Generalizations of quantum-Wajsberg algebras

Abstract

Starting from involutive BE algebras, we redefine the pre-MV and meta-MV algebras, by introducing the notion of pre-Wajsberg and meta-Wajsberg algebras, as generalizations of quantum-Wajsberg algebras. We characterize these algebras, we show that any pre-Wajsberg algebra is a meta-Wajsberg algebra, and we prove that an implicative-orthomodular algebra is a quantum-Wajsberg algebra if and only if it is a meta-Wajsberg algebra. It is shown that the commutative quantum-Wajsberg (implicative-orthomodular, pre-Wajsberg, meta-Wajsberg) algebras coincide with the Wajsberg algebras. We give conditions for the implicative-orthomodular, pre-Wajsberg and meta-Wajsberg algebras to be Wajsberg algebras. The commutative center of an implicative-orthomodular algebra is defined and studied, and it is proved that it is a Wajsberg subalgebra of the implicative-orthomodular algebra.