{
  "abstract": "We study algebras for the $\\mathbf{BM}$-, $\\mathbf{BI}$-, and $\\mathbf{BMI}$-fragments of combinatory logic, where $\\mathbf{B} xyz \\to_{w} x\\left(yz\\right)$, $\\mathbf{M} x \\to_{w} xx$ and $\\mathbf{I} x \\to_{w} x$. We prove that every finite $\\mathbf{BM}$-algebra $\\mathcal{A}$ contains an element $Y$ satisfying $Yx = x\\left(Yx\\right)$ for all $x\\in A$. Moreover, a single closed $\\mathbf{BM}$-term represents such an element in every finite $\\mathbf{BM}$-algebra. On the other hand, we show that there exists an infinite $\\mathbf{BMI}$-algebra with no such element. Consequently, the variety of $\\mathbf{BM}$-algebras is not generated by finite $\\mathbf{BM}$-algebras. We also characterise finite $\\mathbf{BI}$-algebras as finite semigroups with two distinguished left units $B$ and $I$, and finite $\\mathbf{BMI}$-algebras as finite idempotent semigroups with three distinguished left units $B$, $M$ and $I$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37108",
  "author_total": 1,
  "authors": [
    {
      "name": "Takuma Imamura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9gcnbqe95mwn1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37108",
  "fulltext_url": null,
  "id": "9gcnbqe95mwn1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37108",
  "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",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7224134137",
  "pdf_url": "https://arxiv.org/pdf/2610.37108",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9gcnbqe95mwn1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37108",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37108"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Finite $\\mathbf{BM}$-algebras and fixed-point combinators",
  "title_text": "Finite BM-algebras and fixed-point combinators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/9gcnbqe95mwn1",
  "venue": "arXiv",
  "versions": []
}