Ramsey expansions of $\Lambda$-ultrametric spaces

Abstract

For a finite lattice $\Lambda$, $\Lambda$-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When $\Lambda$ is finite and distributive, there exists a generic $\Lambda$-ultrametric space, and we here identify a family of Ramsey expansions for that space. This then allows a description the universal minimal flow of its automorphism group, and also implies the Ramsey property for all known homogeneous finite-dimensional permutation structures, i.e. structures in a language of finitely many linear orders. A point of technical interest is that our proof involves classes with non-unary algebraic closure operations. As a byproduct of some of the concepts developed, we also arrive at a natural description of the known homogeneous finite-dimensional permutation structures, completing our previously begun "census".