A finite obstruction to amalgamation in distributive lattice-ordered monoids

We give a span of commutative idempotent distributive lattice-ordered monoids, with a three-element base and two four-element extensions, which has no amalgam in any distributive lattice-ordered monoid. One extension is a Boolean square with join as multiplication; the other is a chain. The obstruction gives negative answers to the three amalgamation questions posed by Simon Santschi in 2022 and completes the classification of subvarieties of semilinear idempotent distributive lattice-ordered monoids with the amalgamation property: there are exactly seven nontrivial such varieties. The same calculation removes a finite-generation hypothesis from ten known finite obstructions. We also give explicit incompatible existential conditions over the base; failure of model completion follows from the standard implication from model completion to amalgamation for universal theories. GitHub repository