FANDOM


Work in a stable theory (or more generally a simple theory) $ T $. The preweight of a complete type $ \operatorname{tp}(a/C) $ is defined to be the supremum of the cardinals $ \kappa $ such that there is some $ C $-independent set $ \{b_\lambda : \lambda < \kappa\} $ such that $ a $ forks with $ b_\lambda $ for every $ \lambda $, i.e., $ a \not \downarrow_C b_\lambda $ for every $ \lambda $. This is well-defined, and in fact the preweight of $ \operatorname{tp}(a/C) $ is bounded above by the $ \kappa $ appearing in the local character of forking (which is $ \aleph_0 $ for superstable theories).

If $ p $ is a stationary type, the weight of $ p $ is defined to be the largest weight of any non-forking extension of $ p $. Types of Morley rank 1, or more generally, Lascar rank 1 have weight 1. More generally, regular types have weight 1.

Weight is generalized to simple theories in a straightforward way. Weight is generalized to NIP theories by the notion of dp-rank, and is generalized to NTP2 theories by the notion of burden.

Superstable theories have plenty of weight 1 types, in some sense… (Every type is domination equivalent to a product of weight 1 type. Also, every type has finite weight.)