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DISTALITY RANK

Part of: Model theory

Published online by Cambridge University Press:  15 August 2022

ROLAND WALKER Open the ORCID record for ROLAND WALKER [Opens in a new window]
ROLAND WALKER*
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO 851 S MORGAN STREET, 322 SEO, MC 249 CHICAGO, IL 60607-7045, USA
*
E-mail: rwalke20@uic.edu
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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.

Keywords

model theoryclassification theorygeometric stability theoryindiscernible sequencesdependent theoriesrandom graphs

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 03C52: Properties of classes of models
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 2 , June 2023 , pp. 704 - 737
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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