F-Groups

14th Iranian Conference on Fuzzy Systems Sahand University of Technology Tabriz-Iran F-Groups Sergey Davidov1, Jahangir Hatami*2 1 Academic member, Department of Algebra and Geometry, Yerevan State University, Yerevan, Armenia, Davidov@ysu.am 2 Ph. D. student, Department of Algebra and Geometry, Yerevan State University, Yerevan, Armenia, jhatami51@gmail.com Abstract: In this paper, we are going to develop the notion of F-groups. An F-group is a special case of F-algebras (Algebras with fuzzy operations) with the only binary fuzzy operation. In the rest, we investigate the notions fuzzy subgroup, fuzzy normal subgroup and isomorphism theorems. Keywords: F-algebra, F-group, Fuzzy subgroup, Fuzzy normal subgroup, Isomorphism theorems. 1- Introduction The problem of development of algebras with fuzzy operations is formulated in ([3], P: 136). Fuzzy approaches to various universal algebraic concepts started with Rosenfeld’s fuzzy groups [15]. Since then, many fuzzy algebraic structures have been studied (vector spaces, rings, etc.). Also, some authors proposed a general approach to the theory of fuzzy algebras. Another fuzzy approach to universal algebras was initiated by Belohlvek and Vychodil [3,4], who studied the so-called algebras with fuzzy equalities and developed a fuzzy equational logic. These structures have two parts: the functional part, which is an ordinary algebra and the relational part, which is the carrier set of the algebra, equipped with a fuzzy equality which is compatible with all of the fundamental operations of the ordering algebra. In the fuzzy set theory there were many different approaches to the concept of a fuzzy function. In a number of papers various kinds of fuzzy functions based on fuzzy equivalence relations were studied. In particular, such approach was used in definitions of partial fuzzy functions and fuzzy functions, given by Klawonn [12], strong fuzzy functions and perfect fuzzy functions, given by Demirci [6]. Fuzzy functions based on fuzzy equivalence relations have shown oneself to be very useful in many applications in approximate reasoning, fuzzy control, vague algebra and other fields. The content of this paper can be briefly stated as follows: In section 2, we will present the necessary information concerned with the fuzzy operation and algebras with fuzzy operations and in section 3, we define F- groups by fuzzy operations. In the rest of this paper, we define F- congruences on F-groups and F- quotient groups by F- congruences and investigate their properties. 1 www.icfs14.ir 14th Iranian Conference on Fuzzy Systems Sahand University of Technology Tabriz-Iran 2- Group In this paper we will use complete residuated lattices L  L, , , , ,0,1  as the structures of truth values. Definition 1. A fuzzy equivalence relation on a set is a mapping E : X  X  L satisfying (i) E ( x, x)  o (Reflexivity); (ii) E ( x, y)  E( y, x) (Symmetry); (iii) E( x, y)  E( y, z)  E( x, z) (Transitivity), for every x, y, z  X . A fuzzy equivalence E on X where E ( x, y)  1 implies x  y will be called a fuzzy equality. Fuzzy equalities will usually be denoted by  . Theorem 1 [5]. Let M  n M i 1 Also, let i and  Mi be a fuzzy equality on M i for every i {1,..., n} . (a1 ,..., an ) M (b1 ,..., bn )  in1 (ai Mi bi ) and (a1 ,..., an ) M (b1 ,..., bn )  in1 (ai Mi bi ) for every ai , bi  M i . Then M and M are fuzzy equalities on M . Definition 2 [5]. Let  M be a fuzzy equality on M . An (n+1)-ary fuzzy relation  on a set M is called an n-ary fuzzy operation w.r.t.  