Hesitant Anti-Fuzzy Soft Set in BCK-Algebras

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

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

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

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

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