{
  "abstract": "Dyer conjectured that the join of two biclosed sets of positive roots can be described by increasing Bruhat paths whose reflection labels belong to their union. We give a type-uniform proof of this conjecture for all finite Coxeter groups. More generally, for a $2$-coclosed set $C$ of positive roots contained in an inversion set, we show that its $2$-closure is an inversion set $I(w)$ and that the elements reachable from the identity using reflections labeled by roots in $C$ form exactly the set $[e,w]_B w^{-1}$. The proof combines Dyer's closure criteria with a root-selection argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27253",
  "author_total": 3,
  "authors": [
    {
      "name": "Yibo Gao"
    },
    {
      "name": "Yulin Peng"
    },
    {
      "name": "Hanlin Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af32pdyv0hbpf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27253",
  "fulltext_url": null,
  "id": "af32pdyv0hbpf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27253",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214184431",
  "pdf_url": "https://arxiv.org/pdf/2609.27253",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af32pdyv0hbpf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.27253",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27253"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounded joins of biclosed sets",
  "title_text": "Bounded joins of biclosed sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/af32pdyv0hbpf",
  "venue": "arXiv",
  "versions": []
}