On almost orthogonality in simple theories. (English) Zbl 1084.03032

Ben-Yaacov, Itay; Wagner, Frank O.

J. Symb. Log. 69, No. 2, 398-408 (2004).

The authors study the interaction of a type \(p\) over a set \(A\) in a simple theory with a family \(\Sigma\) of partial types over \(A\). In a stable theory internality and finite generation are the same; examples of Pillay show there are simple theories where these concepts differ. The authors show that for a real type \(p\) which is internal in a set \(\Sigma\) of partial types in a simple theory there is a type \(p^{\prime}\) interbounded with \(p\) such that \(p^{\prime}\) is finitely generated over \(\Sigma\) and has a fundamental system of solutions relative to \(\Sigma\). The authors also show that if \(p\) is a possibly hyperimaginary Lascar strong type that is almost \(\Sigma\)-internal and almost orthogonal to \(\Sigma^{\omega}\), then there is a canonical, non-trivial, hyperdefinable polygroup which multi-acts on \(p\) and fixes \(\Sigma\) generically. If \(p\) is \(\Sigma\)-internal and \(T\) is stable, this is the binding group of \(p\) over \(\Sigma\).

Reviewer: J. M. Plotkin (East Lansing)

MSC:

03C45

Classification theory, stability, and related concepts in model theory

Keywords:

simple theory; real type; partial types; fundamental system of solutions

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