We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and Moosa-Pillay and show that they do not hold in the counterexample. In order to do so, a (...) study of imaginaries in additive covers is developed. As a by-product of the presentation, we observe that a pure binding-group-theoretic account of the CBP is unlikely. (shrink)

We are concerned with the following problem. Suppose Γ and Σ are closed permutation groups on infinite sets C and W and ρ: Γ → Σ is a non-split, continuous epimorphism with finite kernel. Describe the possibilities for ρ. Here, we consider the case where ρ arises from a finite cover π: C → W. We give reasonably general conditions on the permutation structure W;Σ which allow us to prove that these covers arise in two possible ways. The first way, (...) reminiscent of covers of topological spaces, is as a covering of some Σ-invariant digraph on W. The second construction is less easy to describe, but produces the most familiar of these types of covers: a vector space covering its projective space. (shrink)