In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank context (...) than [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid. (shrink)

In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.

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)