Amalgamation in classes of involutive commutative residuated lattices
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.