Orbits of subsets of the monster model and geometric theories

Abstract

Let $\mathbb{M}$ be the monster model of a complete first-order theory $T$. If $\mathbb{D}$ is a subset of $\mathbb{M}$, following D. Zambella we consider $e(\mathbb{D})=\{{\mathbb{D}}^{\prime }\mid (\mathbb{M},\mathbb{D})\equiv (\mathbb{M},{\mathbb{D}}^{\prime })\}$ and $o(\mathbb{D})=\{{\mathbb{D}}^{\prime }\mid (\mathbb{M},\mathbb{D})\cong (\mathbb{M},{\mathbb{D}}^{\prime })\}$. The general question we ask is when $e(\mathbb{D})=o(\mathbb{D})$ ? The case where $\mathbb{D}$ is $A$-invariant for some small set $A$ is rather straightforward: it just mean that $\mathbb{D}$ is definable. We investigate the case where $\mathbb{D}$ is not invariant over any small subset. If T is geometric and $(\mathbb{M},\mathbb{D})$ is an $H$-structure (in the sense of A. Berenstein and E. Vassiliev) or a lovely pair, we get some answers. In the case of $SU$-rank one, $e(\mathbb{D})$ is always different from $o(\mathbb{D})$. In the o-minimal case, everything can happen, depending on the complexity of the definable closure.