Pseudo-BI-algebras‎: ‎Non-commutative generalization of BI-algebras

Document Type : Original Article

Authors

1 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran

2 Department of Mathematics, Payame Noor University

Abstract

‎We ‎define and study the pseudo BI-algebras as a generalization of BI-algebras and implication algebras and investigate  some properties‎. Also‎, ‎we define distributive pseudo BI-algebras and construct a BI-algebra related to these‎. ‎Further‎, ‎we  prove ‎there is no proper pseudo BI-algebra of the order less than 4 and that every pseudo BI-algebra of order 4 is a  poset‎, ‎and so is a pseudo BH-algebra‎. ‎‎Beside‎, ‎we introduce exchangeable pseudo BI-algebra and show that the class of  them is a proper subclass of the class pseudo CI-algebras‎. ‎Finally‎, ‎we define the notions of (weak) commutative pseudo  BI-algebras and prove ‎every weak commutative pseudo BI-algebra is a (dual) pseudo BH-algebra‎, ‎but the converse is not  true‎, ‎and show that every exchangeable commutative pseudo BI-algebra is an implication algebra. 

Keywords

References

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Journal of Algebraic Hyperstructures and Logical Algebras
Volume 4, Issue 2
November 2023
Pages 167-187

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