted rough accuracy score value. The authors developed a novel rough neutrosophic MADM startegy with incompletely known or completely unknown attribute weight information based on rough accuracy score function. Pramanik and Mondal [94] presented rough Dice and Jaccard similarity measures between rough neutrosophic sets. The authors proposed weighted rough Dice and Jaccard similarity measures, and proved their basic properties. The authors presented an application of rough neutrosophic Dice and Jaccard similarity measures in medical diagnosis. 116 Mondal and Pramanik [95] defined tangent similarity measure and proved their basic properties. In the same study, Mondal and Pramanik developed a novel MADM strategy for MADM problems in SVNS environment. The authors resented two illustrattive exaxmples, namely selection of educational stream and medical diagnosis to demonstrate the feasibility, and applicability of the proposed MADM strategy. Mondal and Pramanik [96] studied the quality claybrick selection strategy based on MADM with single valued neutrosophic GRA.The authors used neutrosophic grey relational coefficient on Hamming distance between each alternative to ideal neutrosophic estimates reliability solution and ideal neutrosophic estimates unreliability solution. They also used neutrosophic relational degree to determine the ranking order of all alternatives. In 2015, Mondal and Pramanik [97] defined a refined tangent similarity measure strategy of refined neutrosophic sets and proved its basic properties. They presented an application of refined tangent similarity measure in medical diagnosis. Mondal and Pramanik [98] introduced cosine, Dice and Jaccard similarity measures of interval rough neutrosophic sets and proved their basic properties. They developed three MADM strategies based on interval rough cosine, Dice and Jaccard similarity measures and presented an illustrative example, namely selection of best laptop for random use. In 2016, Mondal and Pramanaik [47] defined rough tri-complex similarity measure in rough neutrosophic environment and proved its basic properties. In the same study, Mondal and Pramnaik [47] developed a novel MADM strategy for dealing with MADM problem in rough tri-complex neutrosophic envioronment. Mondal, Pramanik, and Smarandache [48] introduced the rough neutrosophic hypercomplex set and the rough neutrosophic hypercomplex cosine function in 2016, and proved their basic properties. They also defined the rough neutrosophic hyper-complex similarity measure and proved their basic properties. They also developed a new MADM strategy to deal with MADM problems in rough neutrosophic hyper-complex set environment. They presented a hypothetical application to the selection problem of best candidate for marriage for Indian context. Mondal, Pramanik, and Smarandache [99] defined rough trigonometric Hamming similarity measures and proved their basic properties. In the same study, Mondal et al. [99] developed a novel MADM strategies to solve MADM problems in rough Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 117 Neutrosophic Sets and Systems, Vol. 20, 2018 neutrosophic environment. The authors provided an application, namely selection of the most suitable smart phone for rough use. In 2017, Mondal, Pramanik and Smarandache [100] developed a new MAGDM strategy by extending the TOPSIS strategy in rough neutrosphic environment, called rough neutrosophic TOPSIS strategy for MAGDM. They also proposed rough neutrosophic aggregate operator and rough neutrosophic weighted aggregate operator. Finally, the authors solved a numerical example to demonstrate the applicability and effectiveness of the proposed TOPSIS startegy. Mondal, Pramanik, Giri and Smarandache [101] proposed neutrosophic number harmonic mean operator (NNHMO) and neutrosophic number weighted harmonic mean operator NNWHMO and cosine function to determine unknown criterion weights in neutrosophic number (NN) environment. The authors developed two strategies of ranking NNs based on score function and accuracy function. The authors also developed two novel MCGDM strategies based on the proposed aggregation operators. The authors solved a hypothetical case study and compared the obtained results with other existing strategies to demonstrate the effectiveness of the proposed MCGDM strategies. The significance of these stratigies is that they combine NNs with harmonic aggregation operators to cope with MCGDM problem. In 2018, Mondal, Pramanik and Giri [102] inroduced hyperbolic sine similarity measure and weighted hyperbolic sine similarity measure namely, SVNHSSM(A, B) for SVNSs. They proved the following basic properties. 1. 0  SVNHSSM(A, B)  1 2. SVNHSSM(A, B) = 1 if and only ifA = B 3. SVNHSSM (A, B) = SVNHSSM(B, A) 4. If R is a SVNS in X and A  B  R then SVNHSSM(A, R)  SVNHSSM(A, B) and SVNHSSM(A, R)  SVNHSSM(B, R). The authors also defined weighted hyperbolic sine similarity measure for SVNS namely, SVNWHSSM(A, B) and proved the following basicproperties. 1. 