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

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 ${\mathbf{FL}}_{ew}^{\mathbf{FGC}}$, is obtained by adjoining a pair of unary connectives forming a Galois connection and satisfying suitable Frobenius-type compatibility conditions. Firstly, we prove that ${\mathbf{FL}}_{ew}^{\mathbf{FGC}}$ 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 ${\mathbf{FL}}_{ew}^{\mathbf{FGC},d}$, 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.