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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Dependence and isolated extensions
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by Vincent Guingona PDF
Proc. Amer. Math. Soc. 139 (2011), 3349-3357 Request permission


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:
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 3349-3357
  • MSC (2010): Primary 03C45
  • DOI:
  • MathSciNet review: 2811289