Definable operators on stable set lattices

Robert Goldblatt

Published 2018-12-04Version 1

Structures based on polarities provide relational semantics for non-distributive logics. 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. This lifts a fundamental result from modal model theory to the non-distributive level. The proof makes use of a relationship between canonical extensions and MacNeille completions.