Definable structures in o-minimal theories: One dimensional types

Assaf Hasson; Alf Onshuus; Ya'acov Peterzil

doi:10.1007/s11856-010-0085-y

Definable structures in o-minimal theories: One dimensional types

Assaf Hasson
, Alf Onshuus
, Ya'acov Peterzil

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	363-379
Number of pages	17
Journal	Israel Journal of Mathematics
Volume	179
Issue number	1
DOIs	https://doi.org/10.1007/s11856-010-0085-y
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

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10.1007/s11856-010-0085-y

Cite this

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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.",

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doi = "10.1007/s11856-010-0085-y",

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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

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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.

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Definable structures in o-minimal theories: One dimensional types

Abstract

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