TY - JOUR
T1 - Definable structures in o-minimal theories
T2 - One dimensional types
AU - Hasson, Assaf
AU - Onshuus, Alf
AU - Peterzil, Ya'acov
N1 - 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
PY - 2010/12/1
Y1 - 2010/12/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=80054867221&partnerID=8YFLogxK
U2 - 10.1007/s11856-010-0085-y
DO - 10.1007/s11856-010-0085-y
M3 - Article
AN - SCOPUS:80054867221
VL - 179
SP - 363
EP - 379
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
SN - 0021-2172
IS - 1
ER -