From distributive l-monoids to l-groups, and back again

TL;DR

This paper establishes a deep connection between lattice-ordered groups and distributive lattice-ordered monoids, showing their equational theories coincide for inverse-free equations, and provides methods for reducing problems between these structures.

Contribution

It proves the equivalence of inverse-free equation validity in LG and DLM, and introduces an effective reduction method from LG to DLM for equational reasoning.

Findings

DLM has the finite model property.

DLM has a decidable equational theory.

A reduction method from LG to DLM is established.

Abstract

We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive l-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian l-groups but not in all commutative distributive l-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.