Adding linear orders

TL;DR

This paper investigates whether adding linear orders to NIP theories preserves the NIP property, providing counterexamples where it does not, especially in certain categorical and stable theories.

Contribution

It demonstrates that expanding some NIP theories by linear orders can introduce IP, with specific counterexamples in categorical and stable theories.

Findings

Adding linear orders can destroy NIP in certain theories.

Counterexamples include a categorical theory and an -stable NDOP theory.

Expanding some theories by linear orders interprets bounded arithmetic.

Abstract

We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an \omega-stable NDOP theory for which every expansion by a linear order interprets bounded arithmetic.