{
  "abstract": "We introduce and investigate the irreducibility of endomorphisms of finitely generated free semigroups, i.e., we investigate when an endomorphism $ϕ: Σ^+ -> Σ^+$, where $Σ$ is any alphabet, can be nontrivially expressed as a composition $ϕ= ψ_2 \\circ ψ_1$ of endomorphisms $ψ_1, ψ_2: Σ^+ -> Σ^+$. We, hence, study a notion of primality in the endomorphism monoid of the free semigroup---a natural and fundamental concept in this algebraic structure. We establish that irreducibility is a nontrivial property for the class of so-called rank-preserving endomorphisms, and we provide a characteristic condition separating the reducible and irreducible endomorphisms. We also characterise when an endomorphism is a factor of another endomorphism, analyse the non-uniqueness of factorisations of a rank-preserving endomorphism into its irreducible components, and investigate the use of incidence matrices to give insights into the (ir-)reducibility of rank-preserving endomorphisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: EPTCS 451, 2026, pp. 60-74",
  "arxiv": "2610.07122",
  "author_total": 3,
  "authors": [
    {
      "name": "Paul C. Bell"
    },
    {
      "name": "Eva Foster"
    },
    {
      "name": "Daniel Reidenbach"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a3wsfd69gg3hp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07122",
  "fulltext_url": null,
  "id": "a3wsfd69gg3hp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07122",
  "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.FL"
    ],
    [
      "arxiv",
      "math.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.07122",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a3wsfd69gg3hp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07122",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07122"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Irreducibility of Endomorphisms of Finitely Generated Free Semigroups",
  "title_text": "Irreducibility of Endomorphisms of Finitely Generated Free Semigroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/a3wsfd69gg3hp",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.4204/eptcs.451.5",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}