# Orbits of subsets of the monster model and geometric theories

## TL;DR

This paper explores the conditions under which the elementary and orbital types of subsets in a monster model coincide, focusing on non-invariant subsets in geometric theories and analyzing specific cases like o-minimal theories.

## Contribution

It investigates the relationship between elementary and orbital types of subsets in monster models, especially for non-invariant sets in geometric and o-minimal theories, providing new insights.

## Key findings

- For $SU$-rank one, $e(D)$ and $o(D)$ are always different.
- In o-minimal theories, the relationship varies with the complexity of definable closure.
- Results apply to $H$-structures and lovely pairs in 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.