- 1983~2021 : Professor of Jeju National University (Mathematics)2021~ : Emeritus professor.edit
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Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the... more
Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form b i , j b i , l b k , j b k , l with nonzero entries. We know that the isolation number of B is not greater than the rank of it. Thus, we investigate the upper bound of the rank of B and the rank of its support for the given matrix B with isolation number h over antinegative semirings.
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Abstract Let M n ( B ) denote the set of n × n ( 0 , 1 ) -matrices with Boolean arithmetic. The set of primitive matrices of exponent k, denoted E k , is the set of matrices such that A k has all nonzero entries and A j has zero entries... more
Abstract Let M n ( B ) denote the set of n × n ( 0 , 1 ) -matrices with Boolean arithmetic. The set of primitive matrices of exponent k, denoted E k , is the set of matrices such that A k has all nonzero entries and A j has zero entries for all j k . For 3 ≤ k ≤ n , we characterize those linear operators that map E k to E k and E k − 1 to E k − 1 . We also characterize those linear operators that strongly preserve E k for 3 ≤ k ≤ n , that is, that map E k to E k and the complement of E k to the complement of E k .
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The notion of GE-algebra is introduced by Bandaru et al. as a generalization of Hilbert algebra. The consept of interior GE-algebra is introduced by Lee et al., and related properties are investigated. GE-filter study at GE-algebra using... more
The notion of GE-algebra is introduced by Bandaru et al. as a generalization of Hilbert algebra. The consept of interior GE-algebra is introduced by Lee et al., and related properties are investigated. GE-filter study at GE-algebra using interior operator was conducted by Song et al. In this paper, various properties that belligerent interior GE-filter could perform were explored, and necessary conditions have been established for interior GE-filter to turn into belligerent interior GE-filter. As a result, the characterization of belligerent interior GE-filter was established.
To investigate the filter and deductive system of the Schaefer stroke Hilbert algebra using the Dokdo structure, the concept of Dokdo filter and Dokdo deductive system is defined, examples are given, and various properties are... more
To investigate the filter and deductive system of the Schaefer stroke Hilbert algebra using the Dokdo structure, the concept of Dokdo filter and Dokdo deductive system is defined, examples are given, and various properties are investigated. The Dokdo filter is formed by attaching appropriate conditions to the given Dokdo structure. The characterization of Dokdo filter is studied. Dokdo filters related to filters are constructed. Dokdo filter and Dokdo deductive system turn out to be the same concept.
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Research Interests: Mathematics and Soft Set
ABSTRACT The notion of positive implicative Smarandache BCC-ideals is introduced, and related properties are investigated. Relations between Smarandache BCC-ideals and positive implicative Smarandache BCC-ideals are discussed. The... more
ABSTRACT The notion of positive implicative Smarandache BCC-ideals is introduced, and related properties are investigated. Relations between Smarandache BCC-ideals and positive implicative Smarandache BCC-ideals are discussed. The extension property of a positive implicative Smarandache BCC-ideals is given.
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The notions of int-soft hyper [Formula: see text]-ideals, int-soft strong hyper [Formula: see text]-ideals, int-soft [Formula: see text]-weak hyper [Formula: see text]-ideals, int-soft weak hyper [Formula: see text]-ideals and int-soft... more
The notions of int-soft hyper [Formula: see text]-ideals, int-soft strong hyper [Formula: see text]-ideals, int-soft [Formula: see text]-weak hyper [Formula: see text]-ideals, int-soft weak hyper [Formula: see text]-ideals and int-soft reflexive hyper [Formula: see text]-ideals are introduced, and related properties and relations are investigated. Using the notion of inclusive sets, characterizations of an int-soft (weak, strong, reflexive) hyper [Formula: see text]-ideal are provided.
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Let denote the set of matrices with entries in . We write to denote the subset of , all of whose entries are non-negative. Let denote the set of all real symmetric matrices. A matrix is said to be completely positive if there is some... more
Let denote the set of matrices with entries in . We write to denote the subset of , all of whose entries are non-negative. Let denote the set of all real symmetric matrices. A matrix is said to be completely positive if there is some matrix such that . The CP-rank of the matrix is the smallest such that for some . In this article, we shall investigate the linear operators on that preserve sets of matrices defined by the CP-rank. We classify those that preserve the CP-rank function, those that preserve the set of CP-rank-1 matrices, those that preserve the sets of CP-rank-1 matrices and the set of CP-rank-2 matrices, and those that strongly preserve the set of CP-rank-1 matrices.
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ABSTRACT For an matrix over the max algebra , a generalized inverse of is an matrix over satisfying . In this paper, we determine the general form of matrices having generalized inverses. Also, we define a space decomposition of a matrix,... more
ABSTRACT For an matrix over the max algebra , a generalized inverse of is an matrix over satisfying . In this paper, we determine the general form of matrices having generalized inverses. Also, we define a space decomposition of a matrix, and prove that a matrix has a generalized inverse if and only if it has a space decomposition. Using this decomposition, we characterize reflexive -inverses of matrices. Furthermore, we establish necessary and sufficient conditions for a matrix to possess various types of -inverses including Moore–Penrose inverse.
