A pathological o-minimal quotient

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. 1 Elimination of imaginaries and o-minimality In o-minimal expansions of real closed fields, as well as many other o-minimal theories, elimination of imaginaries holds as a corollary of definable choice. As noted in [1], some o-minimal theories fail to eliminate imaginaries. For example, elimination of imaginaries fails in the theory of Q with the ordering and with a 4-ary predicate for the relation x − y = z − w. In [2], Eleftheriou, Peterzil, and Ramakrishnan observe that in this example, elimination of imaginaries holds after naming two parameters. This leads them to pose the following question: Question 1.1. Given an o-minimal structure M and a definable equivalence relation E on a definable set X, both definable over a parameter set A, is there a definable map which eliminates X/E, possibly over B ⊇ A? They answer this question in the affirmative when X/E has a definable group structure, as well as when dim(X/E) = 1. However, we will answer Question 1.1 negatively by giving a counterexample in §2. That is, we will give an o-minimal structure M and a set X/E interpretable in M , which cannot be put in definable bijection with a definable subset of Mk. Question 1.1 can be reformulated in several ways, by the following observation. Lemma 1.2. Let M be a structure, and let M M be any elementary extension, such as a monster model. The following are equivalent: (a) Every M -definable quotient can be eliminated over M .