Search

In this paper, the notions of ideals in NBCK-algebras are presented and some properties of such ideals are analyzed. Essentially, our approach is to define NBCK-algebras of all cosets by considering an operation on the set of all cosets defined via ideals. Afterwards, our aim is to study ideal theorems on NBCK-algebras. [ABSTRACT FROM AUTHOR]

Using the notion of bipolar-valued fuzzy soft set, the concepts of bipolar-valued fuzzy soft (weak, s -weak, strong) hyper BCK ideal are introduced, and related properties and relations are investigated. Characterizations of bipolar-valued fuzzy soft (weak) hyper BCK ideal are considered. Conditions for a bipolar-valued fuzzy soft weak hyper BCK ideal to be a bipolar-valued fuzzy soft s -weak hyper BCK ideal are provided. Conditions for a bipolar-valued fuzzy soft set to be a bipolar-valued fuzzy soft strong hyper BCK ideal are displayed. [ABSTRACT FROM AUTHOR]

The notion of eventful element and eventful set is introduced, and its application is considered in BCI and/or BCK algebras. Several properties are investigated, and (positive implicative) ideal in an eventful BCK algebra and closed ideal in an eventful BCI algebra are discussed. Given the subsets A and B of a BCI and/or BCK algebra, the eventful subset E (A , B) which consists of BCI and/or BCK events with a condition is established. Conditions for the eventful subset E (A , B) to be a (positive implicative) ideal of an eventful BCK algebra are provided, and conditions for the eventful subset E (A , B) to be a (closed) ideal of an event BCI algebra are given. An example to show that the set { 0 ̃ } is not a positive implicative ideal in an eventful BCK algebra is provided, and then the conditions for the set { 0 ̃ } to be a positive implicative ideal in an eventful BCK algebra are considered. [ABSTRACT FROM AUTHOR]

Lattice implication algebra is a new logical algebraic system which is established by combining lattice and implication algebra. As a generalization of intuitionistic fuzzy set, neutrosophic set is introduced which deals with indeterminate membership in addition to degree of membership and degree of non-membership. In this article, the notion of neutrosophic set theory is applied to lattice implication algebras. The concept of implicative neutrosophic LI-ideals of a lattice implication algebra is introduced, and several properties are investigated. Relationship between a neutrosophic LI-ideal and an implicative neutrosophic LI-ideal is discussed, and conditions for a neutrosophic LI-ideal to be an implicative neutrosophic LI-ideal are provided. Characterizations of an implicative neutrosophic LI-ideal are considered, and the extension property of an implicative neutrosophic LI-ideal is studied. [ABSTRACT FROM AUTHOR]

Fuzzy soft set theory is applied to hyper BCK-algebras. The notion of fuzzy soft positive implicative hyper BCK-ideals is introduced, and several properties are investigated. The relation between fuzzy soft positive implicative hyper BCK-ideal and fuzzy soft hyper BCK-ideal is considered. Characterizations of fuzzy soft positive implicative hyper BCK-ideal are provided. Using the notion of positive implicative hyper BCK-ideal, a fuzzy soft weak (strong) hyper BCK-ideal is established. [ABSTRACT FROM AUTHOR]

As a foundation to study the similarity measure with applications in pattern recognition, (normalized) Hamming distances and (normalized) Euclidean distances between hyper structures based on two and three parameters are established. An example to show that these distances are not metrics in general is provided, and conditions for these distances to be metrics are considered. The notions of length fuzzy ideals with four types are introduced, and related properties and their relations are investigated. Relations between length fuzzy ideals and length fuzzy subalgebras are investigated. Conditions for a length fuzzy subalgebra to be a length fuzzy ideal are provided. Characterizations of a length fuzzy ideal are considered. [ABSTRACT FROM AUTHOR]

The concept and properties of cubic sets were first introduced and investigated by Jun et al. in 2011. In this paper, we first introduce the concept of cubic sets in UP-subalgebras and consider the UP-ideals of a UP-algebra. Relationships between the cubic UP-subalgebras and the cubic UP-ideals of a UP-algebra are investigated. In addition, the homomorphic images and the inverse images of the cubic UP-subalgebras and the cubic UP-ideals are also studied. Related properties of UP -algebras and their UP-ideals are explored and discussed. Finally, the Cartesian product of cubic UP-algebras are investigated. [ABSTRACT FROM AUTHOR]

The notion of a neutrosophic positive implicative N-ideal in BCK-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N-ideal and a neutrosophic positive implicative N-ideal are discussed. Characterizations of a neutrosophic positive implicative N-ideal are considered. Conditions for a neutrosophic N-ideal to be a neutrosophic positive implicative N-ideal are provided. An extension property of a neutrosophic positive implicative N-ideal based on the negative indeterminacy membership function is discussed. [ABSTRACT FROM AUTHOR]

