{
  "abstract": "Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\\leq i\\leq j\\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\\V(R_{01})$ and $\\V(R_{02})$ have eleven subvarieties each, while $\\V(R_{11})$ has sixty-six. The lattice $\\Sub(\\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $γ_n$ and $\\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22667",
  "author_total": 3,
  "authors": [
    {
      "name": "Lili Wang"
    },
    {
      "name": "Jinjing Wu"
    },
    {
      "name": "Aifa Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4c72fjfpa0gft.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22667",
  "fulltext_url": null,
  "id": "4c72fjfpa0gft",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22667",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213996994",
  "pdf_url": "https://arxiv.org/pdf/2609.22667",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4c72fjfpa0gft.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22667",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22667"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Compatible additions on a six-element commutative semigroup: equational bases and subvariety lattices",
  "title_text": "Compatible additions on a six-element commutative semigroup: equational bases and subvariety lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/4c72fjfpa0gft",
  "venue": "arXiv",
  "versions": []
}