Dependence and isolated extensions
Author:
Vincent Guingona
Journal:
Proc. Amer. Math. Soc. 139 (2011), 3349-3357
MSC (2010):
Primary 03C45
DOI:
https://doi.org/10.1090/S0002-9939-2011-10739-1
Published electronically:
January 21, 2011
MathSciNet review:
2811289
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we show that if $\varphi (x; y)$ is a dependent formula, then all $\varphi$-types $p$ have an extension to a $\varphi$-isolated $\varphi$-type, $p’$. Moreover, we can choose $p’$ to be an elementary $\varphi$-extension of $p$ and $|\mathrm {dom}(p’) - \mathrm {dom}(p)| \le 2 \cdot \mathrm {ID}(\varphi )$. We show that this characterizes $\varphi$ being dependent. Finally, we give some corollaries of this theorem and draw some parallels to the stable setting.
- Saharon Shelah, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 513226
- ---, Dependent theories and the generic pair conjecture, preprint, September 2009. Communications in Contemporary Mathematics, submitted.
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Additional Information
Vincent Guingona
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
MR Author ID:
942387
Email:
vincentg@math.umd.edu
Keywords:
Definability,
dependent,
NIP,
types
Received by editor(s):
November 6, 2009
Received by editor(s) in revised form:
August 16, 2010
Published electronically:
January 21, 2011
Additional Notes:
Special thanks to Chris Laskowski
The author was partially supported by Laskowski’s NSF grants DMS-0600217 and 0901336
Communicated by:
Julia Knight
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
