Soft Computing

, Volume 22, Issue 4, pp 1189–1201 | Cite as

Commutative deductive systems of pseudo-BCK-algebras

  • Lavinia Corina CiunguEmail author


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.


Pseudo-BCK-algebra Commutative deductive system State operator State-morphism operator State pseudo-BCK-algebra 


Compliance with ethical standards

Conflict of interest

The author declares that she has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA

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