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In 1994, S.G. Matthews introduced the notions of partial metric and weighted quasi-metric and, in addition, he showed that both are dual. In fact, he developed a method for generating weightable quasi-metrics from partial metrics and conversely. Recently, J.J. Miñana and O. Valero introduced a general method with the aim of inducing quasi-metrics from partial metrics via real-valued functions in such a way that the original Matthews method can be retrieved as a special case when an appropriate function is chosen. However, they showed that the aforementioned method does not induce in general weighted quasi-metrics and, thus, they posed the question about what properties must be fulfilled by this type of functions in order to guarantee that the generated quasi-metric is weighted. In this paper, we answer such a question providing a characterization of such functions through the use of what we have called weight dierence function and a new type of quadrangular triplets. Finally, a new characterization of weightable quasi-metric spaces is also given and a few properties of weightable quasi-metric spaces are discussed.

A characterisation of weightable quasi-metric generating functions