As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in BCK/BCI-algebra is considered. The notions of α-internal, β-internal, α-external and b-external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of α-internal and α-external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in BCK/BCI-algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notions of closed intersectional soft BCI-ideals and intersectional soft commutative BCI-ideals are introduced, and related properties are investigated. Conditions for an intersectional soft BCI-ideal to be closed are provided. Characterizations of an intersectional soft commutative BCI-ideal are established, and a new intersectional soft c-BCI-ideal from an old one is constructed. [ABSTRACT FROM AUTHOR]

Abstract: The notions of cubic -ideals and cubic -ideals are introduced, and several related properties are investigated. Characterizations of a cubic -ideal are established. Relations between cubic -ideals, cubic -ideals and cubic -ideals are discussed. The cubic extension property of a cubic -ideal is discussed. [Copyright &y& Elsevier]

Abstract: Xu et al. introduced the concept of vague soft sets, which is an extension to the soft set and the vague set. In this paper, we apply the concept of vague soft sets to hemiring theory. The notion of -vague (soft) left -ideals of a hemiring is introduced and some related properties are investigated. [Copyright &y& Elsevier]

Abstract: On the basis of the theory of a falling shadow, the notion of a falling fuzzy positive implicative ideal of a BCK-algebra is introduced. Relations between falling fuzzy positive implicative ideals and falling fuzzy ideals are given. Relations between fuzzy positive implicative ideals and falling fuzzy positive implicative ideals are provided. [ABSTRACT FROM AUTHOR]

Abstract: A generalization of an -fuzzy ideal of a BCK/BCI-algebra is discussed. Characterizations of an -fuzzy ideal and an -fuzzy ideal are provided. Conditions for an -fuzzy ideal (resp. -fuzzy ideal) to be a fuzzy ideal are provided. Using the notion of a fuzzy ideal with thresholds, characterizations of a fuzzy ideal, an -fuzzy ideal and an -fuzzy ideal are discussed. [Copyright &y& Elsevier]

Abstract: In the paper [P.K. Maji, R. Biswas, A.R. Roy, Fuzzy soft sets, J. Fuzzy Math. 9 (3) (2001) 589–602], Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision making problem. The aim of this manuscript is to apply fuzzy soft set for dealing with several kinds of theories in BCK/BCI-algebras. The notions of fuzzy soft BCK/BCI-algebras, (closed) fuzzy soft ideals and fuzzy soft -ideals are introduced, and related properties are investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we show that Theorem 4.17 and Corollary 4.18 coincide with Theorem 4.11 and Corollary 4.12, respectively, in [W. A. Dudek, M. Shabir, M. Irfan Ali, -fuzzy ideals of hemirings, Comput. Math. Appl. 58 (2009) 310–321]. We discuss more updated results than Theorem 4.17 and Corollary 4.18 in the paper [W. A. Dudek, M. Shabir, M. Irfan Ali, -fuzzy ideals of hemirings, Comput. Math. Appl. 58 (2009) 310–321]. [Copyright &y& Elsevier]

Abstract: Using the ideas of belonging and quasi-coincidence of a fuzzy point with a fuzzy set, the concepts of -fuzzy ideals and -fuzzy generalized bi-ideals, which are generalization of fuzzy ideals and fuzzy generalized bi-ideals, in a semigroup are introduced, and related properties are investigated. We also define the lower and upper parts of fuzzy subsets of a semigroup. Characterizations of regular semigroups by the properties of the lower part of -fuzzy left ideals, -fuzzy quasi-ideals and -fuzzy generalized bi-ideals are given. [Copyright &y& Elsevier]

Abstract: Molodtsov [D. Molodtsov, Soft set theory–First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft -ideals and -idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) -ideals in BCI-algebras. We provide relations between fuzzy -ideals and -idealistic soft BCI-algebras. [Copyright &y& Elsevier]

.We prove that there exists a maximal ideal of a BCI-algebra X in which each subalgebra is an ideal. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The notion of hybrid ideals in B C K / B C I -algebras is introduced, and related properties are investigated. Characterizations of hybrid ideals are discussed. Relations between hybrid ideals and hybrid subalgebras are considered. Characterizations of hybrid ideals are considered. Based on a hybrid structure, properties of special sets are investigated, and conditions for the special sets to be ideals are displayed. [ABSTRACT FROM AUTHOR]

