Dualities in full homomorphisms

In this paper we study dualities of graphs and, more generally, relational structures with respect to full homomorphisms, that is, mappings that are both edgeand non-edge-preserving. The research was motivated, a.o., by results from logic (concerning first order definability) and Constraint Satisfaction Problems. We prove that for any finite set of objects B (finite relational structures) there is a finite duality with B to the left. It appears that surprising richness of these dualities leads to interesting problems of Ramsey type; they are which are explicitly analyzed in the simplest case of graphs.