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DEFINABLE AND INVARIANT TYPES IN ENRICHMENTS OF NIP THEORIES

  • SILVAIN RIDEAU (a1) and PIERRE SIMON (a2)
Abstract
Abstract

Let T be an NIP ${\cal L}$ -theory and $\mathop T\limits^\~ $ be an enrichment. We give a sufficient condition on $\mathop T\limits^\~$ for the underlying ${\cal L}$ -type of any definable (respectively invariant) type over a model of $\mathop T\limits^\~$ to be definable (respectively invariant). These results are then applied to Scanlon’s model completion of valued differential fields.

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[1] Haskell D., Hrushovski E., and Macpherson D., Definable sets in algebraically closed valued fields: elimination of imaginaries . Journal für die Reine und Angewandte Mathematik (Crelles Journal), vol. 597 (2006), pp. 175236.
[2] Hrushovski E. and Pillay A., On NIP and invariant measures . Journal of the European Mathematical Society, vol. 13 (2011), no. 4, pp. 10051061.
[3] Matoušek J., Bounded VC-dimension implies a fractional Helly theorem . Discrete and Computational Geometry, vol. 31 (2004), no. 2, pp. 251255.
[4] Rideau S., Imaginaries in valued differential fields , Journal für die Reine und Angewandte Mathematik (CrellesJournal), to appear.
[5] Scanlon T., A model complete theory of valued D-fields, this JOURNAL, vol. 65 (2000),no. 4, pp. 17581784.
[6] Simon P., A Guide to NIP Theories. Lecture Notes in Logic, vol. 44, Cambridge University Press, Cambridge, Association of Symbolic Logic, Chicago, IL, 2015.
[7] Simon P., Invariant types in NIP theories . Journal of Mathematical Logic, vol. 15 (2015), no. 2.
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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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