{
  "abstract": "For a partition $X = Y \\sqcup Z$ of a set $X$ and an arbitrary map $g \\colon Y \\to {\\rm Hom}_{\\rm Set} (Z, \\, Z)$, we construct a solution $R_g \\colon X \\times X \\to X \\times X$ of the braid equation. We characterize intrinsically the solutions which are isomorphic to some $R_g$ and we give a necessary and sufficient condition for two solutions $R_g$ and $R_{g'}$ to be isomorphic. Consequently, the number of isomorphism classes of the solutions $R_g$ on a set with $n$ elements is $1 + \\sum_{k = 1}^{n-1} \\widetilde{o} (n - k, \\, k)$, where $\\widetilde{o} (m, \\, k)$ is the number of orbits of the symmetric group $S_m$, acting by conjugation, on the set of multisets with $k$ elements of maps $\\{1, \\ldots, m\\} \\to \\{1, \\ldots, m\\}$. This construction is then used to obtain a solution of the braid equation whose automorphism group is exactly the centralizer ${\\rm Aut} (f_1, \\ldots, f_k)$ of a given family $f_1, \\ldots, f_k \\colon X \\to X$ of maps, $k \\geq 1$. For $k = 2$, combined with a result of Agore, Chirvasitu and the author, this gives another proof of the fact that any group is isomorphic to the automorphism group of a solution of the braid equation (equivalently, of the set-theoretic Yang--Baxter equation).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03380",
  "author_total": 1,
  "authors": [
    {
      "name": "Gigel Militaru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctm2w1jkgjafe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03380",
  "fulltext_url": null,
  "id": "ctm2w1jkgjafe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03380",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03380",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctm2w1jkgjafe.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03380",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03380"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Centralizers of maps as automorphism groups of Yang--Baxter solutions",
  "title_text": "Centralizers of maps as automorphism groups of Yang--Baxter solutions",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/ctm2w1jkgjafe",
  "venue": "arXiv",
  "versions": []
}