{
  "abstract": "We introduce the median uniformity $\\mathcal U_{\\mathrm m}$, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology $τ_{\\mathrm m}$ provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When the intrinsic topology $τ_{\\mathrm m}$ is Hausdorff and all intervals are compact, the MMC is the unique proper median compactification and is canonically isomorphic to the Roller--Fioravanti compactification. We apply this uniform framework to continuous actions of a topological group $G$ by median automorphisms. We show that the MMC is a median $G$-compactification. In the finite-rank case, the resulting compact $G$-system is Rosenthal representable and dynamically tame.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.16096",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Megrelishvili"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0y2fc42xxxgz6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.16096",
  "fulltext_url": null,
  "id": "0y2fc42xxxgz6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.16096",
  "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.DS"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7161592739",
  "pdf_url": "https://arxiv.org/pdf/2605.16096",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0y2fc42xxxgz6.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:19Z",
      "native_id": "2605.16096",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.16096"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Intrinsic uniform structure on median algebras",
  "title_text": "Intrinsic uniform structure on median algebras",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/0y2fc42xxxgz6",
  "venue": "arXiv",
  "versions": []
}