Catalog · 10.5281/zenodo.23419577
Amalgamation in an infinite chain of commutative idempotent residuated-lattice varieties
- Martino Lupini · University of Bologna
16 October 2026 · CC BY 4.0 · latest of 2 versions
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Amalgamation in an infinite chain of commutative idempotent residuated-lattice varieties
Martino Lupini
Abstract
We construct a strictly increasing infinite chain of finitely generated varieties of commutative idempotent residuated lattices, each with the amalgamation property. The generator of Vₙ is a quasiprimal (n+3)-element algebra obtained by adjoining an absorbing least element to a bounded lattice with n incomparable interior elements; multiplication is join elsewhere. Homogeneity, congruence extension, and the classical homomorphism-extension method yield amalgamation. For n ≥ 2 the varieties are not semilinear, and for n ≥ 3 they are not lattice-distributive. This answers negatively the unrestricted finiteness question posed by Fussner, Metcalfe, and Santschi in Interpolation and the Exchange Rule, circulated in 2023 and published in 2026. The distributive version remains open. Among finite Stanovský algebras L⁰, amalgamation holds exactly when the interior of L is an antichain. Strong amalgamation holds for V₀ and fails for every Vₙ with n ≥ 1.
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- 10.5281/zenodo.23419577
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@unpublished{Lupini202623419577,
title = {Amalgamation in an infinite chain of commutative idempotent residuated-lattice varieties},
author = {Martino Lupini},
year = {2026},
doi = {10.5281/zenodo.23419577},
note = {Zenodo deposit, latest of 2}
}