Abstract
Let \({\mathcal {M}}\) be a dense o-minimal structure, \({\mathcal {N}}\) an unstable structure interpretable in \({\mathcal {M}}\). Then there exists X, definable in \({\mathcal {N}^{eq}}\), such that X, with the induced \({\mathcal {N}}\)-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable þ-minimal types in structures interpretable in o-minimal theories.
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Hasson A., Kowalski P.: Strongly minimal expansions of \({(\mathbb {C},+)}\) definable in o-minimal fields. Proc. Lond. Math. Soc. (3) 97(1), 117–154 (2008)
Hasson A., Onshuus A.: Embedded o-minimal structures. Bull. Lond. Math. Soc. 42(1), 64–74 (2010)
Hasson, A., Onshuus, A.: Stable types in rosy theories. J. Symb. Logic (to appear)
Hasson, A., Peterzil, Y., Onshuus, A.: One dimensional structures in o-minimal theories. Israel J. Math. (to appear)
Hasson, A., Peterzil, Y., Onshuus, A.: One dimensional types in o-minimal theories. Israel J. Math (to appear)
Hrushovski E.: The Mordell–Lang conjecture for function fields. J. Am. Math. Soc. 9(3), 667–690 (1996)
Hrushovski E.: The Manin–Mumford conjecture and the model theory of difference fields. Ann. Pure Appl. Logic 112(1), 43–115 (2001)
Johns J.: An open mapping theorem for o-minimal structures. J. Symb. Logic 66(4), 1817–1820 (2001)
Lascar D., Poizat B.: An introduction to forking. J. Symb. Logic 44(3), 330–350 (1979)
Onshuus A.: Properties and consequences of thorn-independence. J. Symb. Logic 71(1), 1–21 (2006)
Onshuus A., Peterzil Y.: A note on stable sets and groups in theories with nip. Math. Log. Q. 53(3), 295–300 (2007)
Peterzil Y.: Constructing a group-interval in o-minimal structures. J. Pure Appl. Algebra 94(1), 85–100 (1994)
Peterzil Y., Starchenko S.: A trichotomy theorem for o-minimal structures. Proc. Lond. Math. Soc. 77(3), 481–523 (1998)
Peterzil Y., Starchenko S.: Expansions of algebraically closed fields in o-minimal structures. Selecta Math. (N.S.) 7(3), 409–445 (2001)
Peterzil, Y., Starchenko, S.: Complex analytic geometry and analytic-geometric categories. Journal fur die reine und angewandte Mathematik (Crelle’s Journal) (2007, to appear)
Peterzil, Y., Starchenko, S.: Recovering a local exponential map, preprint (2007)
Rabinovich, E.D.: Definability of a field in sufficiently rich incidence systems. QMW Maths Notes, vol. 14. Queen Mary and Westfield College School of Mathematical Sciences, London. With an introduction by Wilfrid Hodges (1993)
Scanlon T.: Diophantine geometry of the torsion of a Drinfeld module. J. Number Theory 97(1), 10–25 (2002)
Scanlon T.: A positive characteristic Manin–Mumford theorem. Compos. Math. 141(6), 1351–1364 (2005)
Scanlon T.: Local André–Oort conjecture for the universal abelian variety. Invent. Math. 163(1), 191–211 (2006)
Dries, L.V.D.: Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)
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A. Hasson’s work was supported by the EPSRC grant no. EP C52800X 1.
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Hasson, A., Onshuus, A. Unstable structures definable in o-minimal theories. Sel. Math. New Ser. 16, 121–143 (2010). https://doi.org/10.1007/s00029-010-0018-y
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DOI: https://doi.org/10.1007/s00029-010-0018-y