{
  "abstract": "Let $S=(\\N,\\max,+)$ be the max-plus algebra of the nonnegative integers, and let $\\overline S$ be obtained from $S$ by adjoining $-\\infty$ as a semiring zero. We first prove that the varieties generated by the full matrix semirings $\\Mat_n(S)$ and by the upper triangular matrix semirings $\\UT_n(\\overline S)$ form strictly ascending chains. Every interval between distinct members of either chain has cardinality $2^{\\aleph_0}$ and contains a chain and an antichain of that cardinality. The proof uses finite syntactic semirings associated with prime-cycle languages and an independent family of identities. We then construct finite commutative ai-semirings from languages of monomials closed under divisibility and prove their bounded-variable property by cancellation, explicit integer evaluations and independence in a free abelian group. A loop-block identity for upper triangular matrices and dimension-independent cubic-padding identities for full matrices show that every variety in each of the above intervals has infinite axiomatic rank. In fact, every variety between $\\V(S)$ and the variety generated by all full matrix dimensions has infinite axiomatic rank. These varieties are therefore nonfinitely based, also relative to the ai-semiring axioms. Finally, for every fixed full matrix dimension, we prove equational equivalence over all six positive or nonnegative integer, rational and real coefficient domains. The original convex-ideal countermodels and their full-matrix transfer proof are recalled separately for comparison; they are not used in the algebraic proofs. Throughout, the signature consists of two binary operations and contains no constant symbols.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20257",
  "author_total": 2,
  "authors": [
    {
      "name": "Jun Jiao"
    },
    {
      "name": "Mengya Yue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3a95pqv3d7xgy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20257",
  "fulltext_url": null,
  "id": "3a95pqv3d7xgy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20257",
  "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": "W7222039073",
  "pdf_url": "https://arxiv.org/pdf/2610.20257",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3a95pqv3d7xgy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20257",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20257"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The finite basis problem and subvariety intervals for max-plus matrix semirings",
  "title_text": "The finite basis problem and subvariety intervals for max-plus matrix semirings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/3a95pqv3d7xgy",
  "venue": "arXiv",
  "versions": []
}