Adding linear orders (Q2892687)

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: {\parindent=6mm\begin{itemize}\item[1)] there is an \(\omega\)-stable NDOP theory of depth 2 such that every extension by a linear order interprets pseudofinite arithmetic; \item[2)] there is a totally categorical theory for which every expansion by a linear order does not have NIP. NEWLINENEWLINE\end{itemize}} 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.