{
  "abstract": "A dcpo \\(D\\) is called \\(C_σ\\)-unique if for every dcpo \\(Q\\), \\( Γ(D)\\cong Γ(Q) \\) implies \\( D\\cong Q, \\) where \\(Γ(D)\\) denotes the lattice of Scott-closed subsets of \\(D\\). We give an intrinsic characterization of \\(C_σ\\)-uniqueness in terms of Skula-density and Scott closure, identifying precisely when a proper sub-dcpo can preserve the lattice of Scott-closed sets. Based on this characterization, we answer three open problems: (1) every power $\\J^I$ of Johnstone's dcpo is $C_σ$-unique; (2) $C_σ$-uniqueness is not preserved by binary products, even when both factors and their product are sober; and (3) a sober countable complete lattice need not be $C_σ$-unique, even when it is a frame.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40191",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuxu Chen"
    },
    {
      "name": "Xulong He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f6k156c94p73s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40191",
  "fulltext_url": null,
  "id": "f6k156c94p73s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40191",
  "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.GN"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7215106318",
  "pdf_url": "https://arxiv.org/pdf/2609.40191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f6k156c94p73s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$C_σ$-Unique Dcpos: Intrinsic Characterizations and Counterexamples",
  "title_text": "C_σ-Unique Dcpos: Intrinsic Characterizations and Counterexamples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/f6k156c94p73s",
  "venue": "arXiv",
  "versions": []
}