Ultrametric Preserving Functions and Weak Similarities of Ultrametric Spaces$$^*$$

Let $$WS(X, d)$$ be the class of ultrametric spaces which are weakly similar to ultrametric space $$(X, d)$$ . The main results of the paper completely describe the ultrametric spaces $$(X, d)$$ for which the equality $$\rho(x, y) = f(d(\Phi(x), \Phi(y)))$$ holds for every $$(Y, \rho) \in WS(X, d)$$ , every weak similarity $$\Phi \colon Y \to X$$ , and all $$x$$ , $$y \in Y$$ with some ultrametric (pseudoultrametric) preserving function $$f$$ depending on $$\Phi$$ .