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

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Radfar A.; Rezaei A.

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Journal Paper

Paper Information

Journal: JOURNAL OF ALGEBRAIC HYPERSTRUCTURES AND LOGICAL ALGEBRAS Year:2023 | Volume:4 | Issue:2 Page(s): 167-187

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Information Journal Paper

Title

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

Author(s)

Radfar A. | Rezaei A. | Issue Writer Certificate

Keywords

BI-algebra‎

‎(distributive‎

‎commutative‎

‎exchangeable) pseudo BI-algebra‎

‎implication algebra

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.

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