  In this paper we generalize the axiom systems given by M. Pa{\l}asi\'nski, B.
Wo\'zniakowska and by W.H. Cornish 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.