0  SVNWHSSM(A, B)  1 2. SVNWHSSM (A, B) = 1 if and only ifA = B 3. SVNWHSSM (A, B) = SVNWHSSM(B, A) 4. If R is a SVNS in X and A  B  R then SVNWHSSM (A, R)  SVNWHSSM(A, B) and SVNWHSSM (A, R)  SVNWHSSM (B, R). The authors defined compromise function to determine unknown weight of the attributes in SVNS environment. The authors developed a novel MADM strategy based on the proposed weighted similarity measure. Lastly, the authors solved a numerical example and compared the obtained results with the existing strategies to demonstrate the effectiveness of the proposed MADM strategy. Mondal, Pramanik, and Giri [103] defined tangent similarity measure and proved its properties in interval valued neutrosophic environment. The authors developed a novel MADM strategy based on the proposed tangent similarity measure in interval valued neutrosophic environment. The authors also solved a numerical example namely, selection of the best investment sector for an Indian government employee. The authors also presented a comparative analysis. Mondal et al. [104] employed refined neutrosophic set to express linguistic variables. The authors proposed linguistic refined neutrosophic set. The authors developed an MADM strategy based on linguistic refined neutrosophic set. The authors also proposed an entropy method to determine unknown weight of the criterion in linguistic neutrosophic refined set environment. They presented an illustrative example of constructional spot selection to show the feasubility and applicability of the proposed strategy. Mr. Kalyan Mondal is a young and hardworking researcher in neutrosophic field. He acts as an area editor of international journal,“Journal of New Theory” and acts as a reviewer for different international peer reviewed journals. In 2015, Mr. Mondal was awarded Diploma certificate from Neutrosophic Science InternationalAssociation (NISA) for his outstanding performance in neutrosophic research. He was awarded the certificate of outstanding contribution in reviewing for the International Journal “Neutrosophic Sets and Systems’’. His works in neutrosophics draw much attention of the researchers at international level. According to “Researchgate’’, citation of his research exceeds 430. 3.3 Dr. Pranab Biswas Pranab Biswas obtained his Bachelor of Science degree in Mathematics and Master degree in Applied Mathematics from University of Kalyani. He obtained Ph. D. in Science from Jadavpur University, India. Title of his thesis is “Multi-attribute decision making in neutrosophic environment”. Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 He is currently an assistant teacher of Mathematics. His research interest includes multiple criteria decision making, aggregation operators, soft computing, optimization, fuzzy set, intuitionistic fuzzy set, neutrosophic set. Contribution: In 2014, Biswas, Pramanik and Giri [105] proposed entropy based grey relational analysis strategy for MADM problem with single valued neutrosophic attribute values. In neutrosophic environment, this is the first case where GRAwas applied to solve MADM problem. The authors also defined neutrosophic relational degree. Lastly, the authors provided a numerical example to show the feasibility and applicability of the developed strategy. In 2014, Biswas et al. [106] introduced single –valued neutrosophic MADM strategy with incompletely known and completely unknown attribute weight information based on modified GRA.The authors also solved an optimization model to find out the completely unknown attribute weight by ustilizing Lagrange function. At the end, the authors provided an illustrative example to show the feasibility, practicalitry and effectiveness of the proposed strategy. Biswas et al. [69] introduced a new strategy called “Cosine similarity based MADM with trapezoidal fuzzy neutrosophic numbers”.The authors also established expected interval and the expected value for trapezoidal fuzzy neutrosophic number and cosine similarity measure of trapozidal fuzzy neutrosophic numbers. In 2015, Biswas et al. [107] extended TOPSIS strategy for MAGDM in neutrosophic environment. In the study, rating values of alternative are expressed by linguistic terms such as Good, Very Good, Bad, Very Bad, etc. and these terms are scaled with single-valued neutrosophic numbers. Single-valued neutrosophic set-based weighted averaging operator is used to aggregate all the individual decision maker’s opinion into one common opinion for rating the importance of criteria and alternatives. The authors provided an illustrative example to demonstrate the proposed TOPSIS strategy. Biswas et al. [108] further extened the TOPSIS strategy for MAGDM in single-valued neutrosophic environment. The authors developed a non-linear programming based strategy to study 118 MAGDM problem. In the same study, the authors converted the single valued neutrosophic numbers into interval numbers. The authors employed nonlinear programming model to determine the relative closeness co-efficient intervals of alternatives for each decision maker. Then, the closeness co-efficient intervals of each alternative are aggregated according to the weight of decision makers. Further, the authors developed a priority matrix with the aggregated intervals of the alternatives. The authors obtained the ranking order of all alternatives by computing the optimal membership degrees of alternatives with the ranking method of interval numbers. Finally, the authors presented an illustrative example to show the effectiveness of the proposed strategy. In 2015, Pramanik, Biswas, and Giri [109] proposed two new hybrid vector similarity measures of single valued and interval neutrosophic sets by hybriding the concept of Dice and cosine similarity measures.The authors also proved their basic properties. The authors