{
  "abstract": "An explicit finite identity basis is given for the eight-element semiring of upper triangular Boolean \\(2\\times2\\) matrices in the signature \\((+,\\cdot)\\). The basis consists of a known multiplicative basis, the ai-semiring laws, and 30 mixed identities, each using at most eight variables. The proof combines a finite basis for the multiplicative reduct with finite rules for shuffling words, duplicating a marked occurrence, and interchanging adjacent occurrences while adding prescribed witnesses. This converts the one-letter gap criterion into a derivation of every valid semiring identity. We also study the band subvariety, a distinguished 156-element interval of the subvariety lattice, congruences, flat members, and finite representations of free algebras. Every \\(m\\)-letter word has an equivalent subword of length at most \\(m^2\\) for \\(m\\geq2\\), and the free algebras have doubly exponential rank growth. For arbitrary partially ordered sets, we determine the equational theory of Boolean relation semirings and of finite-support incidence semirings over nontrivial bounded distributive lattices: finite height \\(h\\) gives the theory of \\(T_h\\), while unbounded height gives precisely the identities of all additively idempotent semirings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32356",
  "author_total": 2,
  "authors": [
    {
      "name": "Jun Jiao"
    },
    {
      "name": "Xiaolei Shao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wa7gsa5f8ngz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32356",
  "fulltext_url": null,
  "id": "3wa7gsa5f8ngz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32356",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7214724406",
  "pdf_url": "https://arxiv.org/pdf/2609.32356",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wa7gsa5f8ngz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.32356",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32356"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The finite basis problem for $2\\times 2$ triangular Boolean matrix semirings and incidence semirings",
  "title_text": "The finite basis problem for 2× 2 triangular Boolean matrix semirings and incidence semirings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/3wa7gsa5f8ngz",
  "venue": "arXiv",
  "versions": []
}