## 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.
Recall that pseudofinite arithmetic is the (incomplete) theory consisting of formulas true in almost all structures $$\langle \{0, 1, \dots, n \}, + , \cdot \rangle$$ (with $$n$$ a natural number). This is a theory without NIP.

### MathOverflow Questions:

On structures that are not submitted to compatibility conditions

### MSC:

 03C45 Classification theory, stability, and related concepts in model theory 06A05 Total orders
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### 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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