On Quantum-MV algebras - Part II: Orthomodular Lattices, Softlattices and Widelattices

Abstract

‎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 ‎$(A‎, ‎\odot‎, ‎{}^-,1)$‎, ‎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 orthomodular 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.

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