A residual obstruction to amalgamation in cancellative residuated lattices

We give a span of three finitely generated, commutative integral cancellative residuated chains that has no amalgam among commutative cancellative residuated lattices, or among semilinear cancellative residuated lattices. The span settles amalgamation questions posed in 2013 by Gil-Férez, Ledda and Tsinakis (published in 2015) and subsequently repeated by Fussner and Santschi. The obstruction uses the additive monoids 2ℕ₀, ℕ₀∖{1} and ℕ₀, with reverse numerical order. Uniqueness of roots forces two elements in an amalgam to coincide, while their residuals disagree. More generally, the variety generated by the residuated chain of a nonzero additive submonoid of ℕ₀ has amalgamation exactly when that monoid is cyclic. GitHub repository