Let ℝ+ = [0,∞) and let Endℝ+ be a 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 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.