Universal homogeneous two-sorted ultrametric spaces

Abstract

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fra\"iss\'e, and that the limit is the countable rational Urysohn ultrametric space $\mathbb{U}$. The space $\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\overline{\mathbb{U}}$ is dc-universal for all separable ultrametric spaces, which is in contrast with the situation of classical ultrametric spaces and isometric embeddings, where no such universal space can exist. We study further properties of $\mathbb{U}$, of its variants, and of its automorphism group, which is richer than its group of isometries. In particular, we provide two types of tree representations of the two-sorted ultrametric spaces, discuss connections to valued fields, and characterize the automorphism group of $\mathbb{U}$ as the semidirect product of a group of order preserving bijections and a group of isometries. Furthermore, we show universality of $\operatorname{Aut}(\mathbb{U})$ and identify its universal minimal flow.