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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.
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.

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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