{
  "abstract": "We study the equational theory and subvariety structure of the ai-semiring variety $\\V_\\infty$ generated by all flat semirings $S(a_1\\cdots a_k)$, where the letters \\(a_i\\) are pairwise distinct. Using interval semirings of posets, we characterize its subdirectly irreducible members and describe variety membership in terms of jointly separating families of strict order-preserving maps. We obtain explicit finite identity bases for \\(\\V_\\infty\\) and each $\\V_k$ generated by $S(a_1\\cdots a_k)$. Consequently, every flat semiring \\(S(W)\\) associated with a nonempty set \\(W\\) of linear words is finitely based. This yields finitely based ai-semirings with exactly \\(k\\)-nilpotent multiplicative reduct for each \\(k\\geq 1\\). For each \\(k\\geq 1\\), let \\(\\B_k\\) be the subvariety of \\(\\V_\\infty\\) defined by the \\((k+1)\\)-nilpotent identity. We prove that each \\(\\B_k\\) is generated by a finite interval semiring and that every proper subvariety of \\(\\V_\\infty\\) is contained in some \\(\\B_k\\). The variety \\(\\B_3\\) is a Cross variety with exactly \\(11\\) subvarieties, whereas \\([\\V_k,\\B_k]\\), \\([\\V_k,\\V_{k+1}]\\), and \\([\\B_{k-1},\\B_k]\\) each contain continuum many subvarieties for every $k\\geq 4$. In particular, this provides infinitely many finitely based finite semirings $S(a_1\\cdots a_k)$ whose generated variety has continuum many subvarieties for every \\(k\\geq 5\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06351",
  "author_total": 2,
  "authors": [
    {
      "name": "Zidong Gao"
    },
    {
      "name": "Yilin Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4exk6s670bew9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06351",
  "fulltext_url": null,
  "id": "4exk6s670bew9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06351",
  "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": "W7220726476",
  "pdf_url": "https://arxiv.org/pdf/2610.06351",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4exk6s670bew9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06351",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06351"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Equational Theories of Interval Semirings of Posets",
  "title_text": "Equational Theories of Interval Semirings of Posets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4exk6s670bew9",
  "venue": "arXiv",
  "versions": []
}