{
  "abstract": "We consider the problem of determining if a given two-dimensional nonnegative integer matrix $M$ is the product of two such matrices, excluding trivial units. A matrix $M$ with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of $2 \\times 2$ matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26033",
  "author_total": 4,
  "authors": [
    {
      "name": "Paul C. Bell"
    },
    {
      "name": "Eva Foster"
    },
    {
      "name": "Daniel Reidenbach"
    },
    {
      "name": "Pavel Semukhin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3sn83qf6nr2qj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26033",
  "fulltext_url": null,
  "id": "3sn83qf6nr2qj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26033",
  "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.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.26033",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3sn83qf6nr2qj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26033"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Factorisability of Low Dimensional Non-Negative Integer Matrices",
  "title_text": "Factorisability of Low Dimensional Non-Negative Integer Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/3sn83qf6nr2qj",
  "venue": "arXiv",
  "versions": []
}