Beautiful pairs

Cubides Kovacsics Pablo, Hils Martin, Ye Jinhe

Abstract

We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete L-theory T, called beautiful pairs, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with classical Fraïssé properties. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically, p-adically and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When T is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces such as Berkovich and Huber's analytifications. In particular, we recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields.