On Involutive Weak Exchange Algebras

In this paper, involutive weak exchange algebras (for short, involutive WE algebras) are introduced and studied. Their properties and characterizations are investigated. Some important results and examples are given. In particular, it is proven that in involutive WE algebras, the properties (BB), (B), (*), (**) and (Tr) are equivalent. Moreover, involutive BE, involutive GE, involutive pre-BCK and involutive pre-Hilbert algebras are considered, their connections are established. It is shown that involutive WE algebras (respectively, involutive GE algebras) satisfying the commutative property are Wajsberg algebras (respectively, Boolean algebras). Finally, the interrelationships between the classes of involutive algebras considered here are presented.