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The concept of a positive implicative makgeolli ideal in BCK-algebras is introduced, and its properties are investigated. The relationship between a makgeolli ideal and a positive implicative makgeolli ideal is established. The conditions... more
The concept of a positive implicative makgeolli ideal in BCK-algebras is introduced, and its properties are investigated. The relationship between a makgeolli ideal and a positive implicative makgeolli ideal is established. The conditions under which a makgeolli ideal can be a positive implicative makgeolli ideal are explored. Characterizations of a positive implicative makgeolli ideal are discussed, and the extension property for a positive implicative makgeolli ideal is established.
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The concept of a prominent interior GE-filter (of type 1 and type 2) is introduced, and their properties are investigated. The relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an... more
The concept of a prominent interior GE-filter (of type 1 and type 2) is introduced, and their properties are investigated. The relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter are discussed. Examples to show that any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter are provided. Conditions for an interior GE-filter to be a prominent interior GE-filter are given. Also, conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter are considered, and an example describing it is given. The relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1 is discussed.
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In this paper, when n = 6, we will determine the minimum permanent and minimizing matrices on the face of which contains exactly two square zero-submatrices.
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Multiplicities of several types of positive implicative hyper K-ideals are considered, and relations among them are discussed.
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We consider the fuzzification of categorical ideals in BCK-algebras, and study their properties. We also state normal fuzzy categorical ideals in BCK-algebras, and construct an extension of a fuzzy categorical ideal.
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As a generalization of a neutrosophic set, the notion of MBJ-neutrosophic sets is introduced by Mohseni Takallo, Borzooei and Jun, and it is applied to BCK/BCI-algebras. In this article, MBJ-neutrosophic set is used to study commutative... more
As a generalization of a neutrosophic set, the notion of MBJ-neutrosophic sets is introduced by Mohseni Takallo, Borzooei and Jun, and it is applied to BCK/BCI-algebras. In this article, MBJ-neutrosophic set is used to study commutative ideal in BCI-algebras. The concept of closed MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal is introduced and their properties and relationships are studied. The conditions for an MBJ-neutrosophic ideal to be a commutative MBJ-neutrosophic ideal are given. The conditions for an MBJ-neutrosophic ideal to be a closed MBJ-neutrosophic ideal are provided. Characterization of a commutative MBJ-neutrosophic ideal is established. Finally, the extension property for a commutative MBJ-neutrosophic ideal is founded.
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Research Interests: Mathematics and Semiring
Abstract:- The zero-term rank of a matrix is the maximum number of zeros in any generalized diagonal. This article characterizes the linear operators that preserve zero-term rank of m n ´ atrices when the matrices have entries either in a... more
Abstract:- The zero-term rank of a matrix is the maximum number of zeros in any generalized diagonal. This article characterizes the linear operators that preserve zero-term rank of m n ´ atrices when the matrices have entries either in a field with at least mn+1 elements or in a ring whose characteristic is not 2. Key-Words:- Zero-term rank, term rank, linear operator, preserver, ( , ,)P Q B-operator, cell. 1
For any m × n nonbinary Boolean matrix A, its spanning column rank is the minimum number of the columns of A that spans its column space. We have a characterization of spanning column rank-1 nonbinary Boolean matrices. We investigate the... more
For any m × n nonbinary Boolean matrix A, its spanning column rank is the minimum number of the columns of A that spans its column space. We have a characterization of spanning column rank-1 nonbinary Boolean matrices. We investigate the linear operators that preserve the spanning column ranks of matrices over the nonbinary Boolean algebra. That is, for a linear operator T on m × n nonbinary Boolean matrices, it preserves all spanning column ranks if and only if there exist an invertible nonbinary Boolean matrix P of order m and a permutation matrix Q of order n such that T (A) = PAQ for all m × n nonbinary Boolean matrix A. We also obtain other characterizations of the (spanning) column rank preserver.
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In this article, we apply the notion of interval neutrosophic sets to ideal theory in BCK/BCI-algebras.
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The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are... more
The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a quasi (or, pseudo) star-shaped set to be a star-shaped set are provided.
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Intuitionistic permeable values in BCK/BCI-algebras are introduced, and several properties are investigated. A relation between an intuitionistic permeable S-value and an intuitionistic permeable I-value is discussed. Conditions for the... more
Intuitionistic permeable values in BCK/BCI-algebras are introduced, and several properties are investigated. A relation between an intuitionistic permeable S-value and an intuitionistic permeable I-value is discussed. Conditions for the intuitionistic lower (upper) level set to be S-energetic and I-energetic are considered. Conditions for a couple of numbers to be an intuitionistic permeable S-value are studied.
Given i, j, k ∈ {1,2,3,4}, the notion of (i, j, k)-length neutrosophic subalgebras in BCK=BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level... more
Given i, j, k ∈ {1,2,3,4}, the notion of (i, j, k)-length neutrosophic subalgebras in BCK=BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.
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We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I... more
We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .