Commutative deductive systems of pseudo BCK-algebras

TL;DR

This paper extends the theory of commutative pseudo BCK-algebras by generalizing axioms, characterizing their structure, and exploring the properties of deductive systems and state operators within these algebraic frameworks.

Contribution

It introduces generalized axioms for commutative pseudo BCK-algebras, characterizes their structure, and studies the properties of deductive systems and state operators in this context.

Findings

A pseudo BCK-algebra is commutative iff all its deductive systems are commutative.

Normal commutative deductive systems lead to quotient algebras that are commutative.

Properties of state and state-morphism operators are analyzed within commutative pseudo BCK-algebras.

Abstract

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.