A characterisation of weightable quasi-metric generating functions. Issue 11 (2nd November 2022)

• Record Type: Journal Article
• Authors: Gopal, Dhananjay; Valero, Oscar; Yadav, Shubham
• Is Part Of: Quaestiones mathematicae. Volume 45:Issue 11(2022)
• Page Start: 1683
• Page End: 1698
• Publication Date: 2022-11-02
• Subjects: 54A05 -- 54D10 -- 54E35 -- 54E99; Weighted quasi-metric -- partial metric -- weight difference function -- weighted quasi-metric generating function; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2021.1968531
• Languages: English
• ISSNs: 1607-3606

Abstract

Abstract: 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 difference 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.