{
  "abstract": "Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set $Π$ of prime numbers, the $Π$-extension property is equivalent to the $Π$-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the $Π$-extension property, we reformate the property using the concepts of EP problems and $Π$-solutions. We then show that the existence of a $Π$-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a $Π$-solution for any $Π$. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the $p$-extension property and the $p$-LERF fail for any prime $p$. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a $Π$-solution for any $Π$. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04870",
  "author_total": 2,
  "authors": [
    {
      "name": "Mahmood Etedadialiabad"
    },
    {
      "name": "Su Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8aac7ntnaxndj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04870",
  "fulltext_url": null,
  "id": "8aac7ntnaxndj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04870",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.04870",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8aac7ntnaxndj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04870",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04870"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Extension properties for partial permutations",
  "title_text": "Extension properties for partial permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8aac7ntnaxndj",
  "venue": "arXiv",
  "versions": []
}