Strict independence

TL;DR

This paper explores the concepts of strict independence and strict non-forking in model theory, establishing their properties and implications for various classes of theories, including resilient, NTP2, and dependent theories.

Contribution

It introduces new notions of weight to characterize NTP2, dependence, and strong dependence, and provides characterizations of sequences witnessing dividing in resilient theories.

Findings

Strict non-forking is symmetric in resilient theories.

Characterizations of Morley sequences that witness dividing.

Types co-dominated by generically stable types have Morley sequences that are witnesses.

Abstract

We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness.