Strongly quasi-pseudometric aggregation functions

TL;DR

This paper explores the extension of strongly metric-preserving functions to quasi-pseudometrics, characterizing their properties and conditions for aggregation functions to produce consistent metrics.

Contribution

It extends classical metric aggregation concepts to strongly quasi-pseudometrics, providing new characterizations and addressing previously unexplored cases.

Findings

Characterized strongly (quasi-)metric aggregation functions via continuity and zero preimage conditions.

Demonstrated the importance of the supremum topology for fixed set quasi-metrics.

Provided necessary and sufficient conditions for strong quasi-metric aggregation functions.

Abstract

Metric-preserving functions (here, metric aggregation functions) offer a natural method for constructing metrics on Cartesian products of metric spaces or for aggregating multiple metrics defined on a common set. Strongly metric-preserving functions represent a more specialized subset of these functions, ensuring that the new metric aligns with the product topology, in the Cartesian product case. However, these strong functions have not been previously explored for quasi-pseudometrics. Furthermore, in the case where all metrics are defined on the same set, the problem has not been addressed previously. In this paper, we investigate the class of strongly (quasi-)(pseudo)metric aggregation functions, extending the classical concept. We begin by examining the case where the aggregation function produces (quasi-)(pseudo)metrics on Cartesian products, characterizing these functions through continuity at zero and a minimal zero preimage condition. In addition, we will examine the scenario where the aggregation function produces a (quasi-)(pseudo)metric defined on a fixed set. Within this context, we will demonstrate that the appropriate topology to consider is the supremum topology. We will also provide both necessary and sufficient conditions for an (quasi-)(pseudo)metric aggregation function on sets to qualify as a strongly one, thereby addressing a gap in the existing literature.