connections between ge algebras and pre-hilbert algebras

Ge algebras (generalized exchange algebras) and pre-hilbert algebras are a generalization of well-known hilbert algebras. in the paper, connections between these algebras are studied. in particular, it is proven that pi-be algebras are equivalent with ge algebras satisfying the exchange property. some characterizations of transitive ge algebras and exchange pre-hilbert algebras are given. it is shown that the intersection of classes of ge algebras and pre-hilbert algebras is the class of transitive ge algebras. moreover, ge, be and pre-hilbert algebras with the antisymmetry property are investigated. it is proven that transitive ge algebras satisfying the property of antisymmetry coincide with hilbert algebras. finally, positive implicative ge and pre-hilbert algebras are considered, their connections with some algebras of logic are presented. in addition, the hiearchies existing between the classes of algebras studied here are shown.