Definable structures in o-minimal theories: one dimensional types
The paper deals with a structure \({\mathcal N}\) definable in a sufficiently saturated o-minimal structure \({\mathcal M}\) (with a dense underlying order) and proves a Zilber trichotomy principle for complete 1-types \(p\) over \(N\) in \({\mathcal N}\) having dimension 1 in \({\mathcal M}\). In a previous paper the authors showed a weak version of this result, saying that, if \({\mathcal N}\) is stable and one-dimensional, then it is 1-based. Here they prove that exactly one of the following conditions holds: {\parindent6.5mm \begin{itemize}\item[(1)] \(p\) is trivial. \item[(2)] \(p\) is linear, in which case \(p\) is non-orthogonal to a generic type of an \(N\)-definable (possibly locally ordered) group \(G\); moreover the structure induced by \({\mathcal N}\) on \(G\) is linear, that is, given by definable subgroups of the various \(G^n\). \item[(3)] \(p\) is rich, in which case it is non-orthogonal to a generic type of a real closed field definable in \({\mathcal N}\). \end{itemize}} A similar trichotomy theorem is proved for the structures \({\mathcal N}\). In particular, it is shown that \({\mathcal N}\) satisfies exactly one of the following conditions: {\parindent6.5mm \begin{itemize}\item[(1)] \({\mathcal N}\) is degenerate. \item[(2)] \({\mathcal N}\) is linear: There is a subset \(X\) of \(N^{\text{eq}}\) with dimension 1 in \({\mathcal M}\) such that the structure \({\mathcal N}\) induces on \(X\) is either a strongly minimal locally modular group or is elementarily equivalent to a group-interval in an ordered vector space. Furthermore, no field is interpretable in \({\mathcal N}\). \item[(3)] \({\mathcal N}\) is rich and interprets a real closed field. \end{itemize}}
scientific article; zbMATH DE number 5823094
scientific article