On PBZ*-lattices

TL;DR

This paper advances the understanding of PBZ*-lattices by analyzing their subvariety lattice, providing axiomatic bases, and relating some subvarieties to known algebraic structures in nonclassical logics.

Contribution

It offers new axiomatic characterizations of subvarieties of PBZ*-lattices and connects certain distributive subvarieties to well-known algebraic frameworks.

Findings

Axiomatic bases for some subvarieties of PBZ*-lattices.

Identification of distributive subvarieties as term-equivalent to Kleene lattices.

Locating PBZ*-lattices within the landscape of nonclassical logic algebraic structures.

Abstract

We continue our investigation of paraorthomodular BZ*-lattices (PBZ*-lattices), started in \cite{GLP1+,PBZ2,rgcmfp,pbzsums,pbz5}. We shed further light on the structure of the subvariety lattice of the variety $\mathbb{PBZL}^{\ast }$ of PBZ*-lattices; in particular, we provide axiomatic bases for some of its members. Further, we show that some distributive subvarieties of $\mathbb{PBZL}^{\ast }$ are term-equivalent to well-known varieties of expanded Kleene lattices or of nonclassical modal algebras. By so doing, we somehow help the reader to locate PBZ*-lattices on the atlas of algebraic structures for nonclassical logics.