{
  "abstract": "An algebra with binary operations $+,\\cdot,\\uparrow$ and constant $1$ is called an HSI algebra if it satisfies the basic commutative semiring laws for $+,\\cdot,1$ on~$\\mathbb{N}$ as well as the familiar index laws for exponentiation $\\uparrow$. These basic axioms, known as the ``High School Identities'' are known to be incomplete, and an algebra satisfying $\\HSI$ but failing an equation valid on ${\\bf N}:=\\langle \\mathbb{N};+,\\cdot,\\uparrow,1\\rangle$ is called a \\emph{Gurevič algebra}. It is currently unknown if there is an algorithm to recognise finite Gurevič algebras, and the best current result is that there exists a 12-element Gurevič algebra, and that none of the five 2-element HSI algebras are Gurevič algebras. We explain how the work of Alex Wilkie can be used to provide an algorithm for deciding validity of the equational laws of $\\langle \\mathbb{N};+,\\cdot,\\uparrow,1\\rangle$, and use this to show that multiplicatively-idempotent HSI algebras satisfy all valid laws of $\\langle \\mathbb{N};+,\\cdot,\\uparrow,1\\rangle$. As consequences of this result, we show that all 44 HSI algebras on 3 elements lie in the variety of $\\langle \\mathbb{N};+,\\cdot,\\uparrow,1\\rangle$ (that is, are not Gurevič algebras), as well as 597 of the 657 models on 4 elements and 11158 of the 13577 models on 5 elements. A further consequence is that every Brouwerian lattice lies in the variety of $\\langle \\mathbb{N};+,\\cdot,\\uparrow,1\\rangle$ (up to a simple term equivalence), showing that the variety of $\\langle \\mathbb{N};+,\\cdot,\\uparrow,1\\rangle$ has continuum many subvarieties. The constant-free signature is also explored, and it is shown that in this case all $2$-element models of the constant-free High School Laws satisfy all valid constant-free laws in $\\langle \\mathbb{N};+,\\cdot,\\uparrow\\rangle$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16922",
  "author_total": 3,
  "authors": [
    {
      "name": "Tumadhir Alsulami"
    },
    {
      "name": "Marcel Jackson"
    },
    {
      "name": "Michael Kinyon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ygk8s1d3esc7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16922",
  "fulltext_url": null,
  "id": "6ygk8s1d3esc7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16922",
  "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.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7213418197",
  "pdf_url": "https://arxiv.org/pdf/2609.16922",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ygk8s1d3esc7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16922",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16922"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Multiplicatively idempotent HSI algebras satisfy all equations of $\\mathbb{N}$",
  "title_text": "Multiplicatively idempotent HSI algebras satisfy all equations of ℕ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/6ygk8s1d3esc7",
  "venue": "arXiv",
  "versions": []
}