$$\Lambda $$-Ultrametric spaces and lattices of equivalence relations

Abstract

For a finite lattice Λ, Λ-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When Λ is finite and distributive, there exists a generic Λultrametric 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 of the homogeneous structures constructed in [2]. 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 structures in a language consisting of finitely many linear orders, thus completing one of the goals of [2].