{
  "abstract": "The Burnside ring of a finite category, introduced by Webb, generalizes the classical Burnside ring of a finite group. However, unlike the classical case, the Burnside ring of a finite category has finite rank if and only if the category is equivalent to a groupoid. In this article, we introduce a new invariant associated with a finite category $\\mathcal{C}$, called the \\emph{simple Burnside ring} of $\\mathcal{C}$ and denoted by $B^S(\\mathcal{C})$. This construction is obtained from simple $\\mathcal{C}$-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring $B^S(\\mathcal{C})$ always has finite rank. We develop the basic theory of simple $\\mathcal{C}$-sets and study several structural properties of the ring $B^S(\\mathcal{C})$. In particular, we determine all ring homomorphisms from $B^S(\\mathcal{C})$ to $\\mathbb{Z}$, describe its prime spectrum, and obtain a decomposition theorem expressing $B^S(\\mathcal{C})$ as a product of simple Burnside rings of strongly connected subcategories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.25036",
  "author_total": 4,
  "authors": [
    {
      "name": "José Miguel Calderón León"
    },
    {
      "name": "Alberto G. Raggi-Cárdenas"
    },
    {
      "name": "Itzel Rosas"
    },
    {
      "name": "Ramón H. Ruiz-Medina"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fx7t7km6swn24.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.25036",
  "fulltext_url": null,
  "id": "fx7t7km6swn24",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.25036",
  "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.CT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7171683374",
  "pdf_url": "https://arxiv.org/pdf/2607.25036",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fx7t7km6swn24.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2607.25036",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.25036"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Burnside ring of simple $\\mathcal{C}$-sets",
  "title_text": "The Burnside ring of simple 𝒞-sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/fx7t7km6swn24",
  "venue": "arXiv",
  "versions": []
}