On Matthews' Relationship Between Quasi-Metrics and Partial Metrics: An Aggregation Perspective

Abstract

Borsik and Dobo studied the problem of how to merge a family of metric spaces into a single one through a function. They called such functions metric preserving and provided a characterization of them in terms of the so-called triangle triplets. Since then, different papers have extended their study to the case of generalized metric spaces. Concretely, Mayor and Valero (Inf Sci 180:803-812, 2010) provided two characterizations of those functions, called quasi-metric aggregation functions, that allows us to merge a collection of quasi-metric spaces into a new one. In Massanet and Valero (in: Sainz-Palmero et al (eds) Proceedings of the 16th Spanish conference on fuzzy technology and fuzzy logic, European Society for Fuzzy Logic and Techonology, Valladolid, 2012) gave a characterization of the functions, called partial metric aggregation function, that are useful for merging a collection of partial metric spaces into single one as final output. Inspired by the preceding work, Martin et al. (in: Bustince et al (eds) Aggregation functions in theory and in practice. Advances in intelligent systems and computing, vol 228, Springer, Berlin, 2013) addressed the problem of constructing metrics from quasi-metrics, in a general way, using a class of functions that they called metric generating functions. In particular, they solved the posed problem providing a characterization of such functions and, thus, all ways under which a metric can be induced from a quasi-metric from an aggregation viewpoint. Following this idea, we propose the same problem in the framework of partial metric spaces. So, we characterize those functions that are able to generate a quasi-metric from a partial metric, and conversely, in such a way that Matthews' relationship between both type of generalized metrics is retrieved as a particular case. Moreover, we study if both, the partial order and the topology induced by a partial metric or a quasi-metric, respectively, are preserved by the new method in the spirit of Matthews. Furthermore, we discuss the relationship between the new functions and those families introduced in the literature, i.e., metric preserving functions, quasi-metric aggregation functions, partial metric aggregation functions and metric generating functions.