Varieties of De Morgan bisemilattices

TL;DR

This paper provides a comprehensive classification of subvarieties within De Morgan bisemilattices, including their generators, representations, and axiomatizations, advancing the algebraic understanding of these structures.

Contribution

It offers a complete lattice-theoretic description of subvarieties of De Morgan bisemilattices, including generators, representations, and axiomatizations, which was not previously known.

Findings

Complete description of the lattice of subvarieties of DMBL

Identification of finite generators for each subvariety

Characterization of De Morgan-P{ }lonka representations

Abstract

De Morgan bisemilattices are expansions of distributive bisemilattices by an involution satisfying De Morgan properties. They have attracted interest both as algebraic models of analytic containment logics, and as a case study for a certain generalisation of the P{\l}onka sum construction (De Morgan- P{\l}onka sums). In this paper, we provide a complete description of the lattice of subvarieties of the variety DMBL of De Morgan bisemilattices. For each subvariety in the lattice, we identify a finite set of finite generators, a characterisation of the De Morgan-P{\l}onka representations of its members, and a syntactic description of its valid identities. In many cases, we also give an axiomatisation relative to DMBL.