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Definable Operators on Stable Set Lattices

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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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  1. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Robert Goldblatt

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  1. Robert Goldblatt
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Correspondence to Robert Goldblatt.

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Goldblatt, R. Definable Operators on Stable Set Lattices. Stud Logica 108, 1263–1280 (2020). https://doi.org/10.1007/s11225-020-09896-0

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