{
  "abstract": "We study maximal (proper) filters in the complete lattice of partitions of an infinite set~$X$. In the language of uniform spaces, these are precisely the atoms of the lattice of zero-dimensional uniformities on $X$, introduced by Pelant and Reiterman and studied further by Pelant, Reiterman, Rödl and Simon. The first half of this paper recovers, sharpens, and extends their classification in purely partition-theoretic terms. {Call a maximal filter of partitions {\\em type I} if it does not contain all finite partitions, and {\\em type II} if it does.} Type I filters are induced, in an essentially unique way, by ultrafilters on families of pairwise disjoint doubletons. {A type II filter} determines a non-principal ultrafilter on $X$, the \\emph{heart}; the heart determines the filter precisely when {the former} is minimal in the Rudin--Keisler order. Each member $F$ of {a type II filter} gives rise to a closed \\emph{fiber} in $X^{*} = βX \\setminus X$, {consisting of} the set of ultrafilters agreeing with the heart on $F$. We prove a trichotomy describing the topology of arbitrary fibers. We then show that the fibers do not encode the filter: fibers do not form a semilattice under intersection, the closure of an infinite discrete set of ultrafilters of a single Rudin--Keisler type (a \\emph{sparse} set, in our terminology) need not be a fiber, and a partition incompatible with a member of the filter may have a {fiber strictly larger than that member.} A representation that does succeed is nevertheless available in another category: the {type-II filters} with heart $\\mathfrak{u}$ correspond to the maximal proper substructures of the ultrapower $X^X\\!/\\mathfrak{u}$ of the full structure on $X$. The topological representation problem remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26201",
  "author_total": 2,
  "authors": [
    {
      "name": "David Victor Feldman"
    },
    {
      "name": "Alexander Wilce"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47kap080gffmj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26201",
  "fulltext_url": null,
  "id": "47kap080gffmj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26201",
  "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",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214119269",
  "pdf_url": "https://arxiv.org/pdf/2609.26201",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/47kap080gffmj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26201",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26201"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Maximal Filters in the Lattice of Partitions of an Infinite Set",
  "title_text": "Maximal Filters in the Lattice of Partitions of an Infinite Set",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/47kap080gffmj",
  "venue": "arXiv",
  "versions": []
}