Definability in low simple theories (Q2710591)

It is an open question whether the Independence Theorem, true in every stable theory, holds in simple theories, too. In Ann. Pure Appl. Log. 88, No. 2-3, 149-164 (1997; Zbl 0897.03036), \textit{B. Kim} and \textit{A. Pillay} showed in this enlarged setting a weaker version, using Lascar strong types instead of strong types. Later Buechler introduced a class of simple theories, called low, including stable theories as well as certain supersimple theories, where a Lascar strong type is the same as a strong type, and so the Independence Theorem holds in all generality. The paper under review proposes a new approach both to Buechler's and Kim-Pillay's theorems. Definability in low theories is also discussed, as well as equality between Lascar strong types in the simple framework. Finally, the paper compares Lascar strong types and types within simple theories, and provides a (necessary and sufficient) condition ensuring their equivalence.