{
  "abstract": "We give a negative answer to the question, posed by Erné, whether every $3$-distributive lattice is $ω$-distributive. More precisely, we exhibit a continuous frame that is $κ$-distributive for every integer $κ\\geq 2$, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based $T_0$ topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of $ω$-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Erné's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11671",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Luan"
    },
    {
      "name": "Qingguo Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjte5st664k07.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11671",
  "fulltext_url": null,
  "id": "cjte5st664k07",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11671",
  "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": "W7212243281",
  "pdf_url": "https://arxiv.org/pdf/2609.11671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjte5st664k07.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A continuous $3$-distributive frame that is not $ω$-distributive",
  "title_text": "A continuous $3$-distributive frame that is not ω-distributive",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/cjte5st664k07",
  "venue": "arXiv",
  "versions": []
}