Delhomme-Laflamme-Pouzet-Sauer space as groupoid

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.