# Definable operators on stable set lattices

## TL;DR

This paper generalizes a fundamental Boolean modal logic result to non-distributive logics with relational semantics based on polarities, showing that certain classes of structures generate varieties of algebras closed under canonical extensions.

## Contribution

It extends the concept of definable operators on stable set lattices to non-distributive logics with polarity-based semantics, establishing new closure properties.

## Key findings

- Classes of structures closed under ultraproducts generate varieties closed under canonical extensions.
- First-order definable operators on stable set lattices induce algebraic varieties with specific closure properties.
- The proof leverages the relationship between canonical extensions and MacNeille completions.

## Abstract

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.