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

Semilinear idempotent distributive l-monoids

“…Note, however, that in [37] Repnitskii shows that the variety generated by the inverse-free reducts of abelian -groups is not finitely axiomatizable, gives a recursive axiomatization for this variety, and proves that in contrast to the case of all distributive l-monoids, there are equations that hold in this variety but not in all commutative distributive l-monoids. Extending the result of Repnitskii, it is proved in [9] there are equations that hold in all inverse-free reducts of totally ordered -groups that do not hold in all totally ordered distributive -monoids.…”

“…We denote the variety of semilinear idempotent distributive -monoids by SemIdDLM, i.e., SemIdDLM is the variety generated by IdOM. Note that in the literature semilinear distributive -monoids are also called representable (see e.g., [9]) following the nomenclature for -group.…”

“…Corollary 2.8 ([30, Corollary 2], see also [9]). Every commutative distributive -monoid is semilinear.…”

“…Distributive -monoids are monoids with a distributive lattice-order such that multiplication distributes over both binary meets and binary joins. They occur naturally as inverse-free reducts of lattice-ordered groups ( -groups) and indeed have the same equational theory as the latter [9]. More generally, distributive -monoids occur as residual-free reducts of fully distributive residuated lattices (see e.g., [6]) and a systematic study of distributive -monoids can further the investigation of these reducts and the corresponding implication-free fragments of substructural logics [19] (see e.g, [20]).…”

“…Despite providing one of the most natural generalizations of -groups, the structure theory of distributive -monoids is not yet at a very sophisticated state, possibly because the tools and techniques of -group theory do not extend well in the absence of inverses. In particular, there does not yet exist a good characterization of congruences of distributive -monoids (but see [30,4,9], where congruences are constructed from prime lattice ideals of distributive -monoids) and not much is known about subdirectly irreducible members of this class to date (see [38], covering the finite commutative case). On the other hand, there exists a natural analogue of the representation theorem of Holland [24] of -groups as subalgebras of automorphism -groups of chains.…”