Ultrametric-preserving functions as monoid endomorphisms

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