A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. The proof makes use of a relationship between canonical extensions and MacNeille completions.

中文翻译：

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稳定集格上的可定义算子

布尔模态逻辑的一个基本结果表明，一阶可定义的 Kripke 框架类定义了一个由其所有规范框架验证的逻辑。我们将其概括为具有基于极性的结构提供的关系语义的非分配逻辑级别。这种结构与稳定子集的完整格相关联，并且这些结构已被用于构建基于格的代数的规范扩展。我们研究在超积下封闭的结构类别，其稳定的集合格具有在底层结构中一阶可定义的附加算子。我们表明，此类类会生成在规范扩展下封闭的各种代数。该证明利用了规范扩展和 MacNeille 完成之间的关系。