Definable Operators on Stable Set Lattices

Studia Logica 108 (6):1263-1280 (2020)
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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.

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References found in this work

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
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Varieties of complex algebras.Robert Goldblatt - 1989 - Annals of Pure and Applied Logic 44 (3):173-242.
Generalized Kripke Frames.Mai Gehrke - 2006 - Studia Logica 84 (2):241-275.
Lattice Theory.Garrett Birkhoff - 1940 - Journal of Symbolic Logic 5 (4):155-157.

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