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Pairs of partial orders and the amalgamation property

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Paolo Lipparini

Logic · Submitted 10 Mar 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property, Fra\"\i ss\'e limits and, in many cases, an $ \omega$-categorical model completion with quantifier elimination.

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arXiv ID2303.06205 (version 2)
Pith IDO6GMUCOA, Pith's permanent record of this paper and its review
CitationPaolo Lipparini. "Pairs of partial orders and the amalgamation property." arXiv:2303.06205 (2023). https://pith.science/paper/O6GMUCOA
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,742 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.58 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.71 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property, Fra\"\i ss\'e limits and, in many cases, an $ \omega$-categorical model completion with quantifier elimination. The same applies to Kronheimer and Penrose's causal spaces. On the other hand, Urquhart doubly ordered sets do not have the amalgamation property. Our main tool is the superamalgamation property (the strong amalgamation property is not enough), thus we provide further arguments suggesting the usefulness of the superamalgamation property also in pure model theory, not only in algebraic logic.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property, Fra\"\i ss\'e limits and, in many cases, an $ \omega$-categorical model completion with quantifier elimination. result 0.89 0.93 0.79 asserted 1.00
s1 The same applies to Kronheimer and Penrose's causal spaces. result 0.97 0.54 0.06 asserted 1.00
s2 On the other hand, Urquhart doubly ordered sets do not have the amalgamation property. result 0.91 0.59 0.07 asserted 1.00
s3 Our main tool is the superamalgamation property (the strong amalgamation property is not enough), thus we provide further arguments suggesting the usefulness of the superamalgamation property also in pure model theory, not only in algebraic logic. method 0.64 0.87 0.23 hedged possibility 0.70

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property, Fra\"\i ss\'e limits and, in many cases, an $ \omega$-categorical model completion with quantifier elimination. The same applies to Kronheimer and Penrose's causal spaces. On the other hand, Urquhart doubly ordered sets do not have the amalgamation property. Our main tool is the superamalgamation property (the strong amalgamation property is not enough), thus we provide further arguments suggesting the usefulness of the superamalgamation property also in pure model theory, not only in algebraic logic.

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