{
  "abstract": "We show that regarding the model-theoretic properties of distributive lattices isomorphic to finite products of linear orders there is a significant symmetry: if ${\\mathbb X}_i$, $i<n$ are linear orders of size $>1$, ${\\mathcal T}:=\\mathop{\\rm Th}\\nolimits (\\prod _{i<n}{\\mathbb X} _i)$ and ${\\mathcal T}_i:=\\mathop{\\rm Th}\\nolimits ({\\mathbb X}_i)$, for $i<n$, then ${\\mathcal T}$ is finitely axiomatizable (resp. atomic, small) iff ${\\mathcal T}_i$, $i<n$, are finitely axiomatizable (resp. atomic, small); the countable atomic (resp. countably saturated) model of ${\\mathcal T}$, if it exists, is the product of countable atomic (resp. countably saturated) models of ${\\mathcal T}_i$; further, we describe all models of ${\\mathcal T}$: ${\\mathbb Y} \\models {\\mathcal T}$ iff ${\\mathbb Y}\\cong \\prod _{i<n}{\\mathbb Y}_i$, where ${\\mathbb Y} _i \\models {\\mathcal T}_i$, for $i<n$, and show that ${\\mathbb E} \\preccurlyeq \\prod _{i<n}{\\mathbb X}_i$ iff ${\\mathbb E} = \\prod _{i<n}{\\mathbb E}_i$, where ${\\mathbb E}_i \\preccurlyeq {\\mathbb X}_i$, for $i<n$. ($I({\\mathcal T},ω)\\in \\{ 1,{\\mathfrak{c}}\\}$ was proved in a previous paper.) In addition, for each $n\\in {\\mathbb N}$ the class ${\\mathcal C}_n$ of distributive lattices isomorphic to a direct product of $n$ linear orders of size $>1$ is first order definable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24683",
  "author_total": 1,
  "authors": [
    {
      "name": "Miloš S. Kurilić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/420n8m44bhn0j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24683",
  "fulltext_url": null,
  "id": "420n8m44bhn0j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24683",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7223008395",
  "pdf_url": "https://arxiv.org/pdf/2610.24683",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/420n8m44bhn0j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24683",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24683"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Model Theory of Products of Chains",
  "title_text": "Model Theory of Products of Chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/420n8m44bhn0j",
  "venue": "arXiv",
  "versions": []
}