Weak pseudo-BCK algebras

Abstract In this paper we define and study the weak pseudo-BCK algebras as generalizations of weak BCK-algebras, extending some results given by Cı⃖rulis for weak BCK-algebras. We give some characterizations of weak pseudo-BCK algebras and we prove that a weak pseudo-BCK algebra satisfying the right distribution laws is a BCK-algebra. We define the class of commutative weak pseudo-BCK algebras, and we give equivalent definitions and characterization theorems for commutative weak pseudo-BCK algebras. The classes of quasi pseudo-BCK algebras and weak pseudo-BCK(E) algebras are introduced and a characterization theorem for quasi pseudo-BCK algebras is given. We prove that any weak pseudo-BCK(E) algebra is a pseudo-BE algebra and the class of commutative weak pseudo-BCK(E) algebras is equivalent to the class of commutative pseudo-BCK algebras.