Elimination of Hyperimaginaries and Stable Independence in simple CM-trivial theories

International audienceIn 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

Ample Pairs

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of $H$-structures of a trivial theory of rank $1$.Comment: Research partially supported by the program MTM2014-59178-P. The second author conducted research with support of the programme ANR-13-BS01-0006 Valcomo. The third author would like to thank the European Research Council grant 33882

Ample thoughts

International audienceNon-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