A General Approach to Midpoint Theory and Aggregation of Quasimetrics

Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the mipoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of “compromise” or “middle way” between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved.