Algebra universalis

, 79:80 | Cite as

Canonical extensions and ultraproducts of polarities

  • Robert GoldblattEmail author
Part of the following topical collections:
  1. In memory of Bjarni Jónsson


Jónsson 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.


Canonical extension Canonical variety Lattice Completion Lattice-based algebra MacNeille completion Ultraproduct Polarity Galois connection Hanf number 

Mathematics Subject Classification

03G10 06B23 03C20 06A15 06D50 



The author thanks Mai Gehrke and Ian Hodkinson for some very helpful comments, information and improvements.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsVictoria University of WellingtonWellingtonNew Zealand

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