implicative pre-hilbert algebras and their connections with other algebras of logic

In the paper, implicative pre-hilbert algebras are introduced and studied, their characterizations and connections with some algebras of logic are presented. some important results and examples are given. in particular, it is proven that an implicative pre-hilbert algebra is equivalent to an implicative pre-bcc algebra. it is shown that for any hilbert algebra, the implicative property is equivalent to the commutative property. moreover, several old or new characterizations of tarski algebras are established. we prove that tarski algebras coincide with commutative pre-hilbert algebras and with implicative generalized exchange algebras satisfying the property of antisymmetry. finally, the hierarchies existing between all classes of implicative algebras considered here are shown.