A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics

In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set X. In this paper, we adapt such a relationship to the fuzzy context, in the sense of George and Veeramani, by establishing a duality relationship between fuzzy quasi-metrics and fuzzy partial metrics on a set X, defined using the residuum operator of a continuous t-norm ∗. Concretely, we provide a method to construct a fuzzy quasi-metric from a fuzzy partial one. Subsequently, we introduce the notion of fuzzy weighted quasi-metric and obtain a way to construct a fuzzy partial metric from a fuzzy weighted quasi-metric. Such constructions are restricted to the case in which the continuous t-norm ∗ is Archimedean and we show that such a restriction cannot be deleted. Moreover, in both cases, the topology is preserved, i.e., the topology of the fuzzy quasi-metric obtained coincides with the topology of the fuzzy partial metric from which it is constructed and vice versa. Besides, different examples to illustrate the exposed theory are provided, which, in addition, show the consistence of our constructions comparing it with the classical duality relationship.

On partial metric preserving functions and their characterization

In 1981, J. Bors?k and J. Dob?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.