also presented their applications in multi-attribute decision making in neutrosophic environment. Biswas et al. [110] proposed triangular fuzzy number neutrosophic sets by combining triangular fuzzy number with single valued neutrosophic set in 2016. Biswas et al. [110] also defined some of its operational rules. The authors defined triangular fuzzy number neutrosophic weighted arithmetic averaging operator and triangular fuzzy number neutrosophic weighted geometric averaging operator to aggregate triangular fuzzy number nuetrosophic set. The authors also established some of their properties of the proposed operators. The authors also presented an MADM strategy to solve MADM in triangular fuzzy number neutrosophic set environment. In 2016, Biswas et al. [111] defined score value, accuracy value, certainty value, and normalized Hamming distance of single valued neutrosophic hesitant fuzzy sets.The authors also defined positive ideal solution and negative ideal solution by score value and accuracy value. The authors calculated the degree of grey relational coefficent between each alternative and ideal alternative. The authors also determined a relative closeness coefficient to obtain the ranking order of all alternatives. Finally, the authors provided an illustrative example to show the Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 validity and effectiveness of the proposed grey relational analysis based MADM strategy in single valued neutrosophic hesitant fuzzy set environment. Biswas, Pramanik, and Giri [112] proposed a class of distance measures for single-valued neutrosophic hesitant fuzzy sets in 2016 and proved their properties with variational parameters. The authors applied weighted distance measures to calculate the distances between each alternative and ideal alternative in the MADM problems. The authors developed a MADM strategy based on the proposed distance functions in single valued neutrosophic hesitant fuzzy set environment. The authors provided an illustrative example to verify the proposed strategy and to show its fruitfulness. The authors also compared the proposed strategy with other existing startegies for solving MADM in single valued neutrosophic hesitant fuzzy set environment. Biswas et al. [113] introduced single-valued trapezoidal neutrosophic number (SVTrNN), which is a special case of single-valued neutrosophic number and developed a ranking method for ranking SVTrNNs. The authors presented some operational rules as well as cut sets of SVTrNNs. The authors defined the value and ambiguity indices of truth, indeterminacy, and falsity membership functions of SVTrNNs. Using the proposed ranking strategy and proposed indices, the authors developoed a new MADM strategy to solve MADM problem in which the ratings of the alternatives over the attributes are expressed in terms of TrNFNs. Finally, the authors provided an illustrative example to demonstrate the validity and applicability of the proposed MADM strategy with SVTrNNs. In 2016, Biswas et al.[114] introduced the concept of SVTrNN in the form: 𝐴 = 〈(𝑎 , 𝑎 , 𝑎 , 𝑎 ), (𝑏 , 𝑏 , 𝑏 , 𝑏 ), (𝑐 , 𝑐 , 𝑐 , 𝑐 ) 〉 ,where 𝑎 , 𝑎 , 𝑎 , 𝑎 , 𝑏 , 𝑏 , 𝑏 , 𝑏 , 𝑐 , 𝑐 , 𝑐 , 𝑐 are real numbers and satisfy the inequality 𝑐 ≤𝑏 ≤𝑎 ≤𝑐 ≤𝑏 ≤𝑎 ≤𝑎 ≤𝑏 ≤ 𝑐 ≤𝑎 ≤𝑏 ≤𝑐 . The authors defined some arithmetical operational rules. The authors also defined value index and ambiguity index of SVTrNNs and established some of their properties. The authors developed a ranking strategy with the proposed indicess to rank SVTrNNs. The authors developed a new MADM strategy to solve MADM problems in SVTrNN environment. Biswas et al. [115] extended the TOPSIS strategy of MADM problems in single-valued trapezoidal neutrosophic number environment. In their study, the attribute values are expressed in terms of single- 119 valued trapezoidal neutrosophic numbers. The authors deal with the situation where the weight information of attribute is incompletely known or completely unknown. The authors developed an optimization model using maximum deviation strategy to obtain the weight of the attributes. The authors also illustrated and validated the proposed TOPSIS strategy by solving a numerical example of MADM problems. Biswas et al. [116] introduced a new neutrosophic numbers called interval neutrosophic trapezoidal number (INTrN) characterized by interval valued truth, indeterminacy, and falsity membership degrees and defined some arithmetic operations on INTrNs, and normalized Hamming distance between INTrNs. In the same study, Biswas et al. [116] developed a new MADM strategy, where the rating values of alternatives over the attributes and the importance of weight of attributes assume the form of INTrNs. Biswas et al. [116] employed the entropy strategy to determine thr attribute weight and then used it to calculate aggregated weighted distance measure and determined ranking order of alternatives with the help of aggregated weighted distance measures. Biswas et al. [116] also solved an illustrative example to show the feasibility, applicability and effectiveness of the proposed strategy. Dr. Biswas’s work [117] obtained outstanding paper award at “Second Regional Science and Technology Congress, 2017’’ held at University of Kalyani, Nadia, West Bengal, India. His resesrch interest includes fuzzy, intuitionistic fuzzy and neutrosophic decision making. Dr. Pranab Biswas is a young and hardworking researcher in