M if we have the following conditions Extensionality: in11 (ai M bi )   (a1 ,..., an1 )   (b1 ,..., bn1 ) a1, b1,..., an1, bn1  M ; Functionality:  (a1 ,..., an , y)   (a1 ,..., an , y ')  ( y M y ') ai , bi , y, y '  M ; Fully defined:   (a1 ,..., an , y)  1 a1 ,..., an  M . yM We say that  is a fuzzy operation on M with arity n. Definition 3 [5]. An algebra with fuzzy operations of type  , R  , (consisting a binary relation symbol  is called a symbol for fuzzy equality and a set R of symbols of relations), is a triplet M  M , M , RM  such that (i) M is a fuzzy equality on the set M , (ii) R M is a set of fuzzy operations on the set M . To simply, we call F-algebra instead of the algebra with fuzzy operations. 2 www.icfs14.ir 14th Iranian Conference on Fuzzy Systems Sahand University of Technology Tabriz-Iran Definition 4. Let G  G, G , G  be an F-algebra of type  ,   , where  is a ternary relation symbol. Then (i) G is a fuzzy abelian iff G satisfies (a, b, c, c '  G)[(G (a, b, c) G (b, a, c '))  (c G c ')]. (ii) G is a fuzzy associative iff G satisfies [(G (b, c, d ) G (a, d , m) G (a, b, q) G (q, c, w))  (m G w)] for all a, b, c, d , m, q, w  G . Definition 5. Let G  G, G , G  be an F-algebra of type  ,   . Then G  G, G , G  is an F-group iff G is fuzzy associative and satisfies the following conditions (i) There exists an (two sided) identity element e  G such that G (e, a, a)  G (a, e, a)  1 for each a  G . (ii) For a given identity element e  G , and for a given a  G , there exists an element a 1  G such that G (a 1 , a, e)  G (a, a 1 , e)  1. To simply, we denote the F-group G  G, G , G  by G  G, ,   . Theorem 2. Let G  G, ,   be an F-group. Then (i) G has an unique identity element e. (ii) For every a  G , a 1 is an unique element. In the rest, we define and investigate the notions of subgroup, congruence, and morphism on F-groups. In [3], a fuzzy subalgebra in an ordinary algebra M  M , F  is introduced by an L-set A in M , ( A : M  L ), such that for each n-ary operation f M  F M we have A(a1 )  ...  A(an )  A( f M (a1,..., an )) for every a1,..., an  M . In [13], the fuzzy subgroup and normal fuzzy subgroup is defined as the above definition. By considering this notions, we can introduce fuzzy (normal) subgroups. Definition 6. Let G be any F-group. By a fuzzy subgroup  of G is defined as a function  : G  L such that  (a)   (b)   (c) (a, b1, c)  cG for all a, b  G. Moreover a normal fuzzy subgroup  of G is defined as a fuzzy subgroup satisfying the condition  (c ') (b, a, c ')   (c) (a, b, c)  c 'G cG 3 www.icfs14.ir 14th Iranian Conference on Fuzzy Systems Sahand University of Technology Tabriz-Iran for every a, b  G. In the following, we get into the concept of morphisms for F-groups. Definition 7. Let G  G, ,   and G '  G ',  ',  '  be two F-groups. A mapping h : G  G ' is called a morphism of G to G ' if (i) (a  b)  (h(a)  ' h(b)) a, b  G, (ii) (a1, a2 , a3 )   '(h(a1), h(a2 ), h(a3 )) a1, a2 , a3  G. The fact that h : G  G ' is a morphism is denoted by h : G  G '. Furthermore, (a) a morphism such that (a  b)  (h(a)  ' h(b)) a, b  G, is called an embedding, (b) a surjective morphism is called an epimorphism, (c) an injective morphism is called a monomorphism, (d) an epimorphism which is an embedding is called an isomorphism. Theorem 3. Let G  G, ,   and G '  G ',  ',  '  be two F-groups and f : G  G ' be a morphism. Then (i) ( f (e)  ' e ')  1 where e and e ' are the identity elements of G and G ' respectively, (ii) f (b1 )  ( f (b))1 b  G, (iii) ( f (b)  ' e ')  (( f (b))1  ' e ') b  G. Definition 8. Let G  G, ,   and G '  G ',  ',  '  be two F-groups and f : G  G ' be a morphism. For any fuzzy subgroup  of G ', we define a map f 1 ( ) from G to L by f 1 ( )( x)   ( f ( x)) for all x  G, we call a preimage of fuzzy subgroup  under f . For any subgroup  of G we define an image f [ ] of  under f by f [  ]( y )    (u ) u f 1 ( y ) for all y  G '. Theorem 4. Let G  G, ,   and G '  G ',  ',  '  be two F-groups, f : G  G ' be an embedding, and (a1, a2 , a3 )   '(h(a1), h(a2 ), h(a3 )) for every a1, a2 , a3  G. Let  be a fuzzy (normal) subgroup of G ', then the preimage f 1 ( ) is a fuzzy (normal) subgroup of G. 4 www.icfs14.ir 14th Iranian Conference on Fuzzy Systems Sahand University of Technology Tabriz-Iran Theorem 5. Let G  G, ,   and G '  G ',  ',  '  be two F-groups, f : G  G ' be an embedding and monomorphism, and (a1, a2 , a3 )   '(h(a1), h(a2 ), h(a3 )) for every a1, a2 , a3  G. Let  be a fuzzy (normal) subgroup of G, then the image f [  ] is a fuzzy (normal) subgroup of G '. Defenition 9. Let G  G, ,   be an F-group. A binary L-relation (binary fuzzy relation)  on G is called an F-congruence on G if (i)  is a fuzzy equivalence on G, (ii)  is compatible with  ,i.e (a  b)  (a '  b ')  (a, a ')   (b, b ') for every a, a ', b, b '  G, (iii) 3  (ai ,bi ) (a1, a2 ,a3 )  (b1, b2 ,b3 ) for every  i 1 a1, a2 , a3  G. Remark 1. Dfinition 9(ii) holds iff   . [3. Lemma 1.82]. Definition 10. Let  be an F-congruence on an F-group G  G, ,   . An F-quotient group G by  is an F-group G /   G /  , G / , G /  such that (i) [a] G / [b]   (a, b) (ii) G / ([a1 ] ,[a2 ] ,[a3 ] )  [a] ,[b]  G /  ,  ((a1, a2 , c)  (c, a3 )), cG where [a1 ] ,[a2 ] ,[a3 ]  G /  , G /   {[a] | a  G} and [a]  {a '  G |  (a, a ')  1}. Remark 2. An F-quotient group is well-defined. Definition 11. Let G and G ' be two F-groups. let h : G  G ' be a morphism. Then the kernel of h, the binary L-set (fuzzy set) h : G  G  L, is defined by h (a, b)  h(a)  ' h(b). Theorem 6. Let G and G ' be two F-groups. let h : G  G ' be an embedding. Then  h is an F-congruence on G. Definition 12. For every F-group G and  an F-congruence on G, a mapping h : G  G /  , where h (a)  [a] for all a  G, is called a natural mapping. Theorem 7. A natural mapping h from an F-group G to an F-quotient group G /  is an epimorphism. Theorem 8. (first isomorphism theorem). Let h : G  G ' be an embedding of F-groups. Then there is an isomorphism g : G / h  G ' such that hh g  h. 5 www.icfs14.ir 14th Iranian Conference on Fuzzy Systems Sahand University of Technology Tabriz-Iran Definition 13. Let  ,  be two F-congruences of an F-group G and   . Then we let  /  denote a binary L-relation (binary fuzzy relation) on G /  defined by ( /  )([a] ,[b] )   (a, b) for all a, b  G. Theorem 9. Let  ,  be two F-congruences of an F-group G and   . Then  /  is an Fcongruence of (G /  ). Theorem 10. (second isomorphism theorem). Suppose G is an F-group and  ,  are two F-congruences on G and   . Then the mapping h : (G /  ) / ( /  )  G /  Defined by h([[a] ] / )  [a] is an isomorphism. 3- Conclusions Under conditions, the F-groups can be convertible to ordinary groups and vice versa. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] Burris. S, Sankappanavar. H. P, A Course in Universal Algebra, Springer Verlag, New York, 1981. Belohlavek. R, Vychodil. V, Fuzzy Equational Logic, Springer, Berlin, Heidelberg, 2005. Belohlavek. R, Vychodil. V, Algebras with fuzzy equalities, Fuzzy Sets and Systems, 157, 161-201, 2006. Davidov. S. S, Hatami. J, Algebras with fuzzy operations, Mathematical problems of computer science, Yerevan, 38(2012), 49-52. Demirci. 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Fuzzy congruences on groups. Quasigroups and Related Systems 11, 59-70, 2004. Novak. V, Perfilieva. I, Mocor. J, Mathematical Principles of Fuzzy Logic. Kluwer, Boston, 1999. Peter. L. Clark, Summer 2010 course on model theory and its applications.….. Rosenfeld. A, Fuzzy groups, J. Math. Anal. Appl. 35(3), 512-517, 1971. Sostak. A. P, Fuzzy functions and extension of the category L-Top of CHANG-GOGUEN L-Topological spaces, pp 271294. Zadeh. L. A, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8(3) (1975) 199-251 301-347, 9 (1975) 43-80. 6 www.icfs14.ir View publication stats