{
  "abstract": "For an additively idempotent semiring (ai-semiring) $S$, let $S^0$ denote the ai-semiring obtained from $S$ by adjoining a new element $0$. In this paper, we develop an approach to investigate the interval $[\\mathsf{V}(S), \\mathsf{V}(S^0)]$ of ai-semiring varieties between the variety generated by $S$ and that generated by $S^0$. We establish a general sufficient condition under which this interval has the cardinality of the continuum. This is applied in particular to $[\\mathsf{V}(S_7), \\mathsf{V}(S_7^0)]$, where $S_7$ is a $3$-element ai-semiring and is a nonfinitely based algebra of the smallest possible order, thereby resolving an open problem proposed by Jackson, Ren, and Zhao (J. Algebra \\textbf{611} (2022), 211--245). The same conclusion holds for $[\\mathsf{V}(B_2^1), \\mathsf{V}((B_2^1)^0)]$, where $B_2^1$ is the ai-semiring whose multiplicative reduct is the $6$-element Brandt semigroup. We also present a sufficient condition for the nonfinite basis property in ai-semiring varieties. As a corollary, we obtain a new proof of Dolinka's theorem (Internat. J. Algebra Comput. \\textbf{17} (2007), no.~8, 1537--1551) that the $7$-element ai-semiring $(B_2^1)^0$ has no finite basis for its identities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24514",
  "author_total": 3,
  "authors": [
    {
      "name": "Zidong Gao"
    },
    {
      "name": "Miaomiao Ren"
    },
    {
      "name": "Xianzhong Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e73h0rzdsk79q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24514",
  "fulltext_url": null,
  "id": "e73h0rzdsk79q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24514",
  "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": "W7213999120",
  "pdf_url": "https://arxiv.org/pdf/2609.24514",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e73h0rzdsk79q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24514",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24514"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Continuum in the Lattice of Semiring Varieties: The Interval $[\\mathsf{V}(S), \\mathsf{V}(S^0)]$",
  "title_text": "A Continuum in the Lattice of Semiring Varieties: The Interval [V(S), V(S^0)]",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/e73h0rzdsk79q",
  "venue": "arXiv",
  "versions": []
}