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 Pl, 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.