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**Adding linear orders.**
*(English)*
Zbl 1251.03038

It is an open problem whether every unstable NIP theory interprets an infinite linear order. Here the authors deal with a related question, namely whether a NIP theory can be expanded by adding a linear order on the whole domain and preserving NIP. The first part of the paper gives a positive solution when algebraic closure is trivial (meaning that \(\mathrm{acl}(A) = A\) for all \(A\)). But strong negative answers are provided in other cases. In detail, it is shown:

- 1)
- there is an \(\omega\)-stable NDOP theory of depth 2 such that every extension by a linear order interprets pseudofinite arithmetic;
- 2)
- there is a totally categorical theory for which every expansion by a linear order does not have NIP.

Reviewer: Carlo Toffalori (Camerino)

### MSC:

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

06A05 | Total orders |

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\textit{S. Shelah} and \textit{P. Simon}, J. Symb. Log. 77, No. 2, 717--725 (2012; Zbl 1251.03038)

### References:

[1] | DOI: 10.1002/1521-3870(200101)47:1<45::AID-MALQ45>3.0.CO;2-A · Zbl 0967.03032 |

[2] | Decidability and N 0-categoricity of theories of partially ordered sets 45 pp 585– (1980) |

[3] | DOI: 10.1090/S0002-9947-00-02672-6 · Zbl 0960.03027 |

[4] | Ramsey theory (1990) |

[5] | DOI: 10.1080/01621459.1963.10500830 |

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