Delhomme–Laflamme–Pouzet–Sauer space as groupoid

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