{
  "abstract": "We study Scott-product compatibility and sobriety for countable meet-continuous directed-complete partial orders (dcpos). We construct a countable meet-continuous dcpo $\\mathcal R$ whose Scott space is well-filtered but not sober, although the Scott topology on every finite power is the product of the factor Scott topologies. The ideals of $\\mathcal R$ admit a unique finite-parameter description. We also construct a countable meet-continuous dcpo $\\mathbb{H}$ with a greatest element whose Scott space is coherent and well-filtered, but whose Scott square has a topology strictly finer than the ordinary product topology. These constructions answer negatively two questions about countable meet-continuous dcpos. On the positive side, we prove that a meet-continuous $L$-dcpo, meaning a dcpo whose principal ideals are complete lattices, is Scott sober whenever its Scott square carries the ordinary product topology. This answers Jia's core-compactness question affirmatively for $L$-dcpos without any countability assumption. Finally, we characterize well-filteredness of $L$-dcpos by boundedness of closed Rudin sets and formulate related open questions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02642",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    },
    {
      "name": "Wei Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/60tpng3nyt3tr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02642",
  "fulltext_url": null,
  "id": "60tpng3nyt3tr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02642",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7220427615",
  "pdf_url": "https://arxiv.org/pdf/2610.02642",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/60tpng3nyt3tr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02642",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02642"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Scott products and sobriety of countable meet-continuous dcpos",
  "title_text": "Scott products and sobriety of countable meet-continuous dcpos",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/60tpng3nyt3tr",
  "venue": "arXiv",
  "versions": []
}