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05-01-2021 | Foundations

A duality for two-sorted lattices

Authors: Umberto Rivieccio, Achim Jung

Published in: Soft Computing | Issue 2/2021

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Abstract

A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.

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The notation \(\langle \mathbf {L} _+, \mathbf {L} _-, n, p \rangle \) originates in the literature on bilattices and bitopology (Jakl et al. 2016), in which \(\mathbf {L} _+\) and \(\mathbf {L} _-\) are viewed as truth value spaces corresponding to (resp.) positive and negative information concerning, say, a proposition. Thus, the map n (which has the lattice \(L_+\) as source and \(L_-\) as target) allows one to “translate” positive information into negative, and likewise, p provides a translation the other way round.

In classical propositional logic, \(\lnot (\lnot x \rightarrow \lnot y)\) is equivalent to \(\lnot (y\rightarrow x)\), hence our notation \(x \not \leftarrow y\).

Notice that \(\mathbf {B} _-\) has \(\vee \) as meet (whose residuum is \(\not \leftarrow \)) and \(\wedge \) as join.

As observed in Rivieccio and Spinks (2019) and Rivieccio and Spinks (2020), conditions (i)–(iii) entail that p also preserves the Heyting implication.

Abstractly, a Priestley space is defined as a compact ordered topological space \(\langle X, \tau , \le \rangle \) such that, for all \(x,y \in X\), if \(x \not \le y\), then there is a clopen up-set \(U \subseteq X\) with \(x \in U\) and \(y \notin U\). It follows that \(\langle X, \tau \rangle \) is a Stone space.

This is nevertheless a choice, different, for example, from the one made in Jakl et al. (2016, Definition 3.3).

Our formulation of (ii).1 is different from the one in Bezhanishvili and Jansana (2011, Definition 6.2), but the two are easily seen to be equivalent in our context. Also, we always require \(R_n\) and \(R_p\) to be total (Bezhanishvili and Jansana 2011, Definition 6.11) because our maps n, p preserve all lattice bounds.

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Title: A duality for two-sorted lattices
Authors: Umberto Rivieccio
Achim Jung
Publication date: 05-01-2021
Publisher: Springer Berlin Heidelberg
Published in: Soft Computing / Issue 2/2021
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-020-05482-7

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