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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