neutrosophic field. In 2015, Dr. Biswas was awarded “Diploma Certificate” from Neutrosophic Science International Association (NISA) for his outstanding performance in neutrosophic research. He was awarded the certificate of outstanding contribution in reviewing for the International Journal “Neutrosophic Sets and Systems’’ in 2018. According to “Researchgate’’, citation of his research exceeds 375. Research papers of Biswas et al. [105, 112] received the best paper award from “Neutrosophic Sets and Systems’’ for volume 2, 2014 and volume 12, 2016. His works in neutrosophics draw much attention of the researchers in national as well international level. His Ph. D. thesis entilted:“Multi-attribute decision making in neutrosophic environment” was awarded “Doctorate of Neutrosophic theory” by Indian Society for Neutrosophic Study (ISNS) with sponsorship by Neutrosophic Science International Association (NSIA). Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 3.4 Dr.Durga Banerjee 120 Association (NSIA). According to “Researchgate’’, citation of his research exceeds 55. 3.5 Shyamal Dalapati Durga Banerjee passed M. Sc. from Jadavpur University in 2005. In 2017, D. Banerjee obtained Ph. D. Degree in Science from Jadavpur University. Her research interest includes operations research, fuzzy optimization, and neutrosophic decision making. Title of her Ph. D. Thesis [118] is: “Some studies on decision making in an uncertain environment’’. Her Ph. D. thesis comprises of few chapters dealing with MADM in neutrosophic environment. Contribution: In 2016, Pramanik, Banerjee, and Giri [119] introduced refined tangent similarity measure.The authors presented an MAGDM model based on tangent similarity measure of neutrosophic refined set. The authors also introduced simplified form of tangent similarity measure. The authors defined new ranking method based on refined tangent similarity measure. Lastly, the authors solved a numerical example of teacher selectionin in neutrosophic refined set environment to see the effectiveness of the proposed strategy. In 2016, Banerjee et al.[120] developed TOPSIS startegy for MADM in refined neutrosophic environment. The authors also provided a numerical example to show the feasibility and applicability of the proposed TOPSIS strategy. In 2017, Banerjee, Pramanik, Giri and Smarandache [121] at first developed an MADM strategy in neutrosophic cubic set environment using grey relational analysis. The authors discussed about positive and negative grey relational coefficients,and weighted grey relational coefficients, Hamming distances for weighted grey relational coefficients and standard grey relational coefficient. Her Ph. D. thesis [118] entilted:“Multi-attribute decision making in neutrosophic environment” was awarded “Doctorate of Neutrosophic theory” by the Indian Society for Neutrosophic Study (ISNS) with sponsorship by Neutrosophic Science International Shyamal Dalapati qualified CSIR-NET-Junior Research Fellowship (JRF) in 2017. He is a research scholar in Mathematics at the Indian Institute of Engineering Science and Technology (IIEST), Shibpur, West Bengal, India.Title of his Ph. D. thesis is:“Some studies on neutrosophic decision making”. He continues his research in the field of neutrosophic multi attribute group decision making; neutrosophic hybrid systems; neutrosophic soft MADM . Curently, he is an assistant teacher of Mathematics His research interest includes decision making in neutrosophic environemnt and optimization. Contribution: In 2016, Dalapati and Pramanik [122] defined neutrosophic soft weighted average operator.They determined the order of the alternatives and identify the most suitable alternative based on grey relational coefficient. They also presented a numerical example of logistics center location selection problem to show the effectiveness and applicability of the proposed strategy. Dalapati,Pramanik, and Roy [123] proposed modeling of logistics center location problem using the score and accuracy function, hybrid-score-accuracy function of SVNNs and linguistic variables under singlevalued neutrosophic environment, where weight of the decision makers are completely unknown and the weight of criteria are incompletely known. Dalapati, Pramanik, Alam, Roy, and Smaradache [124] defined IN-cross entropy measure in INS environment in 2017. The authors proved the basic properties of the cross entropy measure. The authors also defined weighted IN- cross entropy measure and proved its basic properties. They also introduced a novel MAGDM strategy based on weighted IN-cross entropy. Finally, the authors solved a MAGDM problem to show the feasibility and efficiency of the proposed MAGDM strategy. Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 121 Neutrosophic Sets and Systems, Vol. 20, 2018 Pramanik, Dalapati, Alam, and Roy [125] defined TODIM strategy in bipolar neutrosophic set environment to handle MAGDM. The authors proposed a new strategy for solving MAGDM problems. The authors also solved an MADM problem to show the applicability and effectiveness of the proposed startegy. Pramanik, Dalapati, Alam, and Roy [126] introduced the score and accuracy functions for neutrosophic cubic sets and prove their basic properties in 2017. The authors developed a new strategy for ranking of neutrosophic cubic numbers based on the score and accuracy functions. The authors first developed a TODIM (Tomada de decisao interativa e multicritévio) stratey in the neutrosophic cubic set (NCS) environment strategy. The authors also solved an MAGDM problem to show the applicability and effectiveness of the developed strategy. Lastly, the authors conducted a comparative study to show the usefulness of proposed strategies. In 2018, Pramanik, Dalapati, Alam, and Roy [127]extended the traditional VIKOR strategy to NCVIKOR strategy and developed an NC-VIKOR based MAGDM strategy in neutrosophic cubic set environment. The authors defined the basic concept of neutrosophic cubic set. Then, the authors introduced neutrosophic cubic number weighted averaging operator and applied it to aggregate the individual opinion to one group opinion. The authors presented an NC-VIKOR based MAGDM strategy with neutrosophic cubic set. They also presented a sensitivity analysis. Finally, the authors solved an MAGDM problem to show the feasibility and efficiency of the proposed MAGDM strategy. Pramanik, Dalapati, Alam, and Roy [128] extended the VIKOR strategy to MAGDM with bipolar neutrosophic environment. The authors introduced the bipolar neutrosophic numbers weighted averaging operator and applied it to aggregate the individual opinion to one group opinion. The authors proposed a VIKOR based MAGDM strategy with bipolar neutrosophic set. Lastly, the authors solved an MAGDM strategy to show the feasibility and efficiency of the proposed MAGDM strategy and presented a sensitivity analysis. Pramanik, Dalapati, Alam, and Roy [129] studied some operations and properties of neutrosophic cubic soft sets.The authors defined some operations such as P-union, P-intersection, R-union, R-intersection for neutrosophic cubic soft sets (NCSSs). The authors proved some theorems on neutrosophic cubic soft sets.The authors also discussed various approaches of internal neutrosophic cubic soft sets (INCSSs) and external neutrosophic cubic soft sets (ENCSSs) and also investigated some of their properties. Pramanik, Dalapati, Alam, Smarandache, and Roy [130] defined a new cross entropy measure in SVNS environment.The authors also proved the basic properties of the NS cross entropy measure. The authors defined weighted SN-cross entropy measure and proved its basic properties. At first the authors proposed an MAGDM strategy based on NS- cross entropy measure. Pramanik, Dalapati, Alam, Roy, Smarandache [131] defined similarity measure between neutrosophic cubic sets and proved its basic properties. They developed a new MADM strategy basd on the proposed similarity measure. They also provided an illustrative example for MADM strategy to show its applicability and effectiveness. Mr. Dalapati’s neutrosophic paper [132] was awarded as the outstanding research paper at the “1st Regional Science and Technology Congress, 2016 in mathematics. Mr. Shamal Dalapati is a young and hardworking researchers in neutrosophic field. In 2017, Mr. Dalapati was awarded “Diploma Certificate” from Neutrosophic Science InternationalAssociation (NISA) for his outstanding performance in neutrosophic research. His research articles receive more than sevent citations. 3.6 Prof.Tapan Kumar Roy Prof. T. K. Roy, Ph. D. in mathematics, is a Professor of mathematics in Indian Institute of Engineering Science and Technology (IIEST), Shibpur. His main research interest includes neutrosophic optimization, neutrosophic game theory, decision making in neutrosophic environment, neutrosophy, etc. Contribution: In 2014, Pramanik and Roy [133] presented the framework of the application of game theory to Jammu Kashmir conflict between India and Pakistan. Pramanik and Roy [20] extended the concept of game Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 122 Neutrosophic Sets and Systems, Vol. 20, 2018 theoretic model [133] of the Jammu and Kashmir conflict in neutrosophic environment. At first, Roy and Das[134] presented multi-objective non –linear programming problem based on neutrosophic optimization technique and its application in Riser design problem in 2015. Roy, Sarkar, and Dey [133] presented a multiobjective neutrosophic optimization technique and its application to structural design in 2016. In 2017, Roy and Sarkar [135-138] also presented several applications of neutrosophic optimization technique. In 2017, Pramanik, Roy, Roy, and Smarandache [139] presented multi criteria decision making using correlation coefficient under rough neutrosophic environment. The authors defined correlation coefficient measure between any two rough neutrosophic sets and also proved some of its basic properties. In 2018, Pramanik, Roy, Roy, and Smarandache [140] defined projection and bidirectional projection measures between interval rough neutrosophic sets and proved their basic properties. The authors developed two new MADM strategies based on interval rough neutrosophic projection and bidirectional projection measures. Then the authors solved a numerical example to show the feasibility, applicability and effectiveness of the proposed strategies. In 2018, Pramanik, Roy, Roy, and Smarandache [141] proposed the sine, cosine and cotangent similarity measures of interval rough neutrosophic sets and proved their basic properties. The authors presented three MADM strategies based on proposed similarity measures. To demonstrate the applicability, the authors solved a numerical example. Prof. Roy did research work on decision making in SVNS, INS, neutrosophic hybrid environment [124-132, 139-141] with S. Pramanik, S. Dalapati, S. Alam and Rumi Roy. His paper [142] together with S. Pramanik and S. Chackrabarti was awarded as the best research paper in 15th West Bengal State Science & Technology Congress, 2008 held on 28th February-29th February, 2008, at Bengal Engineering and Science University, Shibpur. Prof. Roy is a great motivator and a very hardworking person. He works with Prof. Florentin Smarandache. According to “Googlescholar” his research gets citation over 2635. 