A pathological o-minimal quotient

TL;DR

This paper presents a counterexample in o-minimal structures showing certain quotients cannot be eliminated, while also demonstrating that imaginaries are interdefinable with real elements over independent parameters.

Contribution

It provides the first example of a definable quotient in an o-minimal structure that cannot be eliminated over any parameter set, answering a longstanding question.

Findings

Existence of a definable quotient that cannot be eliminated

Imaginaries in o-minimal structures are interdefinable with real tuples over independent parameters

Interpretable sets are locally like definable sets

Abstract

We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in an o-minimal structure is interdefinable over an independent set of parameters with a tuple of real elements. This can be interpreted as saying that interpretable sets look "locally" like definable sets, in a sense which can be made precise.