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

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.