Ultrametric-preserving functions as monoid endomorphisms

Let R⁺=[0, ∞) and let EndR⁺ be the set of all endomorphisms of the monoid (R⁺, ). The set EndR⁺ is a monoid with respect to the operation of the function composition g ∘ f. It is shown that g : R⁺ → R⁺ is pseudoultrametric-preserving iff g ∈ EndR⁺. In particular, a function f : R⁺ → R⁺ is ultrametrics-preserving iff it is an endomorphism of (R⁺,) with kernel consisting only the zero point. We prove that a given A ⊆ EndR⁺ is a submonoid of (End, ∘) iff there is a class X of pseudoultrametric spaces such that A coincides with the set of all functions which preserve the spaces from X. An explicit construction of such X is given.