Rank Functions and Partial Stability Spectra for Tame Abstract Elementary Classes

Abstract

We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition known as tameness, where the connections between stability and total transcendence are most evident. As a byproduct, we obtain a partial upward stability transfer result for tame abstract elementary classes stable in a cardinal $\lambda$ satisfying ${\lambda }^{{\aleph }_0}>\lambda$, a substantial generalization of a result of Baldwin, Kueker, and VanDieren.