Varieties of De Morgan bisemilattices

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 Płonka sum construction (De Morgan- Płonka 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-Płonka representations of its members, and a syntactic description of its valid identities. In many cases, we also give an axiomatisation relative to DMBL.

Figures (7)

• Figure 1: The 4-element De Morgan lattice $\mathbf{DM}_4$
• Figure 2: The subdirectly irreducible involutive semilattices
• Figure 3: The lattice of subvarieties of $\mathcal{ISL}$ and their relative axiomatisations
• Figure 4: Some subvarieties of the variety $\mathcal{DMBL}$
• Figure 5: Subdirectly irreducible De Morgan bisemilattices in $\mathcal{S}$

Theorems & Definitions (41)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Proposition 3.1
• proof
• Corollary 3.1
• proof