Distality Rank

TL;DR

This paper introduces the concept of distality rank in first-order theories, explores its properties, invariance, and implications for NIP and stable theories, linking it to existing notions like $m$-dependence and forking geometry.

Contribution

It defines distality rank for theories, demonstrates its invariance in NIP theories, and connects it to $m$-determinacy, stability, and Shelah's $m$-dependence, expanding the understanding of model-theoretic hierarchy.

Findings

Distality rank is well-defined for each $m$ with $1 \\leq m \\leq \\omega$.

In NIP theories, distality rank remains invariant under base change.

In stable theories, $m$-distality relates to maximum cycle size in forking geometry.

Abstract

Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called $m$-determinacy and show that theories of distality rank $m$ require certain products to be $m$-determined. Furthermore, for NIP theories, this behavior characterizes $m$-distality. If we narrow the scope to stable theories, we observe that $m$-distality can be characterized by the maximum cycle size found in the forking "geometry," so it coincides with $(m-1)$-triviality. On a broader scale, we see that $m$-distality is a strengthening of Saharon Shelah's notion of $m$-dependence.