Exchange pre-Hilbert algebras and their connections with other algebras of logic

Abstract

In the paper, as a generalization of well-known Hilbert algebras, exchange pre-Hilbert algebras are introduced. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, connections between exchange pre-Hilbert algebras, generalized exchange algebras and BE algebras are presented. Finally, implicative and positive implicative algebras are considered. It is shown that implicative (resp. positive implicative) exchange pre-Hilbert algebras are equivalent to implicative BE algebras with (*) (resp. generalized Hilbert algebras).