Recent trends in modern information processing have focused on polarizing information, and and bipolar fuzzy sets can be useful. Bipolar fuzzy sets are one of the important tools that can be used to distinguish between positive information and negative information. Positive information, for example, already observed or experienced, indicates what is guaranteed to be possible, and negative information indicates that it is impossible, prohibited, or certainly false. The purpose of this paper is to apply the bipolar fuzzy set to BCK/BCI-algebras. The notion of (translated) k-fold bipolar fuzzy sets is introduced, and its application in BCK/BCI-algebras is discussed. The concepts of k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are introduced, and related properties are investigated. Characterizations of k-fold bipolar fuzzy subalgebra/ideal are considered, and relations between k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are displayed. Extension of k-fold bipolar fuzzy subalgebra is discussed. [ABSTRACT FROM AUTHOR]

A fuzzy set is an extension of an existing set using fuzzy logic. Soft set theory is a generalization of fuzzy set theory. Fuzzy and soft set theory are good mathematical tools for dealing with uncertainty in a parametric manner. The aim of this article is to introduce the concept of makgeolli structures using fuzzy and soft set theory and to apply it to BCK/BCI-algebras. The notion of makgeolli algebra and makgeolli ideal in BCK/BCI-algebras is defined, and several properties are investigated. It deals with the relationship between makgeolli algebra and makgeolli ideal, and several examples are given. Characterization of makgeolli algebra and makgeolli ideal are discussed, and a new makgeolli algebra from old one is established. A condition for makgeolli algebra to be makgeolli ideal in BCK-soft universe is considered, and we give example to show that makgeolli ideal is not makgeolli algebra in BCI-soft universe. Conditions for makgeolli ideal to be makgeolli algebra in BCI-soft universe are provided. [ABSTRACT FROM AUTHOR]

Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision-making problem. Maji et al. also introduced the notion of intuitionistic fuzzy soft sets in the paper [P.K. Maji, R. Biswas and A.R. Roy, Intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics, 9 (2001), no. 3, 677–692]. The aim of this manuscript is to apply the notion of intuitionistic fuzzy soft set to hyper BCK algebras. The notions of intuitionistic fuzzy soft hyper BCK ideal, intuitionistic fuzzy soft weak hyper BCK ideal, intuitionistic fuzzy soft s-weak hyper BCK-ideal and intuitionistic fuzzy soft strong hyper BCK-ideal are introduced, and related properties and relations are investigated. Characterizations of intuitionistic fuzzy soft (weak) hyper BCK ideal are considered. Conditions for an intuitionistic fuzzy soft weak hyper BCK ideal to be an intuitionistic fuzzy soft s-weak hyper BCK ideal are provided. Conditions for an intuitionistic fuzzy soft set to be an intuitionistic fuzzy soft strong hyper BCK ideal are given. [ABSTRACT FROM AUTHOR]

A neutrosophic set is initiated by Smarandache, and it is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache. Using this idea, Jun et al. introduced the notion of neutrosophic quadruple B C K / B C I -numbers, and studied neutrosophic quadruple B C K / B C I -algebras. As a continuation of Jun et al.'s paper, the notion of implicative neutrosophic quadruple B C K -algebras is introduced, and several properties are investigated. Given a set Y, conditions for the neutrosophic quadruple Y-set N q (Y) to be a neutrosophic quadruple B C I -algebra are provided. Conditions for the neutrosophic quadruple Y-set N q (Y) to be an implicative neutrosophic quadruple B C K -algebra are provided. Given subsets I and J of a B C K -algebra Y, conditions for the neutrosophic quadruple (I , J) -set N q (I , J) to be an implicative ideal of the neutrosophic quadruple B C K -algebra N q (Y) are discussed. [ABSTRACT FROM AUTHOR]

Intuitionistic falling shadow is introduced, and applied to B C K / B C I -algebras. Falling intuitionistic subalgebra and falling intuitionistic ideal of B C K / B C I -algebras are introduced, and related properties are investigated. Relations between falling intuitionistic subalgebra and falling intuitionistic ideal are discussed. A characterization of falling intuitionistic ideal is established. [ABSTRACT FROM AUTHOR]

The concept of a ( ∈ , ∈ ) -neutrosophic ideal is introduced, and its characterizations are established. The notions of neutrosophic permeable values are introduced, and related properties are investigated. Conditions for the neutrosophic level sets to be energetic, right stable, and right vanished are discussed. Relations between neutrosophic permeable S- and I-values are considered. [ABSTRACT FROM AUTHOR]

The notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups are introduced, and several properties are investigated. Using these notions, characterizations of subsemigroups and left (resp., right) ideals are discussed. The concept of hybrid product is also introduced, and characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals are considered by using the notion of hybrid product. Relations between hybrid intersection and hybrid product are displayed. [ABSTRACT FROM AUTHOR]