On monoids of metric preserving functions

TL;DR

This paper characterizes monoids of metric-preserving functions and shows that certain classes of functions, like amenable subadditive increasing functions, can be exactly represented as metric-preserving functions on some class of metric spaces.

Contribution

It establishes a necessary and sufficient condition for a set of functions to be the set of all metric-preserving functions for some class of metric spaces, linking it to monoid structures.

Findings

The set of metric-preserving functions forms a monoid under composition.

Existence of a class of metric spaces for which a given set of functions equals the metric-preserving functions.

Positive answer to the question about representing amenable subadditive increasing functions as metric-preserving functions.

Abstract

Let $\mathbf{X}$ be a class of metric spaces and let $\mathbf{P}_{\mathbf{X}}$ be the set of all $f:[0, \infty)\to [0, \infty)$ preserving $\mathbf{X},$ $(Y, f\circ\rho)\in\mathbf{X}$ whenever $(Y, \rho)\in\mathbf{X}.$ For arbitrary subset $\mathbf{A}$ of the set of all metric preserving functions we show that the equality $\mathbf{P}_{\mathbf{X}}=\mathbf{A}$ has a solution iff $\mathbf{A}$ is a monoid with respect to the operation of function composition. In particular, for the set $\mathbf{SI}$ of all amenable subadditive increasing functions there is a class $\mathbf{X}$ of metric spaces such that $\mathbf{P}_{\mathbf{X}}=\mathbf{SI}$ holds, which gives a positive answer to the question of paper [1].