$\beta$-Relations On Implicative Bounded Hyper BCK-algebras

In this paper we consider the notion of hyper implicative bounded BCKalgebras, give some examples and introduce the relation β on them. Then we let β ∗ be the transitive closure of β. 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 β ∗ is the fundamental relation on a hyper implicative bounded BCK-algebra. Finally, we state conditions that are equivalent with the transitivity of this relation.

Hyper BCK-algebra, Strongly regular relation, Fundamental relation