Ultrametric-preserving functions as monoid endomorphisms

Let ${\mathbb{R}}^{+}=[0,\mathrm{\infty })$ and let ${\mathbf{E}\mathbf{n}\mathbf{d}}_{{\mathbb{R}}^{+}}$ be the set of all endomorphisms of the monoid $({\mathbb{R}}^{+},\vee )$. The set ${\mathbf{E}\mathbf{n}\mathbf{d}}_{{\mathbb{R}}^{+}}$ 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 $g\in {\mathbf{E}\mathbf{n}\mathbf{d}}_{{\mathbb{R}}^{+}}$. 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 $\mathbf{A}\subseteq {\mathbf{E}\mathbf{n}\mathbf{d}}_{{\mathbb{R}}^{+}}$ is a submonoid of $(\mathbf{E}\mathbf{n}\mathbf{d},\circ )$ 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.