Algebraic Perspective of Cubic Multi-Polar Structures on BCK/BCI-Algebras

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 P F ) 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 P F ) 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 k P F 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 P F 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.