Duality for lattice-ordered algebras and for normal algebraizable logics

Studia Logica 58 (3):403-450 (1997)
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Abstract

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality.

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Citations of this work

Stone duality for lattice expansions.Chrysafis Hartonas - 2018 - Logic Journal of the IGPL 26 (5):475-504.
B-frame duality.Guillaume Massas - 2023 - Annals of Pure and Applied Logic 174 (5):103245.

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References found in this work

Algebraizable Logics.W. J. Blok & Don Pigozzi - 2022 - Advanced Reasoning Forum.
Logics without the contraction rule.Hiroakira Ono & Yuichi Komori - 1985 - Journal of Symbolic Logic 50 (1):169-201.
Varieties of complex algebras.Robert Goldblatt - 1989 - Annals of Pure and Applied Logic 44 (3):173-242.

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