In this paper, we introduce the notion of $\sup$-hesitant fuzzy ideals of semigroups, which is the general notion of hesitant fuzzy ideals and interval-valued fuzzy ideals. In addition, $\sup$-hesitant fuzzy ideals are characterized in terms of sets, fuzzy sets, hesitant fuzzy sets and interval-valued fuzzy sets. Finally, $\sup$-hesitant fuzzy translations and $\sup$-hesitant fuzzy extensions of $\sup$-hesitant fuzzy ideals of semigoups are discussed, and their relations are investigated.
The concept of the direct product of finite family of B-algebras is introduced by Lingcong and Endam [J. A. V. Lingcong and J. C. Endam, Direct product of B-algebras, Int. J. Algebra,10(1):33-40, 2016.]. In this paper, we introduce the concept of the direct product of infinite family of B-algebras, we call the external direct product, which is a generalization of the direct product in the sense of Lingcong and Endam. Also, we introduce the concept of the weak direct product of B-algebras. Finally, we provide several fundamental theorems of (anti-)B-homomorphisms in view of the external direct product B-algebras.
The main aim of this article is to introduce the concept of a sup-hesitant fuzzy ideal, which is a generalization of a hesitant fuzzy ideal and an interval-valued fuzzy ideal, in a ternary semigroup. Some characterizations of a sup-hesitant fuzzy ideal are examined in terms of a fuzzy set, a hesitant fuzzy set, and an interval valued fuzzy set. Further, we discuss the relation between an ideal and a generalization of a characteristic hesitant fuzzy set and a characteristic interval-valued fuzzy set.
The notion of BP-algebras was introduced by Ahn and Han [2] in 2013, which is related to several classes of algebra. It has been examined by several researchers. In the group, the concept of the direct product (DP) [21] was initially developed and given some features. Then, other algebraic structures are subjected to DP groups. Lingcong and Endam [16] examined the idea of the DP of (0-commutative) B-algebras and B-homomorphisms in 2016 and discovered several related features, one of which is a DP of two Balgebras that is a B-algebra. Later on, the concept of the DP of B-algebra was expanded to include finite family B-algebra, and some of the connected issues were researched. In this work, the external direct product (EDP), a general concept of the DP, is established, and the results of the EDP for certain subsets of BP-algebras are determined. In addition, we define the weak direct product (WDP) of BP-algebras. In light of the EDP BP-algebras, we conclude by presenting numerous essential theorems of (anti-)BP-homomorphisms.
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