{
  "abstract": "For every semigroup $S$, the set $\\mathcal{P}(S)$ of all subsets of $S$ and the set $\\mathcal{P}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively. We investigate the finite basis problem for the full and nonempty power semirings $\\mathcal{P}(S_7)$ and $\\mathcal{P}^{+}(S_7)$ of the multiplicative reduct of $S_7$, where $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For $\\mathcal{P}^{+}(S_7)$, we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval $[\\mathsf{V}(\\mathcal{P}^{+}(S_7)), \\mathsf{V}(\\mathcal{P}(S_7))]$ in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19957",
  "author_total": 3,
  "authors": [
    {
      "name": "Mengya Yue"
    },
    {
      "name": "Miaomiao Ren"
    },
    {
      "name": "Zidong Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/34eezj8qqyqcv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19957",
  "fulltext_url": null,
  "id": "34eezj8qqyqcv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19957",
  "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": "W7213588738",
  "pdf_url": "https://arxiv.org/pdf/2609.19957",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/34eezj8qqyqcv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19957",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19957"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Explicit equational bases for the power semirings of $S_7$",
  "title_text": "Explicit equational bases for the power semirings of S_7",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/34eezj8qqyqcv",
  "venue": "arXiv",
  "versions": []
}