{
  "abstract": "The main aim of this paper is to introduce a Frobenius--Galois expansion of the substructural logic $\\mathbf{FL}_{ew}$ and develop its algebraic and categorical semantics. The resulting logic, denoted by $\\mathbf{FL}^{\\mathbf{FGC}}_{ew}$, is obtained by adjoining a pair of unary connectives forming a Galois connection and satisfying suitable Frobenius-type compatibility conditions. Firstly, we prove that $\\mathbf{FL}^{\\mathbf{FGC}}_{ew}$ is algebraizable in the sense of Blok and Pigozzi and identify its equivalent algebraic semantics with the variety of Frobenius-adjoint residuated lattices, establishing the conservativity over $\\mathbf{FL}_{ew}$ and relating its finite model property to residual finiteness of finitely generated free algebras. Secondly, for the distributive setting, we lift the Kalman construction to the Frobenius-adjoint algebraic framework. More precisely, we establish a categorical equivalence between Frobenius-adjoint residuated distributive lattices and Frobenius-adjoint $c$-differential residuated distributive lattices with the condition $\\mathbf{CK}$. This equivalence yields a positive-cone representation of the former structures and, in turn, a logical counterpart of the categorical correspondence. Finally, for the distributive extension $\\mathbf{FL}^{\\mathbf{FGC},d}_{ew}$, we prove that derivability, validity over Frobenius-adjoint residuated distributive lattices, and validity over the corresponding positive cones determine the same consequence relation, and further show that this correspondence preserves equational and quasi-equational consequence, conservativity, and finite countermodels. These results provide a unified algebraic, categorical, and logical framework for Frobenius--Galois expansions of substructural logics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09529",
  "author_total": 3,
  "authors": [
    {
      "name": "Juntao Wang"
    },
    {
      "name": "jieqiong Shi"
    },
    {
      "name": "Mei Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1x5y7bkqcnrjt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09529",
  "fulltext_url": null,
  "id": "1x5y7bkqcnrjt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09529",
  "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.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7212154136",
  "pdf_url": "https://arxiv.org/pdf/2609.09529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1x5y7bkqcnrjt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics",
  "title_text": "Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/1x5y7bkqcnrjt",
  "venue": "arXiv",
  "versions": []
}