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Dependence and isolated extensions

Author: Vincent Guingona
Journal: Proc. Amer. Math. Soc. 139 (2011), 3349-3357
MSC (2010): Primary 03C45
Published electronically: January 21, 2011
MathSciNet review: 2811289
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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.

References [Enhancements On Off] (What's this?)

  • 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

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.