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Stable embeddedness and NIP

  • Anand Pillay (a1)

Abstract

We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are that (or rather its theory) is “rosy”. P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]).

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[1]Adler, H., A geometric introduction to forking and thorn forking, 2007, preprint.
[2]Adler, H., Introduction to theories without the independence property, Archive for Mathematical Logic, to appear.
[3]Chernikov, A. and Kaplan, I., Forking and dividing in NTP2 theories, to appear in this Journal, number 147 on the MODNET preprint server.
[4]Ealy, C., Krupinski, K., and Pillay, A., Superrosy dependent groups having finitely satisfiable generics, Annals of Pure and Applied Logic, vol. 151 (2008), pp. 121.
[5]Ealy, C. and Onshuus, A., Characterizing rosy theories, this Journal, vol. 72 (2007), pp. 919940.
[6]Hasson, A. and Onshuus, A., Embedded o-minimal structures, Bulletin of the London Mathematical Society, vol. 42 (2010), pp. 6474.
[7]Hrushovski, E. and Pillay, A., On NIP and invariant measures, preprint 2009 (revised version), http://arxiv.org/abs/0710.2330.
[8]Onshuus, A., Properties and consequences of thorn independence, this Journal, vol. 71 (2006), pp. 121.
[9]Poizat, B., A course in model theory; an introduction to contemporary mathematical logic, Springer, 2000.
[10]Shelah, S., Dependent first order theories, continued, Israel Journal of Mathematics, vol. 173 (2009), pp. 160.
[11]Usvyatsov, A., Morley sequences in dependent theories, preprint, http://arxiv.org/abs/0810.0733.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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