arxiv: 2604.25836 · v1 · submitted 2026-04-28 · 🧮 math.GN

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Strongly quasi-pseudometric aggregation functions

Alejandro Fructuoso-Bonet, Jes\'us Rodr\'iguez-L\'opez

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Pith reviewed 2026-05-07 13:44 UTC · model grok-4.3

classification 🧮 math.GN

keywords quasi-pseudometricaggregation functioncontinuity at zerominimal zero preimagesupremum topologyproduct topologystrong preservation

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The pith

Strongly quasi-pseudometric aggregation functions are characterized by continuity at zero and a minimal zero preimage condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the notion of strongly metric-preserving functions to quasi-pseudometrics. For aggregation on Cartesian products, these functions are those continuous at zero that satisfy a minimal zero preimage condition. This ensures the aggregated quasi-pseudometric induces the product topology. For the case of multiple quasi-pseudometrics on the same set, the paper identifies the supremum topology as the one to preserve and gives necessary and sufficient conditions for the aggregation function to be strong.

Core claim

We characterize strongly (quasi-)(pseudo)metric aggregation functions for Cartesian products by continuity at zero and a minimal zero preimage condition. For the fixed set case, we show that the supremum topology is appropriate and provide both necessary and sufficient conditions for an aggregation function to be strongly one.

What carries the argument

The continuity at zero and minimal zero preimage condition, along with the supremum topology in the fixed-set case.

Load-bearing premise

The assumption that classical metric results do not directly transfer to the quasi-pseudometric setting and that the supremum topology is the appropriate one to preserve in the fixed-set case.

What would settle it

A counterexample function that is continuous at zero but fails to produce a quasi-pseudometric inducing the product topology on the Cartesian product, or an aggregation on a fixed set that satisfies the conditions but does not preserve the supremum topology.

Figures

Figures reproduced from arXiv: 2604.25836 by Alejandro Fructuoso-Bonet, Jes\'us Rodr\'iguez-L\'opez.

**Figure 1.** Figure 1: summarizes the relationships among the families of aggregation functions on products considered so far. Each region corresponds to a family of aggregation functions, and its label indicates the underlying structure being aggregated. pseudometrics quasi-pseudometrics metrics quasi-metrics Strongly quasi-metrics Strongly quasi-pseudometrics Strongly metric Strongly pseudometrics view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper characterizes strongly quasi-pseudometric aggregation functions via continuity at zero and zero-preimage conditions for products, plus supremum-topology conditions for fixed sets, but these read as direct extensions of the metric literature.

read the letter

The main contribution is filling the gap for quasi-pseudometrics. They characterize when an aggregation function on a product of quasi-pseudometric spaces yields a strong one that respects the product topology, using continuity at zero plus a minimal zero-preimage condition. For the fixed-set case they show the supremum topology is the right target and give necessary and sufficient conditions for the aggregation to be strong under it. Both parts address cases the abstract says were previously untouched, so the characterizations are new on their face.

Referee Report

0 major / 2 minor

Summary. The paper extends the theory of strongly metric aggregation functions to the quasi-pseudometric setting. For the Cartesian-product case it characterizes the strongly quasi-pseudometric aggregation functions by continuity at zero together with a minimal zero-preimage condition. For the fixed-set case it identifies the supremum topology as the appropriate one and supplies necessary and sufficient conditions for an aggregation function to be strongly quasi-pseudometric.

Significance. The characterizations fill a documented gap in the literature on quasi-pseudometric aggregation. They are direct, non-circular adaptations of the classical metric results that correctly incorporate the asymmetry of quasi-pseudometrics through the same zero-set and continuity requirements already present in the symmetric case. The paper thereby supplies new necessary-and-sufficient conditions grounded in standard definitions rather than in its own outputs.

minor comments (2)

Abstract: the repeated future-tense phrasing ('we will examine', 'we will demonstrate') is unnecessary in a completed manuscript; replace with present tense.
Notation: the parenthetical '(quasi-)(pseudo)metric' is used throughout; a single consistent abbreviation introduced once in the preliminaries would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending acceptance. The report correctly identifies the main contributions: the characterizations via continuity at zero and minimal zero-preimage conditions for the Cartesian-product case, and the use of the supremum topology together with necessary-and-sufficient conditions for the fixed-set case.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives characterizations of strongly (quasi-)(pseudo)metric aggregation functions directly from standard definitions of continuity at zero, minimal zero-preimage sets, and the supremum topology. These are explicit necessary-and-sufficient conditions adapted from the metric case to handle asymmetry, without any reduction of a claimed prediction or result back to a fitted parameter, self-defined quantity, or load-bearing self-citation. The central claims rest on independent topological arguments that do not presuppose the target statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard definitions of quasi-pseudometrics and aggregation functions from prior topology literature; no new free parameters or invented entities are introduced.

axioms (1)

standard math A quasi-pseudometric satisfies non-negativity, d(x,x)=0, and the triangle inequality (possibly asymmetric).
Invoked throughout as the background definition for the objects being aggregated.

pith-pipeline@v0.9.0 · 5523 in / 1098 out tokens · 47466 ms · 2026-05-07T13:44:47.684391+00:00 · methodology

discussion (0)

Reference graph

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