From L-Groups to Distributive L-Monoids, and Back Again

Colacito, Almudena; Galatos, Nikolaos; Metcalfe, George; Santschi, Simon (2022). From L-Groups to Distributive L-Monoids, and Back Again. Journal of algebra, 601, pp. 129-148. Elsevier 10.1016/j.jalgebra.2022.02.012

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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 latticeordered 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 are valid 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.

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