Unstable structures definable in o-minimal theories

TL;DR

This paper investigates the properties of unstable structures definable within o-minimal theories, showing that such structures contain interpretable o-minimal subsets and characterizing their types based on dimension.

Contribution

It establishes that unstable definable structures in o-minimal theories contain o-minimal interpretable subsets and characterizes types based on dimension and minimality.

Findings

Existence of o-minimal interpretable subsets in unstable structures

Types of 1-N-dimensional structures are either stable or finite by o-minimal

N-minimal structures are 1-M-dimensional

Abstract

Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.