On Quantum-MV algebras - Part II: Orthomodular lattices, softlattices and widelattices

Orthomodular lattices generalize the Boolean algebras; they have arisen in the study of quantum logic. Quantum-MV  algebras were introduced as non-lattice theoretic generalizations of MV algebras and as non-idempotent generalizations of orthomodular lattices.In this paper, we continue the research in the “world” of involutive algebras of the form (A,⊙,−,1), with 1−= 0, 1 being the last element. We clarify now some aspects concerning the quantum-MV (QMV) algebras as non-idempotent generalizations of orthomodular lattices. We study in some detail the orthomodular lattices (OMLs) and we introduce and study two generalizations of them, the orthomodular softlattices (OMSLs) and the ortho-modular widelattices (OMWLs). We establish systematically connections between OMLs and OMSLs/OMWLs and QMV, pre-MV, metha-MV, orthomodular algebras and ortholattices, orthosoftlattices/orthowidelattices - connections illustrated in 22 Figures. We prove, among others, that the transitive OMLs coincide with the Boolean algebras, that the OMSLs coincide with the OMLs, that the OMLs are included in OMWLs and that the OMWLs are a proper subclass of QMV algebras. The transitive and/or the antisymmetric case is also studied.

On Quantum-MV algebras - Part II: Orthomodular lattices, softlattices and widelattices Keywords:

m-MEL algebra , m-BE algebra , m-pre-BCK algebra , m-BCK algebra , MV algebra , quantum-MV algebra , pre-MV algebra , metha-MV algebra , orthomodular algebra , ortholattice , orthosoftlattice , orthowidelattice , Boolean Algebra