Delhomme-Laflamme-Pouzet-Sauer space as groupoid

TL;DR

This paper characterizes the structure of monomorphisms, automorphisms, and endomorphisms of the ultrametric space $(R^+, d^+)$, revealing their correspondence with ultrametric-preserving functions and self-homeomorphisms.

Contribution

It provides a detailed description of the groupoid structure of $(R^+, d^+)$, linking its morphisms to ultrametric-preserving functions and topological homeomorphisms.

Findings

Monomorphisms are exactly the injective ultrametric-preserving functions.

Automorphisms correspond to self-homeomorphisms of $R^+$.

Endomorphisms are explicitly characterized within the structure.

Abstract

Let $\mathbb{R}^{+}=[0, \infty)$ and let $d^+$ be the ultrametric on $\mathbb{R}^+$ such that $d^+ (x,y) = \max\{x,y\}$ for all different $x,y \in \mathbb{R}^+$. It is shown that the monomorphisms of the groupoid $(\mathbb{R}^+, d^+)$ coincide with the injective ultrametric-preserving functions and that the automorphisms of $(\mathbb{R}^+, d^+)$ coincide with the self-homeomorphisms of $\mathbb{R}^+$. The structure of endomorphisms of $(\mathbb{R}^+, d^+)$ is also described.