LAUSR

In this paper, we generalize the axiom systems given by M. Pałasiński and B. Woźniakowska for commutative BCK-algebras to the case of commutative pseudo-BCK-algebras. A characterization of commutative pseudo-BCK-algebras is also given. We define the commutative deductive systems of pseudo-BCK-algebras, and we generalize some results proved by Yisheng Huang for commutative ideals of BCI-algebras to the case of commutative deductive systems of pseudo-BCK-algebras. We prove that a pseudo-BCK-algebra A is commutative if and only if all the deductive systems of A are commutative. We show that a normal deductive system H of a pseudo-BCK-algebra A is commutative if and only if A / H is a commutative pseudo-BCK-algebra. We introduce the notions of state operators and state-morphism operators on pseudo-BCK-algebras, and we apply these results on commutative deductive systems to investigate the properties of these operators.