New results about aggregation functions of quasi-pseudometric modulars
Alejandro Fructuoso-Bonet, Jesús Rodríguez-López
TL;DR
This work characterizes aggregation functions for quasi-pseudometric modulars on Cartesian products by embedding metric-modular structures into the quantale-enriched framework and using lax morphisms as generalized aggregation operators. A key result is that quasi-pseudometric modular aggregation on products is equivalent to the corresponding set-aggregation notion, expressible via a translating map $F_ abla: abla^{I} o abla$ that preserves the quantale structure; this aligns with and extends prior results BFVale24 within the broader theory FBRL. For pseudometric modulars, product and set aggregations coincide under symmetrically preserving maps, while for quasi-metric modulars the product- versus set-aggregation distinction persists. The approach leverages the isomorphism between quasi-pseudometric modular spaces and enriched categories over the nabla quantale, unifying aggregation theory across different constructions and providing a robust, categorical route to characterize and compare aggregation functions $F$ on both products and sets, with implications for decision theory and clustering contexts wherever modular proximity is relevant.
Abstract
In recent studies, Bibiloni-Femenias, Miñana and Valero characterized the functions that aggregate a family of (quasi-)(pseudo)metric modulars defined over a fixed set $X$ into a single one. In this paper, we adopt a related but different approach to examine those functions that allow us to define a (quasi-)(pseudo)metric modular in the Cartesian product of (quasi-)(pseudo)metric modular spaces. We base our research on the recent development of a general theory of aggregation functions between quantales. This enables to shed light between the two different ways of aggregation (quasi-)(pseudo)metric modulars.