{
  "abstract": "We present an explicit infinite equational basis for the six-element additively idempotent semiring $TR_6$ and prove that $TR_6$ is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by $TR_6$, showing that it forms a four-element chain. Our results demonstrate that the variety generated by $TR_6$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, $\\mathsf{V}(TR_6)$ is the first explicit limit subvariety of the variety generated by the max-plus algebra $\\mathbf{N}$. Moreover, $TR_6$ is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring $SR_6$, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by $SR_6$ and $TR_6$, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09220",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiaolei Shao"
    },
    {
      "name": "Miaomiao Ren"
    },
    {
      "name": "Zidong Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9n75g25pagpsb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09220",
  "fulltext_url": null,
  "id": "9n75g25pagpsb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09220",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7212157987",
  "pdf_url": "https://arxiv.org/pdf/2609.09220",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9n75g25pagpsb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09220",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09220"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety",
  "title_text": "A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/9n75g25pagpsb",
  "venue": "arXiv",
  "versions": []
}