A rank based on Shelah trees

TL;DR

This paper introduces a new global rank for partial types based on Shelah trees, linking it to dividing sequences and chains, and demonstrating its role in characterizing simple and supersimple types.

Contribution

It generalizes Shelah trees to define a global rank that aligns with dividing concepts and remains invariant under non-forking extensions, advancing stability theory.

Findings

Rank characterizes simple and supersimple types

Rank is equivalent to the depth of dividing sequences

Rank satisfies Lascar-style inequalities

Abstract

We define a global rank for partial types based in a generalization of Shelah trees. We prove an equivalence with the depth of a localized version of the constructions known as dividing sequence and dividing chain. This rank characterizes simple and supersimple types. Moreover, this rank does not change for non-forking extensions under certain hypothesys. We also prove this rank satisfies Lascar-style inequalities.