A particular Galois connection between relations and set functions

Motivated by a recent paper of U. Höhle and T. Kubiak on regular sup-preserving maps, we investigate a particular Galois-type connection between relations on one set X to another Y and functions on the power set P(X) to P(Y ) . Since relations can largely be identified with union-preserving set functions, the results obtained can be used to provide some natural generalizations of most of the former results on relations and relators (families of relations). The results on inverses seem to be the only exceptions.