Additive covers and the Canonical Base Property

TL;DR

This paper introduces a new perspective on the failure of the Canonical Base Property (CBP) by representing the known counterexample as an additive cover of an algebraically closed field, and explores related weakenings and imaginaries.

Contribution

It provides an alternative presentation of the CBP counterexample using additive covers and analyzes the imaginaries involved, revealing limitations of Galois-theoretic explanations.

Findings

Counterexample can be represented as an additive cover

Weakening of CBP do not hold in the counterexample

Elimination of finite imaginaries connects to CBP

Abstract

We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacin and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive covers of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis, and show that they do not hold in the counterexample. In order to do so, a study of imaginaries in additive covers is developed, for elimination of finite imaginaries yields a connection to the CBP. As a by-product of the presentation, we notice that no pure Galois-theoretic account of the CBP can be provided.