Year: 2019
Type: preprint
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 $1\leq m \leq ω$. 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.
Source: arXiv (Cornell University)
Author: Walker, Roland
Institution: University of Illinois Chicago
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4300537954