Canonical extensions and ultraproducts of polarities

TL;DR

This paper explores the theory of canonical extensions of lattice-based algebras, showing how certain algebraic properties relate to ultraproducts and stable sets of polarities, with implications for modal logic and algebraic closure.

Contribution

It demonstrates that the failure of a variety to be closed under canonical extensions can be identified by a specific free algebra, and shows that classes of polarities closed under ultraproducts generate varieties closed under canonical extensions.

Findings

Failure of canonical closure is witnessed by a specific free algebra.

Classes of polarities closed under ultraproducts generate canonically closed varieties.

Generalizes earlier results relating Kripke frames and modal logic.

Abstract

J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.