Semilinear idempotent distributive ℓ-monoids

We prove a representation theorem for totally ordered idempotent monoids via a nested sum construction. Using this representation theorem we obtain a characterization of the subdirectly irreducible members of the variety of semilinear idempotent distributive ℓ-monoids and a proof that its lattice of subvarieties is countably infinite. For the variety of commutative idempotent distributive ℓ-monoids we give an explicit description of its lattice of subvarieties and show that each of its subvarieties is finitely axiomatized. Finally we give a characterization of which spans of totally ordered idempotent monoids have an amalgam in the class of totally ordered monoids, showing in particular that the class of totally ordered commutative idempotent monoids has the strong amalgamation property and that various classes of distributive ℓ-monoids do not have the amalgamation property. We also show that exactly seven non-trivial finitely generated subvarieties of the variety of semilinear idempotent distributive ℓ-monoids have the amalgamation property; we are able to determine for all but three of its subvarieties whether they have the amalgamation property or not.