Ultrametric preserving functions and weak similarities of ultrametric spaces

TL;DR

This paper characterizes ultrametric spaces where distances can be expressed through a specific ultrametric-preserving function applied to weak similarities, providing a complete description of such spaces.

Contribution

It offers a full characterization of ultrametric spaces that admit a distance representation via ultrametric-preserving functions under weak similarities.

Findings

Identifies conditions for ultrametric spaces to have distances expressed through ultrametric-preserving functions.

Provides a complete description of spaces satisfying the distance equality with weak similarities.

Advances understanding of the structure of ultrametric spaces and their similarity relations.

Abstract

Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to ultrametric space $(X, d)$. The main results of the paper completely describe the ultrametric spaces $(X, d)$ for which the equality $$ \rho(x, y) = f(d(\Phi(x), \Phi(y))) $$ holds for every $(Y, \rho) \in WS(X, d)$, every weak similarity $\Phi \colon Y \to X$, and all $x$, $y \in Y$ with some ultrametric (pseudoultrametric) preserving function $f$ depending on $\Phi$.