Ample thoughts

TL;DR

This paper generalizes the concept of non-n-ampleness to Sigma-ampleness, proving that it is preserved under analysability and applying it to establish the weak Canonical Base Property for types of finite SU-rank in simple theories, with applications to groups.

Contribution

It introduces Sigma-ampleness as a generalization of non-n-ampleness and proves its preservation under analysability, providing a new proof of the weak CBP for simple theories.

Findings

Sigma-ampleness generalizes non-n-ampleness.

Preservation of Sigma-ampleness under analysability.

Application to groups demonstrating the weak CBP.

Abstract

Non-n-ampleness as defined by Pillay and Evans is preserved under analysability. Generalizing this to a more general notion of Sigma-ampleness, we obtain an immediate proof for all simple theories of CHatzidakis weak Canonical Base Property (CBP) for types of finite SU-rank. This is then applied to the special case of groups.