pseudo-bi-algebras‎: ‎non-commutative generalization of bi-algebras

‎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.