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Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics — arXiv:2609.09529

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.09529
License (as archived): CC0-1.0
Archived: 2026-10-16T10:56:30.839Z
SHA-256 of the archived text: dc4bde35fa5216878eff291f16f5b801fab7aa0d1a1423fdf5f8841dcb216f01
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How to cite

Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics — arXiv:2609.09529 — arXiv (en), https://arxiv.org/abs/2609.09529 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_620ddb8ceff4 (Metraipolis archive dc_620ddb8ceff4, sha256 dc4bde35fa5216878eff291f16f5b801fab7aa0d1a1423fdf5f8841dcb216f01)

BibTeX

@misc{dc620ddb8ceff4,
  title = {Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics — arXiv:2609.09529},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.09529},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_620ddb8ceff4; sha256 dc4bde35fa5216878eff291f16f5b801fab7aa0d1a1423fdf5f8841dcb216f01},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.09529. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-10 in math.LO).
Categorie arXiv: Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2609.09529 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T09:31:02.200Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-10?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).