On partial metric preserving functions and their characterization

Abstract

In 1981, J. Bors\'ik and J. Dob\v{o}s characterized those functions that allow to transform a metric into another one in such a way that the topology of the metric to be transformed is preserved. Later on, in 1994, S.G. Matthews introduced a new generalized metric notion known as partial metric. In this paper, motivated in part by the applications of partial metrics, we characterize partial metric-preserving functions, i.e., those functions that help to transform a partial metric into another one. In particular we prove that partial metric-preserving functions are exactly those that are strictly monotone and concave. Moreover, we prove that the partial metric-preserving functions preserving the topology of the transformed partial metric are exactly those that are continuous. Furthermore, we give a characterization of those partial-metric preserving functions which preserve completeness and contractivity. Concretely, we prove that completeness is preserved by those partial metric-preserving functions that are non-bounded, and contractivity is kept by those partial metric-functions that satisfy a distinguished functional equation involving contractive constants. The relationship between metric-preserving and partial metric-preserving functions is also discussed. Finally, appropriate examples are introduced in order to illustrate the exposed theory.