Choice-free Dualities for Lattice Expansions: Application to Logics with a Negation Operator

TL;DR

This paper develops a unified, choice-free duality framework for lattice expansions, enabling canonical extensions and completeness proofs for various non-classical logics with negation-like operators.

Contribution

It extends relational duality results to all normal lattice expansions, including those with negation operators, without relying on choice axioms.

Findings

Unified duality framework for lattice expansions

Choice-free canonical extensions for negation-involving lattices

Applicability to De Morgan algebras, ortholattices, and Boolean algebras

Abstract

Constructive dualities have been recently proposed for some lattice based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining "choice-free spatial dualities for other classes of algebras [$\ldots$], giving rise to choice-free completeness proofs for non-classical logics''. We present in this article a way to complete the Holliday-Bezhanishvili project (uniformly, for any normal lattice expansion) by recasting recent relational representation and duality results in a choice-free manner. These results have some affinity with the Moshier and Jipsen duality for bounded lattices with quasi-operators, except for aiming at representing operators by relations, extending the J\'{o}nsson-Tarski approach for BAOs, and Dunn's follow up approach for distributive gaggles, to contexts where distribution may not be assumed. To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining canonical extensions in relational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases.