# Hereditary G-compactness

## TL;DR

This paper introduces hereditary G-compactness, provides conditions for non-compactness in posets, and explores its implications for NIP theories, stability, and definable groups, with a focus on model-theoretic properties.

## Contribution

It defines hereditary G-compactness, establishes criteria for non-compactness, and links it to stability and definability in model theory.

## Key findings

- Any linear order is not hereditarily G-compact.
- Under a conjecture, NIP theories are hereditarily G-compact iff stable.
- In hereditarily G-compact theories, G^{00}_A equals G^{000}_A.

## Abstract

We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable (and by a result of Simon, this holds unconditionally for $\aleph_0$-categorical theories). We show that if $G$ is definable over $A$ in a hereditarily G-compact theory, then $G^{00}_A=G^{000}_A$. We also include a brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some (classes of) theories.