Banach–Stone theorems for disjointness preserving relations

The concept of disjointness preserving mappings has proved to be a useful unifying idea in the study of Banach–Stone type theorems. In this paper, we examine disjointness preserving relations between sets of continuous functions (valued in general topological spaces). Under very mild assumptions, it is shown that a disjointness preserving relation is completely determined by a Boolean isomorphism between the Boolean algebras of regular open sets in the domain spaces. Building on this result, certain Banach–Stone type theorems are obtained for disjointness preserving relations. From these, we deduce a generalization of Kaplansky’s classical theorem concerning order isomorphisms to sets of continuous functions with values topological lattices. As another application, we prove some results on the characterization of nonvanishing preservers. Throughout, the domains of the function spaces need not be compact.