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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]

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]

The notions of falling fuzzy (implicative, positive implicative and fantastic) filters of a BL-algebra are introduced based on the theory of falling shadows and fuzzy sets. The relations between fuzzy (implicative, positive implicative and fantastic) filters and falling fuzzy (implicative, positive implicative and fantastic) filters are provided. Finally, we apply the concept of falling fuzzy inference relations to BL-algebras and obtain some related results. [ABSTRACT FROM AUTHOR]

Based on the theory of falling shadows and fuzzy sets, the notions of falling fuzzy (implicative) filters of R0-algebras are introduced. The relations between fuzzy (implicative) filters and falling fuzzy (implicative) filters are provided. Finally, we apply the concept of falling fuzzy inference relations to R0-algebras and obtain some related results. [ABSTRACT FROM AUTHOR]

Abstract: The fuzzification of generalized Tarski filters of generalized Tarski algebras is considered, and related properties are investigated. Characterizations of a fuzzy generalized Tarski filter are established. Based on the theory of a falling shadow, the notion of a falling fuzzy generalized Tarski filter of a generalized Tarski algebra is introduced. Relations between fuzzy generalized Tarski filters and falling fuzzy generalized Tarski filters are provided. A characterization of a falling fuzzy generalized Tarski filter is established. [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: The notions of fuzzy hyper-MV-subalgebras, (weak, strong) fuzzy hyper-MV-deductive systems and previously weak fuzzy hyper-MV-deductive systems are introduced, and their relations/properties are investigated. [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]

Abstract: The notions of (transitive) soft -algebras, soft edge -algebras, soft -algebras, soft -ideals, soft -ideals, soft -ideals, and -idealistic (-idealistic, or -idealistic) soft -algebras are introduced. Also, their related properties are surveyed. [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]

When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and the key to a solution cannot be easily found, we feel the need to approach them for hours and from various directions. As mathematicians, we wish we had the mathematical tools that apply to these processes. If these mathematical tools were developed, we would be able to apply them to algebra, topology, graph theory, etc., from a close point of view, and we would be able to apply these research results to decision-making and/or coding theory, etc., from a distant point of view. In light of this view, the purpose of this study is to introduce the notion of a multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), and to apply it to algebraic structure, in particular, a BCK/BCI-algebra. The notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra are introduced, and related properties are investigated. Relations between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal are discussed. Characterizations of a k-polar intuitionistic fuzzy subalgebra/ideal are provided, and conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal are provided. In a BCI-algebra, relations between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal are discussed. A characterization of a closed k-polar intuitionistic fuzzy ideal is considered, and conditions for a k-polar intuitionistic fuzzy ideal to be closed are provided. [ABSTRACT FROM AUTHOR]

In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion of co-filter on hoops is introduced and related properties are investigated. Then by using of co-filter, a congruence relation on hoops is defined, and the associated quotient structure is studied. Thus Brouwerian semilattices, Heyting algebras, Wajsberg hoops, Hilbert algebras and BL-algebras are obtained. [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]

In this paper, the notion of cubic intuitionistic q-ideals in B C I -algebras is introduced. A relationship between a cubic intuitionistic subalgebra, a cubic intuitionistic ideal, and a cubic intuitionistic q-ideal is discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic q-ideal are provided. Characterizations of a cubic intuitionistic q-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic q-ideal is established. Furthermore, the product of cubic intuitionistic subalgebras, ideals, and q-ideals are investigated. [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]

Abstract: The concepts of -fuzzy (positive implicative, implicative and commutative) ideals and -fuzzy (positive implicative, implicative and commutative) ideals in BCI-algebras are introduced. Some new characterizations are investigated. In particular, we prove that a fuzzy set of a BCI-algebra is an -fuzzy implicative ideal of if and only if it is both an -fuzzy positive implicative ideal and an -fuzzy commutative ideal. Finally, we give some characterizations of three particular classes of BCI-algebras by these generalized fuzzy ideals. [Copyright &y& Elsevier]