|Canonical Extensions and Relational Representations of Lattices with Negation
Studia Logica, 91 (2009), 171--199
This work is part of a wider investigation into lattice-structured algebras and associated dual representations obtained via the methodology of canonical extensions. To this end, here we study lattices, not necessarily distributive, with negation operations.
We consider equational classes of lattices equipped with a negation operation $\neg$ which is dually self-adjoint (the pair ($\neg$,$\neg$) is a Galois connection) and other axioms are added so as to give classes of lattices in which the negation is De Morgan, orthonegation, antilogism, pseudocomplementation or weak pseudocomplementation. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. The fact that the negation is dually self-adjoint plays an important role here, as it implies that it sends arbitrary joins to meets and that will allow us to define the dual structures in a uniform way.
Among these classes, all but one---that of lattices with a negation which is an antilogism---were previously studied by W. Dzik, E. Or\lowska and C. van Alten using Urquhart duality.
In some cases in which a given axiom does not imply that negation is dually self-adjoint, canonicity is proven with the weaker assumption of antitonicity of the negation.