Ultrametric-preserving functions as monoid endomorphisms

Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\RR^+}$ be the set of all endomorphisms of the monoid $(\RR^+, \vee)$. The set $\mathbf{End}_{\RR^+}$ is a monoid with respect to the operation of the function composition $g \circ f$. It is shown that $g : \RR^+ \to \RR^+$ is pseudoultrametric-preserving iff $g \in \mathbf{End}_{\RR^+}$. In particular, a function $f : \RR^+ \to \RR^+$ is ultrametrics-preserving iff it is an endomorphism of $(\RR^+,\vee)$ with kelnel consisting only the zero point. We prove that a given $\mathbf{A} \subseteq \mathbf{End}_{\RR^+}$ 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.