Some Implicativities for Groupoids and BCK-Algebras

Abstract

In this paper, we generalize the notion of an implicativity discussed in B C K -algebras, and apply it to some groupoids and B C K -algebras. We obtain some relations among those axioms in the theory of groupoids.

Transposition Regular TA-Groupoids and Their Structures

“…Figure 4 shows the main results of the TA-groupoids obtained in this paper. In [28], Hwang et al defined the levels of implicativities on the groupoid. In future research, we will use the levels of implicativities the TA-groupoids to study the relationships between the TA-groupoids and the related logic algebras (as shown in [29,30]).…”

On ringoids

“…Let R be the set of all real numbers. We define a binary operation " * " on R by x * y := A + Bx + Cy, for all x, y ∈ R, where A, B, C ∈ R. We call such a groupoid (R, * ) is a linear groupoid [7,9] over reals.…”

“…The theory of groupoids [3,4] has been introduced by some researchers. It has been combined with the theory of general algebraic structures [7,9,10]. One of the methods for the generalization of axioms is to employ special functions, i.e., by using of proper mappings, we may generalize axioms in mathematical structures.…”

On block commutative groupoids

“…The theory of groupoids [3,4] has been introduced by some researchers. It has been combined with the theory of general algebraic structures [7,10,11]. One of the methods for the generalization of axioms is to employ special functions, i.e., by using of proper mappings, we may generalize axioms in mathematical structures.…”