发表机构
School of Science, Xi’an Petroleum University; School of Mathematics and Data Science, Shaanxi University of Science and Technology; School of Science, Xi Hang University（西安石油大学理学院; 陕西科技大学数学与数据科学学院; 西航大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入子结构逻辑FL_ew的Frobenius-Galois扩张，证明其可代数化，建立Kalman等价与正锥语义，统一了代数、范畴与逻辑框架。
AI 中文摘要
本文的主要目的是引入子结构逻辑 $\mathbf{FL}_{ew}$ 的Frobenius-Galois扩张，并发展其代数与范畴语义。所得逻辑记为 $\mathbf{FL}^{\mathbf{FGC}}_{ew}$，通过附加一对形成Galois连接并满足适当Frobenius型相容条件的一元联结词而获得。首先，我们证明 $\mathbf{FL}^{\mathbf{FGC}}_{ew}$ 在Blok和Pigozzi意义下是可代数化的，并将其等价代数语义等同于Frobenius-伴随剩余格簇，建立了对 $\mathbf{FL}_{ew}$ 的保守性，并将其有限模型性质与有限生成自由代数的剩余有限性联系起来。其次，对于分配性设定，我们将Kalman构造提升到Frobenius-伴随代数框架。更确切地说，我们在具有条件 $\mathbf{CK}$ 的Frobenius-伴随剩余分配格与Frobenius-伴随 $c$-微分剩余分配格之间建立了范畴等价。这一等价产生了前者的正锥表示，并进而产生了范畴对应的逻辑对应物。最后，对于分配性扩张 $\mathbf{FL}^{\mathbf{FGC},d}_{ew}$，我们证明可推导性、在Frobenius-伴随剩余分配格上的有效性以及在相应正锥上的有效性决定相同的推论关系，并进一步表明这一对应保持等式与拟等式推论、保守性以及有限反模型。这些结果为子结构逻辑的Frobenius-Galois扩张提供了一个统一的代数、范畴与逻辑框架。
英文摘要
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.
