This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, with a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn's Positive Modal Logic (PML). Unlike PML, we consider logics that may drop distribution and which are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of Intuitionistic Modal Logic.

翻译：本文开启了对分布自由的正规模态逻辑系统的语义研究，为该领域奠定了语义基础并展望了未来的研究方向。本文的研究范围与Bezhanishvili、de Groot、Dmitrieva和Morachini近期的一篇论文大致相同，但采用了不同的方法，他们研究了Dunn正模态逻辑（PML）的一个分布自由版本。与PML不同，我们考虑的是可能放弃分配律，并同时配备蕴含连接词和模态算子的逻辑。我们采用统一的关系统语义方法，该方法依赖于近期关于正规模态格扩张的表示性和对偶性的研究成果。我们证明了最小分布自由正规模态逻辑（仅假设K公理）及其通过添加D、T、B、S4或S5公理中任意一个而得到的公理化扩展在关系统语义下的典范性和完备性。加入分配律可以很容易地实现，并且作为一个附带结果，我们也获得了对直觉主义模态逻辑的一种新的语义处理。