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.
Categories: math.GN