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# Commutative deductive systems of pseudo BCK-algebras

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### 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.

### Author and article information

###### Journal
2016-03-16
###### Article
1603.05241