Ultrametric-preserving functions as monoid endomorphisms

TL;DR

This paper characterizes ultrametric-preserving functions as endomorphisms of a monoid on non-negative reals and explores their structure and relation to pseudoultrametric spaces.

Contribution

It establishes a correspondence between ultrametric-preserving functions and monoid endomorphisms, providing a new algebraic perspective and explicit constructions.

Findings

Ultrametric-preserving functions are exactly monoid endomorphisms of $(R^+, igvee)$.

Submonoids correspond to classes of pseudoultrametric spaces.

Explicit construction of space classes associated with submonoids.

Abstract

Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\mathbb{R}^+}$ be the set of all endomorphisms of the monoid $(\mathbb{R}^+, \vee)$. The set $\mathbf{End}_{\mathbb{R}^+}$ is a monoid with respect to the operation of the function composition $g \circ f$. It is shown that $g : \mathbb{R}^+ \to \mathbb{R}^+$ is pseudoultrametric-preserving iff $g \in \mathbf{End}_{\mathbb{R}^+}$. In particular, a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ is ultrametrics-preserving iff it is an endomorphism of $(\mathbb{R}^+,\vee)$ with kernel consisting only the zero point. We prove that a given $\mathbf{A} \subseteq \mathbf{End}_{\mathbb{R}^+}$ is a submonoid of $(\mathbf{End}, \circ)$ iff there is a class $\mathbf{X}$ of pseudoultrametric spaces such that $\mathbf{A}$ coincides with the set of all functions which preserve the spaces from $\mathbf{X}$. An explicit construction of such $\mathbf{X}$ is given.