  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.
