2 research outputs found
An independence theorem for NTP2 theories
We establish several results regarding dividing and forking in NTP2 theories.
We show that dividing is the same as array-dividing. Combining it with
existence of strictly invariant sequences we deduce that forking satisfies the
chain condition over extension bases (namely, the forking ideal is S1, in
Hrushovski's terminology). Using it we prove an independence theorem over
extension bases (which, in the case of simple theories, specializes to the
ordinary independence theorem). As an application we show that Lascar strong
type and compact strong type coincide over extension bases in an NTP2 theory.
We also define the dividing order of a theory -- a generalization of Poizat's
fundamental order from stable theories -- and give some equivalent
characterizations under the assumption of NTP2. The last section is devoted to
a refinement of the class of strong theories and its place in the
classification hierarchy
Dividing and chain conditions
We obtain a chain condition for dividing in an arbitrary theory and a new and shorter proof of a chain condition result of Shelah for simple theories.