Skip to content
DraconiticResearch
Search Sources API
203,948 studies31 sourcesUpdated 31 min ago
PreprintCC BY 4.0arXiv

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

  • Juntao Wang
  • jieqiong Shi
  • Mei Wang
Open at sourcePDFDOICite

Abstract

The main aim of this paper is to introduce a Frobenius--Galois expansion of the substructural logic FLew and develop its algebraic and categorical semantics. The resulting logic, denoted by FLFGCew, is obtained by adjoining a pair of unary connectives forming a Galois connection and satisfying suitable Frobenius-type compatibility conditions. Firstly, we prove that FLFGCew 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 FLew 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 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 FLFGC,dew, 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.

Related

  1. Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields
    Shams Ara, Md Fazlul Hoque, Ian Marquette
    arXivPreprintarXiv licenceFull textMathematical physicsQuantum physics & quantum informationarXiv
  2. Connecting Base-Extension Semantics, Beth Semantics, and Ecumenical Logic
    Victor Barroso-Nascimento, Joaquim T. Waddington, Elaine Pimentel
    arXivPreprintCC BY 4.0Full textLogic & foundationsarXiv
  3. Closure Semantics for Weak Modal Logics: Completeness and Representations
    Yuki Nishimura
    arXivPreprintCC BY 4.0Full textLogic & foundationsarXiv
  4. How to prove equations using diagrams: A categorical and operational semantics for e-graphs
    Evan Patterson
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Formal Bipolar Weighted Semantics for Generalized Argumentation Frameworks
    Stefano Ferilli
    MathematicsPeer-reviewedCC BY 4.0General mathematicsCrossref

Cite

BibTeXRIS
BibTeX
@misc{wang2026frobenius,
  title = {{Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics}},
  author = {Juntao Wang and jieqiong Shi and Mei Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09529},
  eprint = {2609.09529},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09529},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics
AU  - Juntao Wang
AU  - jieqiong Shi
AU  - Mei Wang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09529
UR  - https://arxiv.org/abs/2609.09529
AB  - The main aim of this paper is to introduce a Frobenius--Galois expansion of the substructural logic FL_ew and develop its algebraic and categorical semantics. The resulting logic, denoted by FL^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 FL^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 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 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 FL^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.
ER  -