Duality Results for (Co)Residuated Lattices

We present dualities (discrete duality, duality via truth and Stone duality) for implicative and (co)residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orowska and Rewitzky's research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as well as full Stone duality for (co)residuated lattices. Our results have immediate applications to the semantics of related substructural (resource consious) logical calculi.