Amalgamation in classes of involutive commutative residuated lattices

[2012.14181] Amalgamation in classes of involutive commutative residuated lattices Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Logic arXiv:2012.14181 (math) [Submitted on 28 Dec 2020 ( v1 ), last revised 15 Aug 2026 (this version, v11)] Title: Amalgamation in classes of involutive commutative residuated lattices Authors: Sándor Jenei View a PDF of the paper titled Amalgamation in classes of involutive commutative residuated lattices, by S\'andor Jenei View PDF HTML (experimental) Abstract: 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 $\kappa_J$-layer imports the failure of amalgamation for discrete abelian ordered groups with normal embeddings. Even when $\kappa_J$ 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. Subjects: Logic (math.LO) MSC classes: Primary 97H50, 20E06, Secondary 06F05, 06F20, 03B47 Cite as: arXiv:2012.14181 [math.LO] (or arXiv:2012.14181v11 [math.LO] for this version) https://doi.org/10.48550/arXiv.2012.14181 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sándor Jenei [ view email ] [v1] Mon, 28 Dec 2020 10:51:50 UTC (54 KB) [v2] Wed, 13 Jan 2021 09:42:58 UTC (55 KB) [v3] Wed, 20 Jan 2021 10:15:16 UTC (60 KB) [v4] Mon, 25 Jan 2021 12:10:42 UTC (59 KB) [v5] Wed, 27 Jan 2021 19:12:35 UTC (59 KB) [v6] Wed, 3 Feb 2021 10:19:51 UTC (59 KB) [v7] Sun, 14 Nov 2021 19:06:54 UTC (1,360 KB) [v8] Fri, 15 Sep 2023 18:24:43 UTC (115 KB) [v9] Mon, 5 Aug 2024 08:14:48 UTC (63 KB) [v10] Sat, 12 Jul 2025 06:33:07 UTC (120 KB) [v11] Sat, 15 Aug 2026 11:36:09 UTC (33 KB) Full-text links: Access Paper: View a PDF of the paper titled Amalgamation in classes of involutive commutative residuated lattices, by S\'andor Jenei View PDF HTML (experimental) TeX Source view license Current browse context: math.LO < prev | next > new | recent | 2020-12 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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