  Based on implicative involutive BE algebras, we redefine the orthomodular
lattices, by introducing the notion of implicative-orthomodular lattices, and
we study their properties. We characterize these algebras, proving that the
implicative-orthomodular lattices are quantum-Wajsberg algebras. We also define
and characterize the implicative-modular algebras as a subclass of
implicative-orthomodular lattices. The orthomodular softlattices and
orthomodular widelattices are also redefined, by introducing the notions of
implicative-orthomodular softlattices and implicative-orthomodular
widelattices. Finally, we prove that the implicative-orthomodular softlattices
are equivalent to implicative-orthomodular lattices and that the
implicative-orthomodular widelattices are special cases of quantum-Wajsberg
algebras.
