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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### References:

[1] | A course in model theory; an introduction to contemporary mathematical logic (2000) |

[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 |

[6] | Archive for Mathematical Logic |

[7] | Characterizing rosy theories 72 pp 919– (2007) |

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