{
  "abstract": "For ordered sets $X$ and $Y$, let $Y^X$ denote the ordered set of order-preserving maps from $X$ to $Y$, ordered pointwise. In 1984, D. Duffus asked whether $A^A\\cong B^B$ forces $A\\cong B$ when $A$ and $B$ are finite ordered sets. We prove that it does. The proof combines direct decompositions with an auxiliary ordering of connected ordered sets and a careful choice of intervals of maximum height. We then extend the reconstruction result to several forms of repeated exponentiation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20817",
  "author_total": 2,
  "authors": [
    {
      "name": "Jonathan David Farley"
    },
    {
      "name": "George Grätzer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgs040vvhh36s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20817",
  "fulltext_url": null,
  "id": "bgs040vvhh36s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20817",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222443687",
  "pdf_url": "https://arxiv.org/pdf/2610.20817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgs040vvhh36s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The $A^A$ problem for finite ordered sets",
  "title_text": "The A^A problem for finite ordered sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/bgs040vvhh36s",
  "venue": "arXiv",
  "versions": []
}