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Stable embeddedness and NIP. (English) Zbl 1220.03020

Summary: We give some sufficient conditions for a predicate \(\mathcal P\) in a complete theory \(T\) to be “stably embedded”. Let \(\mathcal P\) be \(P\) with its “induced \(\emptyset \)-definable structure”. The conditions are that \(\mathcal P\) (or rather its theory) is “rosy”, \(P\) has NIP in \(T\) and that \(P\) is stably 1-embedded in \(T\). This generalizes a recent result of A. Hasson and A. Onshuus [Bull. Lond. Math. Soc. 42, No. 1, 64–74 (2010; Zbl 1187.03034)] which deals with the case where \(P\) is o-minimal in \(T\). Our proofs make use of the theory of strict nonforking and weight in NIP theories.

MSC:

03C45 Classification theory, stability, and related concepts in model theory

Citations:

Zbl 1187.03034
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References:

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[2] Properties and consequences of thorn independence 71 pp 1– (2006) · Zbl 1103.03036
[3] DOI: 10.1112/blms/bdp098 · Zbl 1187.03034
[4] DOI: 10.1007/s11856-009-0082-1 · Zbl 1195.03040
[5] DOI: 10.1016/j.apal.2007.09.004 · Zbl 1144.03025
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