DeepAI AI Chat
Log In Sign Up

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

by   Chrysafis Hartonas, et al.

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 […], 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 Jónsson-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.


page 1

page 2

page 3

page 4


Choice-free Topological Duality for Implicative Lattices and Heyting Algebras

We develop a common semantic framework for the interpretation both of 𝐈𝐏...

Duality for Normal Lattice Expansions and Sorted, Residuated Frames with Relations

We revisit the problem of Stone duality for lattices with various quasio...

Complexity of the universal theory of bounded residuated distributive lattice-ordered groupoids

We prove that the universal theory and the quasi-equational theory of bo...

Cut-free Calculi and Relational Semantics for Temporal STIT Logics

We present cut-free labelled sequent calculi for a central formalism in ...

Extension of Quasi-Overlap and Quasi-Grouping Functions Defined on Bounded Lattices Via Retractions

In this paper, we propose a method of extending quasi-overlap and groupi...

Compatibility and accessibility: lattice representations for semantics of non-classical and modal logics

In this paper, we study three representations of lattices by means of a ...