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 (α,β)-complex fuzzy sets and then define α,β-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of α,β-complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

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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.

Cubic multipolar structure with finite degree (briefly, cubic k-polar (CkP) structure) is a new hybrid extension of both k-polar fuzzy (kPF) structure and cubic structure in which CkP structure consists of two parts; the first one is an interval-valued k-polar fuzzy (IVkPF) structure acting as a membership grade extended from the interval P[0,1] to 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 kPF 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 CkP algebraic structure lies in its large range comparative to both kPF algebraic structure and cubic algebraic structure. The aim of this manuscript is to apply the theory of CkP structure on BCK/BCI-algebras. We originate the concepts of CkP subalgebras and (closed) CkP ideals. Moreover, some illustrative examples and dominant properties of these concepts are studied in detail. Characterizations of a CkP subalgebra/ideal are given, and the correspondence between CkP subalgebras and (closed) CkP ideals are discussed. In this regard, we provide a condition for a CkP subalgebra to be a CkP ideal in a BCK-algebra. In a BCI-algebra, we provide conditions for a CkP subalgebra to be a CkP ideal, and conditions for a CkP subalgebra to be a closed CkP ideal. We prove that, in weakly BCK-algebra, every CkP ideal is a closed CkP ideal. Finally, we establish the CkP extension property for a CkP ideal.