$beta$-relations on implicative bounded hyper BCK-algebras

Abstract

In this paper we consider the notion of hyper implicative bounded BCK- algebras, give some examples and introduce the relation $beta$ on them. Then we let $beta^∗$ be the transitive closure of $beta$ . In hyper implicative bounded BCK-algebra theory, the fundamental relation is defined as the smallest equivalence relation so that the quotient would be the (fundamental) BCK-algebra. We show that$beta^∗$ is the fundamental re- lation on a hyper implicative bounded BCK-algebra. Finally, we state conditions that are equivalent with the transitivity of this relation.