Semilinear De Morgan monoids and epimorphisms

TL;DR

This paper proves a representation theorem for certain semilinear De Morgan monoids, establishes their local finiteness, and shows that epimorphisms are surjective in these and related varieties, impacting substructural logic definability.

Contribution

It introduces a new representation theorem for negatively generated semilinear De Morgan monoids and proves epimorphism-surjectivity in these and related algebraic varieties.

Findings

De Morgan monoids satisfying semilinearity and negative generation form a locally finite variety.

Epimorphisms are surjective in all varieties of negatively generated semilinear De Morgan monoids.

Results resolve questions about Beth-style definability in substructural logics.

Abstract

A representation theorem is proved for De Morgan monoids that are (i) semilinear, i.e., subdirect products of totally ordered algebras, and (ii) negatively generated, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.