3.7Prof.Bibhas C. Giri Prof. Bibhas C.Giri is a Prof. of mathematics in Jadavpur University. He did his M.S. in Mathematics and Ph. D. in Operations Research both from Jadavpur University, Kolkata, India. His research interests include inventory/supply chain management, production planning and scheduling, reliability and maintenance. He was a JSPS Research Fellow at Hiroshima University, Japan during the period 2002-2004 and Humboldt Research Fellow at Mannheim University, Germany during the period 2007-2008, Fulbright Senior Research Fellow at Louisiana State University in the year 2012. Contribution: Prof. Giri works with S. Pramanik, P. Biswas and P. P. Dey in neutrosophic environment. His neutrosophic paper [143] coauthored with Kalyan Mondal and Surapati Pramanik received the outstanding research paper award at the“1st Regional Science and Technology Congress, 2016 in mathematics. His neutrosophic paper [144] together with Kalyan Mondal and Surapati Pramanik received the best research paper in 25 th West Bengal State Science and Technology Congress 2018 in mathematics. His neutrosophic research work and vast contribution can be found in [71-80, 82, 101-119]. Prof. Giri is a great motivator. According to “Googlescholar’, his research receives more than 4920 citations having h-index-31 and i-10 index-78. 3.8 Prof. Anjan Mukherjee Anjan Mukherjee was born in 1955. He completed his B. Sc. and M. Sc. in Mathematics from University of Calcutta and Ph. D. from Tripura University. Currently, he is a Professor and Pro -Vice Chancellor Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 123 Neutrosophic Sets and Systems, Vol. 20, 2018 of Tripura University. Under his guidance, 12 candidates obtained Ph. D. award. He has 30 years of research and teaching experience. His main research interest includes topology, fuzzy set theory, rough sets, soft sets, neutrosophic set, neutrosophic soft set, etc. Contribution: In 2014, Anjan Mukherjee and Sadhan Sarkar [145] defined the Hamming and Euclidean distances between two interval valued neutrosophic soft sets (IVNSSs). The authors also introduced similarity measures based on distances between two interval valued neutrosophic soft sets.The authors proved some basic properties of the similarity measures between two interval valued neutrosophic soft sets. They established an MADM strategy for interval valued neutrosophic soft set setting using similarity measures. Mukherjee and Sarkar [146] also defined several distances between two interval valued neutrosophoic soft sets in 2014. The authors proposed similarity measure between two interval valued neutrosophic soft sets. The authors also proposed similarity measure between two interval valued neutrosophic soft sets based on set theoretic approach. They also presented a comparative study of different similarity measures. Mukherjee and Sarkar [147]defined several distances between two neutrosophoic soft sets.The authors also defined similarity measure between two neutrosophic soft sets.The authors developed an MADM strategy based on the proposed similarity measure. Mukherjee and Sarkar [148] proposed a new method of measuring degree of similarity and weighted similarity between two neutrosophic soft sets and studied some properties of similarity measure. Based on the comparison between the proposed strategy [148] and existing strategies introduced by Mukherjee and Sarkar[147], the authors found that the proposed strategy [148] offers strong similarity measure. The authors also proposed a decision making strategy based on similarity measure. Prof. Anjan Mukherjee evaluated many Ph. D. theses. Among them, the Ph. D. thesis of Durga Banerjee [118] dealing with neutrosophic decision making was evaluated by Prof. Anjaan Mukherjee. Research of Prof. Mukherjee receives more than 700 citations for his works. Prof. Mukherjee is working with his group members with neutrosophic soft sets and its applications. 3.9 Dr.Pabitra Kumar Maji Dr. Pabitra Kumar Maji, M. Sc., Post Doc., is an Assistant Professor of mathematics in Bidhan Chandra College, Asansol, West bengal. He works on soft set, fuzzy soft set, intuitionistic fuzzy set, fuzzy set, neutrosophic set, neutrosophic soft set, etc., Contribution: In 2011, Maji [149] presented an application of neutrosophic soft set in object recognition problem based on multi-observer input data set. The author also introduced an algorithm to choose an appropriate object from a set of objects depending on some specified parameters. In 2014, Maji, Broumi, and Smarandache [150] defined intuitionistic neutrosophic soft set over ring and proved some properties related to this concept. They also defined intersection, union, AND and OR operations over ring (INSSOR). Finally, the authors defined the product of two intuitionistic neutrosophic soft set over ring. In 2015, Maji [151] presented weighted neutrosophic