Fine Structure of Dependence in Superstable Theories of Finite Rank

Superstable theories of finite rank can be built using realizations of semiminimal types. In his paper 'Vaught's Conjecture for Superstable Theories of Finite Rank', Buechler gave a level-partitioning of semiminimal constructions which allows for a fine analysis of dependence between finite-rank sets. Using this partitioning, we study when dependence on finite-rank sets above the first level has a modular-like behavior, a property we formalize and call the Level Dependence Property (LDP). We prove that LDP is equivalent to the Canonical Base Property (CBP) of Moosa and Pillay for every superstable theory of finite rank. Pillay has shown that CBP (and hence also LDP) holds in compact complex spaces, and together with Ziegler he has shown that CBP holds in differentially closed fields and difference fields. Our main results prove that LDP holds in superstable theories of finite rank under additional assumptions. The first result involves certain orthogonality relations. As a consequence we obtain that any 'reduced' counterexample to LDP must be unidimensional, which is also implied by a result of Chatzidakis. The second result proves LDP under certain rank restrictions. Moosa and Pillay have conjectured that LDP holds of all theories of finite rank. Based on an idea of Hrushovski, we present an explicit $aleph_1$-categorical structure that may fail to satisfy LDP; moreover, the model is of the simplest possible kind allowed by our positive results.