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Definable structures in o-minimal theories: One dimensional types

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Abstract

Let N be a structure definable in an o-minimal structure M and p ∈ S N (N), a complete N-1-type. If dim M (p) = 1, then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N-definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to a generic type of an N-definable real closed field.

As a result, we obtain a similar trichotomy for definable one-dimensional structures in o-minimal theories.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Assaf Hasson

  2. Department of mathematics, Ben Gurion University of the Negev, P.O.B. 653, Beer Sheva, 84105, Israel

    Assaf Hasson

  3. Departemento de Matemáticas, Universidad de los Andes, Cra. 1 No 18A-10, Bogotá, Colombia

    Alf Onshuus

  4. Department of Mathematics, University of Haifa, Haifa, Israel

    Ya’acov Peterzil

Authors
  1. Assaf Hasson
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  2. Alf Onshuus
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  3. Ya’acov Peterzil
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Correspondence to Assaf Hasson.

Additional information

Supported by the EPSRC grant no. EP C52800X 1

Partially supported by the EC FP6 through the Marie Curie Research Training Network MODNET (MRTN-CT-2004-512234)

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Hasson, A., Onshuus, A. & Peterzil, Y. Definable structures in o-minimal theories: One dimensional types. Isr. J. Math. 179, 363–379 (2010). https://doi.org/10.1007/s11856-010-0085-y

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