Ultrametric-preserving functions as monoid endomorphisms

2024/06/11 by Oleksiy Dovgoshey, Dovgoshey, Oleksiy · 1 citation
Mathematics · #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.07166

Abstract

Let ℝ⁺=[0, ∞) and let End_ℝ⁺ be the set of all endomorphisms of the monoid (ℝ⁺, \vee). 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⁺,\vee) with kernel consisting 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 which preserve the spaces from X. An explicit construction of such X is given.

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