Abstract
Let N be a structure definable in an o-minimal structure M and p ∈ SN (N), a complete N-1-type. If dimM(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.
Original language | English |
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Pages (from-to) | 363-379 |
Number of pages | 17 |
Journal | Israel Journal of Mathematics |
Volume | 179 |
Issue number | 1 |
DOIs | |
State | Published - 2010 |
Bibliographical note
Funding Information:∗ Supported by the EPSRC grant no. EP C52800X 1 ∗∗Partially supported by the EC FP6 through the Marie Curie Network MODNET (MRTN-CT-2004-512234) Received April 14, 2007 and in revised form January 1, 2009
ASJC Scopus subject areas
- General Mathematics