Small sets in Mann pairs

Pantelis E Eleftheriou

doi:10.1007/s00153-020-00747-2

Abstract

Let widetilde{{mathcal {M}}}=langle {{{mathcal {M}}}}, Grangle be an expansion of a real closed field {{{mathcal {M}}}} by a dense subgroup G of langle M^{>0}, cdot rangle with the Mann property. We prove that the induced structure on G by {{{mathcal {M}}}} eliminates imaginaries. As a consequence, every small set X definable in {{{mathcal {M}}}} can be definably embedded into some G^l, uniformly in parameters. These results are proved in a more general setting, where widetilde{{mathcal {M}}}=langle {{{mathcal {M}}}}, Prangle is an expansion of an o-minimal structure {{mathcal {M}}} by a dense set Psubseteq M, satisfying three tameness conditions.

Highlights

This note is a natural extension of the work in [6]
Expansions M = M, P of an o-minimal structure M by a dense predicate P ⊆ M were studied, and under three tameness conditions, it was shown that the induced structure Pind on P by M eliminates imaginaries
The tameness conditions were verified for dense pairs of real closed fields, for expansions of M by an independent set P, and for expansions of a real closed field M by a dense subgroup P of M>0, · with the Mann property, assuming P is divisible

Summary

Introduction

This note is a natural extension of the work in [6]. In that reference, expansions M = M, P of an o-minimal structure M by a dense predicate P ⊆ M were studied, and under three tameness conditions, it was shown that the induced structure Pind on P by M eliminates imaginaries. Van den Dries–Günaydin [5, Theorem 7.2] showed that in a Mann pair, where G is divisible (such as 2Q), every definable set X ⊆ Gn is a full trace; in particular, (ind)D from [6] holds. Under the mild assumption that for every prime p, G[p] has finite index in G, [5, Theorem 7.5] provides a near model completeness result, which is used in [1] to prove that every definable set X ⊆ Pn is a finite union of traces on ∅-definable subsets of Pn (Fact 3.10 below) Note this mild assumption is still satisfied by all multiplicative subgroups of R>0, · of finite rank (as noted in [9]). Using Theorem 1.1, we further reduce the study of P-bound sets to that of definable subsets of Pl. Corollary 1.3 Assume (OP), (dcl)D and (ind)D hold for every D ⊆ M which is dclindependent over P.

Preliminaries

Elimination of imaginaries

The induced structure

Proofs of the results