# On ultrametric-preserving functions

## TL;DR

This paper characterizes functions that preserve ultrametrics and pseudoultrametrics, explores their structural properties, and introduces a new concept to better understand ultrametric spaces.

## Contribution

It provides new characterizations of ultrametric-preserving functions and their structural properties, including a dual characterization and the concept of k-separating families.

## Key findings

- Characterizations of pseudoultrametric-preserving functions
- Structural properties of pseudoultrametrics as compositions
- Introduction of k-separating family concept

## Abstract

Characterizations of pseudoultrametric-preserving functions and semimetric-preserving functions are found. The structural properties of pseudoultrametrics which can be represented as a composition of an ultrametric and ultrametric-pseudoultrametric-preserving function are found. A dual form of Pongsriiam-Termwuttipong characterization of the ultrametric-preserving functions is described. We also introduce a concept of $k$-separating family of functions and use it to characterize the ultrametric spaces.