Small sets in Mann pairs
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Mathematical Logic
ORIGINAL ARTICLE
Small sets in Mann pairs Pantelis E. Eleftheriou1 Received: 16 July 2019 / Accepted: 10 August 2020 © The Author(s) 2020
Abstract = M, G be an expansion of a real closed field M by a dense subgroup G Let M of M >0 , · with the Mann property. We prove that the induced structure on G by M eliminates imaginaries. As a consequence, every small set X definable in M can be definably embedded into some G l , uniformly in parameters. These results are proved = M, P is an expansion of an o-minimal in a more general setting, where M structure M by a dense set P ⊆ M, satisfying three tameness conditions. Keywords Mann pairs · Elimination of imaginaries · Small sets Mathematics Subject Classification Primary 03C64; Secondary 06F20
1 Introduction This note is a natural extension of the work in [6]. In that reference, expansions = M, P of an o-minimal structure M by a dense predicate P ⊆ M were M 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 (henceforth called Mann pairs), assuming P is divisible. As pointed out in [6, Remark 4.10], without the divisibility assumption in the last example, the third tameness condition no longer holds, and in [6, Question 4.11] it was asked whether in that case Pind still eliminates imaginaries. In this note, we prove that it does. Indeed, we replace the third tameness condition by a weaker one, which we verify for arbitrary
Research supported by a Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.
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Pantelis E. Eleftheriou [email protected] Department of Mathematics and Statistics, University of Konstanz, Box 216, 78457 Constance, Germany
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P. E. Eleftheriou
Mann pairs, and prove that together with the two other tameness conditions it implies elimination of imaginaries for Pind . Let us fix our setting. Throughout this text, M = M, 0 , · of finite rank [8], such as 2Q and 2Z 3Z . Van den Dries–Günaydin [5, Theorem 7.2] showed that in a Mann pair, where moreover G is divisible (such as 2Q ), every definable set X ⊆ G n is a full trace; in particular, (ind) D from [6] holds. Without the divisibility assumption, however, this is no longer true. Consider for example G = 2Z 3Z and let X be the subgroup of G consisting of all elements divisible by 2. That is, X = {22m 32n : m, n ∈ Z}. This set is clearly dense and co-dense in R, and cannot be a trace on any subset of G. A substitute to [5, Theorem 7.2] was proved by Berenstein-Ealy-Günaydin [1], as follows. Consider, for every d ∈ N, the set G [d] of all elements of G divisible by d, G [d] = {x ∈ G : ∃y ∈ G, x = y d }. Under the mild assumption that for every prime p, G [ p] has finite index in G, [5, Theorem 7.5] provi
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