We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $\Pi_1$-persistence and show that every weak positive modal logic is $\Pi_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result.

翻译：我们为不必满足分配律的（模态）格发展了一种对偶性，并利用它来研究超越分配性的正规模态逻辑，我们称之为弱正规模态逻辑。该对偶性建立在Hofmann、Mislove和Stralka关于交半格的对偶性基础之上。我们引入了$\Pi_1$-持久性的概念，并证明了每个弱正规模态逻辑都是$\Pi_1$-持久的。这一方法为弱正规模态逻辑提供了一种新的关系语义学，并针对该语义学证明了Sahlqvist对应定理的类似结果。