Canonical extensions and relational representations of lattices with negation

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 classes of lattices equipped with a negation operation which is a self-adjoint residuation and other axioms are added so as to give classes of lattices in which the negation is De Morgan, ortho-negation, antilogism, pseudo-complementation or weak pseudo-complementation. Among these, all but one --- the one of lattices with a negation which is an antilogism --- were previously studied by E. Orlowska, W. Dzik and C. van~Alten using Urquhart's duality. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. In some cases in which a given axiom does not imply that negation is a residuation, canonicity is proven with the weaker assumption of antitonicity of the negation.