Work in a stable theory (or more generally a simple theory) . The preweight of a complete type
is defined to be the supremum of the cardinals
such that there is some
-independent set
such that
forks with
for every
, i.e.,
for every
. This is well-defined, and in fact the preweight of
is bounded above by the
appearing in the local character of forking (which is
for superstable theories).
If is a stationary type, the weight of
is defined to be the largest weight of any non-forking extension of
. 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.) ::: ::: :::