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GENERIC STABILITY AND STABILITY

  • HANS ADLER (a1), ENRIQUE CASANOVAS (a2) and ANAND PILLAY (a3)

Abstract

We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p (m) for all m. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If P(x) ε S(B) is stable and does not fork over A then prestrictionA is stable. (They had solved some special cases.)

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[1] Casanovas, E., More on NIP and Related Topics, September 2011. Lecture Notes of Model Theory Seminar, University of Barcelona, available at http://www.ub.edu/modeltheory/documentos/nip2.pdf.
[2] Haskell, D., Hrushovski, E., and Macpherson, D., Stable domination and independence in algebraically closed valued fields, Lecture Notes in Logic, vol. 30. Cambridge University Press, Cambridge, 2008.
[3] Hasson, A. and Onshuus, A., Stable types in rosy theories. this Journal, vol. 75 (2010), no. 4, pp. 12111230.
[4] Hrushovski, E. and Pillay, A., On NIP and invariant measures. Journal of the European Mathematical Society, vol. 13 (2011), pp. 10051061.
[5] Pillay, A. and Tanović, P., Generic stability, regularity, and quasi-minimality, Models, logics and higher-dimensional categories, vol. 53, CRM Proceedings and Lecture Notes, AMS, Providence, RI, 2011, pp. 189211.
[6] Shelah, S., Classification theory for elementary classes with the dependence property—A modest beginning. Scientiae Mathematicae Japonicae, vol. 59 (2004), no. 2, pp. 265316.
[7] Poizat, B., A course in model theory. Springer-Verlag, New York, 2000.
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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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