Amalgamation in classes of involutive commutative residuated lattices
Sándor Jenei · arXiv (Cornell University) · 2020
We study amalgamation in odd and even involutive commutative residuated chains through their categorical representation by bunches of linearly ordered abelian groups. The representation separates three obstructions. A discrete $κ_J$-layer imports the failure of amalgamation for discrete abelian ordered groups with normal embeddings. Even when $κ_J$ is empty, distinguished layer subgroups obstruct amalgamation in the unrestricted idempotent-symmetric classes. Moreover, purity of every induced layer embedding does not suffice when the two targets enlarge the positive-idempotent skeleton in different ways. We then isolate a sufficient positive regime. For every finite $n\geq1$, the idempotent-symmetric odd chains, the idempotent-symmetric even chains with idempotent falsum, and their union have the Amalgamation Property after further restriction to algebras with exactly $n$ positive idempotents and divisible canonical layer groups. Fixed $n$ identifies the skeletons, divisibility makes the layer embeddings pure, ordinary abelian-group pushouts remain torsion-free, and a simultaneous order-extension theorem produces an amalgamating bunch. Although failure of SAP already follows formally from failure of AP, we also give an independent one-layer witness to failure of SAP in the unrestricted idempotent-symmetric odd and even classes. At the variety level, the essential counterexamples yield a general transfer criterion for failure of AP in semilinear varieties; in particular, AP fails for the varieties generated by the idempotent-symmetric odd and even chains, and for every semilinear variety containing the variety of odd semilinear involutive commutative residuated lattices. Finally, although every fixed-$n$, layer-divisible chain class considered here has AP, among the varieties they generate only the odd one-layer variety has AP.