soft sets. The author presented an application of weighted neutrosophic soft sets in MADM problem. According “Googlescholar’’, his publication includes 20 research paper having citations 5948. Maji [152] studied the concept of weighted neutrosophic soft sets. The author considered a multiobserver decision-making problem as an application of weighted neutrosophic soft sets. We have considered here a recognition strategy based on multi-observer input parameter data set. 3.10 Dr. Harish Kumar Garg Dr. Harish Garg is an Assistant Professor in the School of Mathematics, Thapar Institute of Engineering &Technology (Deemed University) Patiala. He completed his post graduation (M.Sc) in Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Neutrosophic Sets and Systems, Vol. 20, 2018 124 Mathematics from Punjabi University Patiala, India in 2008 and Ph.D. from Department of Mathematics, Indian Institute of Technology (IIT) Roorkee, India in 2013. His research interest includes neutrosophic decision-making, aggregation operators, reliability theory, soft computing technique, fuzzy and intuitionistic fuzzy set theory, etc. environment.The authors proposed some prioritized weighted and ordered weighted averaging as well as geometric aggregation operators for a collection of linguistic single-valued neutrosophic numbers and established their basic properties. The authors also proposed MADM strategy and solved a numerical example. Contribution: Dr. Garg research receives more than 2000 citations. Dr. Garg acts an active reviewer for reputed international journals and received certificate of outstanding in reviewing from “Computer & Industrial Engineering’’, “Engineering Applications of Artificial Intelligence’’, “Applied Soft Computing’’, “Applied Mathematical Modeling’’, etc. Dr. Garg acts as editor for many international journals. In 2016, Garg and Nancy [153] defined some operations of SVNNs such as sum, product, and scalar multiplication under Frank norm operations. The authors also defined some averaging and geometric aggregation operators and established their basic properties.The authors also established a decision-making strategy based on the proposed operators and presented an illustrative numerical example. 3.11 Dr.Sukanto Bhattacharya In 2017, Garg and Nancy [154] developed a nonlinear programming (NP) model based on TOPSIS to solve decision-making problems. At first, the authorsconstructed a pair of the nonlinear fractional programming model based on the concept of closeness coefficient and then transformed it into the linear programming model. Garg and Nancy [155] defined some new types of distance measures to overcome the shortcomings of the existing measures for SVNSs. The authors presented a comparison between the proposed and the existing measures in terms of counter-intuitive cases for showing validity. The authors also demonstrated the defined measures with hypothetical case studies of pattern recognition as well as medical diagnoses. Garg and Nancy [156] studied the entropy measure of order α for single valued neutrosophic numbers. The authors established some desirable properties of entropy measure. The author also developed a MADM strategy based on entropy measures and solved a numerical example of investment problem. Nancy and Garg [157] proposed an improved score function for ranking the single as well as intervalvalued neutrosophic sets by incorporating the idea of hesitation degree between the truth and false degrees. The authors also presented an MADM strategy based on proposed function and solved a numerical example to show its practicality and effectiveness. Garg and Nancy [158] introduced some new linguistic prioritized aggregation operators in the linguistic single-valued neutrosophic set (LSVNS) Sukanto Bhattacharya is a faculy member and associated with Deakin Business School, Deakin University. Sukanto Bhattacharya [159] is the first researcher who employed utility theory to financial decisionmaking and obtained Ph. D. for applying neutrosophic probability in finance. His Ph. D. thesis covers a substantial mosaic of related concepts in utility theory as applied to financial decisionmaking. The author reviewed some of the classical notions of Benthamite utility and the normative utility paradigm. The author proposed some key theoretical constructs like the neutrosophicnotion of perceived risk and the entropic utility measure. Khoshnevisan, and Bhattacharya [160] added a neutrosophic dimension to the problem of determining the conditional probability that a financial misrepresentation of the data set. Prof. Bhattacharya is an active researcher and his works in neutrosophics are found in [159-163]. His research receives more than 380 citations. 4. Conclusions We have presented a brief overview of the contributions of some selected Indian researchers who Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview 125 Neutrosophic Sets and Systems, Vol. 20, 2018 conducted research in neutrosophic decision making. We briefly presented the contribution of the selected Indian neutrosophic researchers in MADM. In future, the contribution of Indian researchers such as W. B. V. Kandasamy, Pinaki Majumdar,Surapati Pramanik, Samarjit Kar, and other Indian mathematicians in developing neutrosophics can be studied. The study can also be extended for mathematicians from other countries who contributed in developing neutrosophic science. Decision making in neutrosophic hybrid environment is gaining much attention. So it is a promising field of research in different neutrosophic hybrid environment and the real cahllenge lies in the applications of the developed theories. Since some of the selected researchers are young, it is hoped that the researchers will do more creative works and new research regarding their contributions will have to be conducted in future. Acknowledgements The authors would like to acknowledge the constructive comments and suggestions of the anonymous referees. References [1] C. L. 