Small sets in dense pairs

Abstract

Let $\widetilde{\cal M} = \langle {\cal M},P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set P ⊆ M, such that three tameness conditions hold. We prove that the induced structure on P by $\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small set X definable in $\widetilde{\cal M}$ can be definably embedded into some P^l, uniformly in parameters, settling a question from [8]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of ℳ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [13]. The above results are in contrast to recent literature, as it is known in general that $\widetilde{\cal M}$ does not eliminate imaginaries, and neither it nor the induced structure on P admits definable Skolem functions.