Semigroup Structures and Commutative Ideals of BCK-Algebras Based on Crossing Cubic Set Structures

First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between crossing cubic ideal and commutative crossing cubic ideal is discussed. An example to show that crossing cubic ideal is not commutative crossing cubic ideal is given, and then the conditions in which crossing cubic ideal can be commutative crossing cubic ideal are explored. Characterizations of commutative crossing cubic ideal are discussed, and the relationship between commutative crossing cubic ideal and crossing cubic level set is considered. An extension property of commutative crossing cubic ideal is established, and the translation of commutative crossing cubic ideal is studied. Conditions for the translation of crossing cubic set structure to be commutative crossing cubic ideal are provided, and its characterization is processed.

Fuzzy soft set theory applied to commutative ideals of BCK-algebras

In the present paper, we apply the fuzzy soft set theory to commutative ideals of BCK-algebras. In fact, the notion of fuzzy soft commutative ideals over BCK-algebras is introduced, and related properties are investigated. Relations between fuzzy soft ideals and fuzzy soft commutative ideals are discussed, and conditions for a fuzzy soft ideal to be a fuzzy soft commutative ideal are provided. The “AND” operation, “extended intersection” and “union” of fuzzy soft (commutative) ideals are studied. Furthermore, characterizations of fuzzy soft (commutative) ideals are considered.

()-Fuzzy Commutative Ideals in BCH-Algebras

The purpose of this paper is to introduce the concept of ()-fuzzy commutative ideal in BCH-algebra and investigate some of their related properties.

Uni-Hesitant Fuzzy Set Approach to the Ideal Theory of BCK-Algebras

In this chapter, the author studies the uni-hesitant fuzzy set-theoretical approach to the ideals of BCK-algebras. For a hesitant fuzzy set H on S and a subset of [0,1], the set L(H;ʎ):={x∈S|xH⊆ʎ}, is called the uni-hesitant level set of H. Moreover, the author discusses the relations between uni-hesitant fuzzy commutative ideals and uni-hesitant fuzzy ideals. Further, he considered the characterizations of uni-hesitant fuzzy commutative ideals in BCK-algebras. Finally, he proved some conditions for a uni-hesitant fuzzy ideal to be a uni-hesitant fuzzy commutative ideal.

On (α,β)-US Sets in BCK/BCI-Algebras

Molodtsov originated soft set theory, which followed a general mathematical framework for handling uncertainties, in which we encounter the data by affixing the parameterized factor during the information analysis. The aim of this paper is to establish a bridge to connect a soft set and the union operations on sets, then applying it to B C K / B C I -algebras. Firstly, we introduce the notion of the ( α , β ) -Union-Soft ( ( α , β ) -US) set, with some supporting examples. Then, we discuss the soft B C K / B C I -algebras, which are called ( α , β ) -US algebras, ( α , β ) -US ideals, ( α , β ) -US closed ideals, and ( α , β ) -US commutative ideals. In particular, some related properties and relationships of the above algebraic structures are investigated. We also provide the condition of an ( α , β ) -US ideal to be an ( α , β ) -US closed ideal. Some conditions for a Union-Soft (US) ideal to be a US commutative ideal are given by means of ( α , β ) -unions. Moreover, several characterization theorems of (closed) US ideals and US commutative ideals are given in terms of ( α , β ) -unions. Finally, the extension property for an ( α , β ) -US commutative ideal is established.

Uni-Soft Commutative Ideals and Closed Uni-Soft Ideals in BCI-Algebras

The notions of uni-soft commutative ideals and closed uni-soft ideals in [Formula: see text]-algebras are introduced, and related properties are investigated. Conditions for a uni-soft ideal to be closed are provided. Relations between a uni-soft ideal and a uni-soft commutative ideal are investigated. Conditions for a uni-soft ideal to be a uni-soft commutative ideal are provided. A new uni-soft commutative ideal from an old one is constructed.

Symmetric ring property on nil ideals

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring [Formula: see text] will be said to be right (respectively, left) nil-ideal-symmetric if [Formula: see text] implies [Formula: see text] (respectively, [Formula: see text]) for nil ideals [Formula: see text] of [Formula: see text]. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

Commutative Energetic Subsets of BCK-Algebras

The notions of a C-energetic subset and (anti) permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an (anti) permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal.