A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of $(\alpha ,\beta )$-complex fuzzy sets and then define $\left(\alpha ,\beta \right)$-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an $(\alpha ,\beta )$-complex fuzzy subgroup and define $(\alpha ,\beta )$-complex fuzzy normal subgroups of given group. We extend this ideology to define $(\alpha ,\beta )$-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that $(\alpha ,\beta )$-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of $(\alpha ,\beta )$-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the $(\alpha ,\beta )$-complex fuzzy subgroup of the classical quotient group and show that the set of all $(\alpha ,\beta )$-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of $\left(\alpha ,\beta \right)$-complex fuzzy subgroups and investigate the $(\alpha ,\beta )$-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

In this paper, the concepts of fuzzy translation and fuzzy multiplication on a BRK-algebra are introduced. We investigated fuzzy translation and fuzzy multiplication (BRK-subalgebras & BRK-ideals) in BRK-algebras and discussed related properties. Finally, we presented the nation fuzzy magnified--translation on BRK-algebra.

We introduce the notions of hesitant anti‐fuzzy soft set (subalgebras and ideals) and provide relation between them. However, we study new types of hesitant anti‐fuzzy soft ideals (implicative, positive implicative, and commutative). Also, we stated and proved some theorems which determine the relationship between these notions.

Cubic multipolar structure with finite degree (briefly, cubic k-polar ($C_{k}P$) structure) is a new hybrid extension of both k-polar fuzzy (${}_{k}PF$) structure and cubic structure in which $C_{k}P$ structure consists of two parts; the first one is an interval-valued k-polar fuzzy ($I{V}_{k}PF$) structure acting as a membership grade extended from the interval $\mathbb{P}[0,1]$ to $\mathbb{P}{[0,1]}^k$ (i.e., from interval-valued of real numbers to the k-tuple interval-valued of real numbers), and the second one is a ${}_{k}PF$ structure acting as a nonmembership grade extended from the interval $[0,1]$ to ${[0,1]}^k$ (i.e., from real numbers to the k-tuple of real numbers). This approach is based on generalized cubic algebraic structures using polarity concepts and therefore the novelty of a $C_{k}P$ algebraic structure lies in its large range comparative to both ${}_{k}PF$ algebraic structure and cubic algebraic structure. The aim of this manuscript is to apply the theory of $C_{k}P$ structure on BCK/BCI-algebras. We originate the concepts of $C_{k}P$ subalgebras and (closed) $C_{k}P$ ideals. Moreover, some illustrative examples and dominant properties of these concepts are studied in detail. Characterizations of a $C_{k}P$ subalgebra/ideal are given, and the correspondence between $C_{k}P$ subalgebras and (closed) $C_{k}P$ ideals are discussed. In this regard, we provide a condition for a $C_{k}P$ subalgebra to be a $C_{k}P$ ideal in a BCK-algebra. In a BCI-algebra, we provide conditions for a $C_{k}P$ subalgebra to be a $C_{k}P$ ideal, and conditions for a $C_{k}P$ subalgebra to be a closed $C_{k}P$ ideal. We prove that, in weakly BCK-algebra, every $C_{k}P$ ideal is a closed $C_{k}P$ ideal. Finally, we establish the $C_{k}P$ extension property for a $C_{k}P$ ideal.

This paper presents the concept of the notions of bipolar fuzzy commutative ideals of a BCK-algebra. We also study various properties regarding this concept. We prove various characterisations for bipolar fuzzy commutative ideals. Moreover, a relationship between bipolar fuzzy commutative ideals and bipolar fuzzy positive implicative ideals is presented. In addition, we present the notation of bipolar fuzzy characteristic commutative ideal. Then a relationship between a bipolar fuzzy characteristic commutative ideal and its level-cut commutative ideals is provided.

Many extension and generalization of fuzzy sets have been introduced and studied in the literature. Picture fuzzy set acquired more concentration in the domain of decision making, as many real time circumstances might have choices of more than one and researchers are looking for optimum choice/decision. Spherical fuzzy digraph is a generalization of intuitionistic fuzzy set and fuzzy graph. In this paper, we redefine some preliminary operations of Spherical fuzzy graph and it is referred as spherical fuzzy digraph. We discuss some arithmetic operations and relations among spherical fuzzy digraph. We further proposed a method to solve a shortest path problem using score function.