An independence theorem for NTP2 theories

TL;DR

This paper advances the understanding of dividing and forking in NTP2 theories, establishing key properties like the chain condition and an independence theorem, and exploring the dividing order and classification of strong theories.

Contribution

It proves that dividing equals array-dividing, establishes the chain condition over extension bases, and introduces the dividing order in NTP2 theories, extending stability theory concepts.

Findings

Dividing equals array-dividing in NTP2 theories

Forking satisfies the chain condition over extension bases

Lascar and compact strong types coincide over extension bases

Abstract

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.