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Neutrosophic Sets and Systems, 15 (2017),70-79.. S. Pramanik, P. P. Dey, B. C. Giri, and F. Smarandache. An extended TOPSIS for multi-attribute decision making problems with neutrosophic cubic information. Neutrosophic Sets and Systems,17(2017), 20-28. P. P. Dey, S. Pramanik, J. Ye, and F. Smarandache. Cross entropy measures of bipolar and interval bipolar neutrosophic sets and their application for multiattribute decision making. Axioms,7(2)(2018). doi:10.3390/axioms7020021 S. Pramanik, P. P. Dey, and B. C. Giri. Hybrid vector similarity measure of single valued refined neutrosophic sets to multi-attribute decision making problems. In F. Smarandache,& S. Pramanik (Eds), New trends in neutrosophic theoryand applications, Vol II. Pons Editions, Brussells, 2018, 156-174. S. Pramanik, P. P. Dey, and F. Smaradache. Correlation coefficient measures of interval bipolar neutrosophic sets for solving multi-attribute decision making problems. Neutrosophic Sets and Systems, 19 (2018), 70-79. S. Pramanik, P. P. Dey, and F. Smarandache. MADM strategy based on some similarity measures in interval bipolar neutrosophic set environment. Preprints. 2018, 2018040012.doi:10.20944/preprints201804.0012.v1. S. Pramanik and P. P. Dey. Fuzzy goal programming for multilevel linear fractional programming problems. Presented at 18th West Bengal State Science & Technology Congress held on 28th February -1st March, 2011, Ramakrishna Mission Residential College, Narendrapur, Kolkata 700 103. K. Mondal and S. Pramanik. (2014). Multi-criteria group decision making approach for teacher recruitment in higher education under simplified neutrosophic environment. Neutrosophic Sets and Systems, 6, 28-34. S. Pramanik, and K. Mondal. Cosine similarity measure of rough neutrosophic sets and its application in medical diagnosis. Global Journal of Advanced Research, 2(1) (2015), 212-220. 128 [88] S. Pramanik, and K. Mondal. 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TOPSIS strategy for MADM with trapezoidal neutrosophic numbers. Neutrosophic Sets and Systems, 19 (2018), 29-39. [116] P. Biswas, S. Pramanik, and B. C. Giri. Distance measure based MADM strategy with interval trapezoidal neutrosophic numbers. Neutrosophic Sets and Systems, 19 (2018), 240-46. [117] P. Biswas, S. Pramanik, and B. C. Giri. Students’ progress reports evaluation based on fuzzy hybrid vector similarity measure. Presented at Second Regional Science and Technology Congress, 2017, held at University of Kalyani, December 14-15, 2017. [118] D. Banerjee. Some studies on decision making in an uncertain environment. Unpublished Ph. D. Thesis. Jadavpur University, 2017. [119] S. Pramanik, D. Banerjee, and B.C. Giri. Multi– criteriagroup decision making model in neutrosophic refined setand its application. Global Journal of Engineering Scienceand Research Management, 3(6) (2016), 12-18. [120] S. Pramanik, D. Banerjee, and B.C. Giri. 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To be and not to be–an introduction to neutrosophy: A novel decision paradigm. Neutrosophic theory and its applications. Collected Papers,1, 424-39. http://fs.gallup.unm.edu/ToBeAndNotToBe.pdf [163] M. Khoshnevisan, and S. Bhattacharya. Neutrosophic information fusion applied to the options market. Investment management and financial innovations 1, (2005),139-145. Received : March 2, 2018. Accepted : April 30, 2018. Surapati Pramanik, Rama Mallick, Anindita Dasgupta, Contributions of Selected Indian Researchers to Multi Attribute Decision Making in Neutrosophic Environment: An Overview Abstract Contributors to current issue (listed in papers' order): Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri, Seon Jeong Kim, Seok-Zun Song, Young Bae Jun, G. Muhiuddin, Hashem Bordbar, Florentin Smarandache, Mehmat Ali Ozturk, Tuhin Bera, Nirmal Kumar Mahapatra, Emad Marei, M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. Antonysamy, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy, Rama Mallick, Anindita Dasgupta. Papers in current issue (listed in papers' order): Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy; Hybrid Binary Logarithm Similarity Measure for MAGDM Problems under SVNS Assessments; Generalizations of Neutrosophic Subalgebras in BCK/BCI-Algebras Based on Neutrosophic Points; Further results on (∈, ∈)-neutrosophic subalgebras and ideals in BCK/BCI-algebras; Commutative falling neutrosophic ideals in BCK-algebras; On Neutrosophic Soft Prime Ideal; Single Valued Neutrosophic Soft Approach to Rough Sets, Theory and Application; A novel approach to nano topology via neutrosophic sets; NC-VIKOR Based MAGDM Strategy under Neutrosophic Cubic Set Environment; Contributions of Selected Indian Researchers to Multi-Attribute Decision Making in Neutrosophic Environment: An Overview. $39.95