In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\), as well as for its extensions with analytic structural rules satisfying certain additional properties.

在本文中，我们将代数证明理论的研究项目从完整 Lambek 演算的公理扩展扩展到由某些形式的正态格展开（正态 LE 逻辑）代数捕获的逻辑。具体来说，我们将[34]中的剩余框架推广到正常晶格展开（LE）的任意签名。这种概括为证明 LE 逻辑的重要属性的完全一致性提供了一个有价值的工具。我们证明了与基本正常 LE 逻辑及其公理扩展与分析归纳公理相关的显示演算 D.LE 的语义削减消除。我们还证明了每个此类微积分 D.LE 的有限模型属性 (FMP)，以及满足某些附加属性的解析结构规则的扩展。