On monoids of 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].

15 pages