Small sets in Mann pairs

TL;DR

This paper proves that in certain o-minimal expansions of real closed fields with dense subgroups, small definable sets can be embedded into finite powers of the subgroup, extending previous results to a more general setting.

Contribution

It establishes elimination of imaginaries for the induced structure on dense subgroups with the Mann property in a broad o-minimal context, generalizing prior work.

Findings

Small sets in the structure can be embedded into finite powers of the dense subgroup.

Elimination of imaginaries holds for the induced structure on the dense subgroup.

Results apply to a general class of o-minimal expansions with tameness conditions.

Abstract

Let $\widetilde{\mathcal M}=\langle \mathcal M, G\rangle$ 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 $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ is an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, satisfying three tameness conditions.

Equations

$${\mathcal{L}}_{ind(\mathcal{D})}=\{{\mathcal{R}}_{\varphi (x)}(x):\varphi (x)\in {\mathcal{L}}_{\mathcal{D}}\},$$

$$P_{ind(D)}\models R_{\varphi }(a)\iff \mathcal{M}\models \varphi (a).$$

$$A=\mathrm{dcl}(BD)\cap \mathrm{dcl}(CD)\cap P.$$

$$\mathrm{dcl}(AD)=\mathrm{dcl}(BD)\cap \mathrm{dcl}(CD).$$

$$X=\bigcup _i(Y_i\cap Q_i).$$

$$a_1{x}_1+\cdots +a_r{x}_r=1$$

$$G^{[d]}=\{x\in G:\exists y\in G,x=y^d\}.$$

$$\{b\in M^m:(b,x)\in X\}.$$

$${\mathrm{int}}_Y(X)=\{x\in X:\text{ there is open B⊆Mn containing x with B∩Y⊆X}\}.$$

$$\dim (X\setminus {\mathrm{int}}_Y(X))<\dim Y.$$

$$E(x,y)\iff f(x)=f(y).$$

$$P_{ind(D)}=⟨P,\{R\cap P^l:R\subseteq M^l\text{ LD-definable},l\in \mathbb{N}\}⟩.$$

$$X=\bigcup _{i=1}^m(Z_i\cap R_i),$$

$$X=\bigcup _{i=1}^l(Y_i\cap Q_i),$$

$$Q_{\sigma }=\bigcup _{i\in \sigma }{R}_i$$

$$Y_{\sigma }=\left(\bigcap _{i\in \sigma }{Z}_i\right)\setminus \left(\bigcup _{j\notin \sigma }{Z}_j\right).$$

$$X=\bigcup _{\sigma \subseteq \{1,\dots ,m\}}(Y_{\sigma }\cap Q_{\sigma }),$$

$$X=\bigcup _{i=1}^l{W}_i\cap S_i.$$

$$X=\bigcup _{i=1}^l(Z_i\cap R_i).$$

$$T_i=\{x\in \overline{X}:\text{ there is relatively open V⊆X around x, with V∩Ri⊆X}\}.$$

$$\dim\big{(}(Z_{i}\cap X)\setminus(T_{i}\cap R_{i})\big{)}<k.$$

$$(Z_{i}\cap X)\setminus(T_{i}\cap R_{i})=\big{(}(Z_{i}\cap X)\setminus\operatorname{int}_{\overline{X}}(Z_{i}\cap X)\big{)}\cup\big{(}\operatorname{int}_{\overline{X}}(Z_{i}\cap X)\setminus(T_{i}\cap R_{i})\big{)},$$

$${\mathrm{int}}_{\overline{X}}(Z_i)\cap X\subseteq T_i\cap R_i.$$

$$X=\left(X\setminus \bigcup _i(T_i\cap R_i)\right)\cup \bigcup _i(T_i\cap R_i),$$

$$X=\bigcup _i(W_i\cap S_i).$$

$$X_i=\{x\in \pi (W_i):(W_i{)}_x\text{ is singleton}\}$$

$$W_i^{\prime }=\left(\bigcup _{x\in X_i}\{x\}\times (W_i{)}_x\right)\cup \{(x,0):x\in M^n\setminus X_i\}.$$

$$X=\bigcup _{i=1}^l{W}_i\cap S_i.$$

$$\mathrm{dcl}(BD)\cap \mathrm{dcl}(CD)\cap P=c{l}_D(B)\cap c{l}_D(C)=A,$$

$$X=\bigcup _i(Y_i\cap Q_i).$$

Full text

Research supported by an Independent Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.

1. Introduction

This note is a natural extension of the work in [6]. In that reference, expansions $\widetilde{\cal M}=\langle{\cal M},P\rangle$ of an o-minimal structure ${\cal M}$ by a dense predicate $P\subseteq M$ were studied, and under three tameness conditions, it was shown that the induced structure $P_{ind}$ on $P$ by ${\cal M}$ eliminates imaginaries. The tameness conditions were verified for dense pairs of real closed fields, for expansions of $\cal M$ by an independent set $P$, and for expansions of a real closed field $\cal M$ by a dense subgroup $P$ of $\langle M^{>0},\cdot\rangle$ 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 $P_{ind}$ 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 Mann pairs, and prove that together with the two other tameness conditions it implies elimination of imaginaries for $P_{ind}$.

Let us fix our setting. Throughout this text, $\cal M=\langle M,<,+,0,\dots\rangle$ denotes an o-minimal expansion of an ordered group with a distinguished positive element $1$. We denote by $\cal L$ its language, and by $\operatorname{dcl}$ the usual definable closure operator in M. An ‘$\cal L$-definable’ set is a set definable in $\cal M$ with parameters. We write ‘$\cal L_{A}$-definable’ to specify that those parameters come from $A\subseteq M$. It is well-known that $\cal M$ admits definable Skolem functions and eliminates imaginaries ([4, Chapter 6]).

Let $D,P\subseteq M$. The $D$-induced structure on $P$ by M, denoted by $P_{ind(D)}$, is a structure in the language

[TABLE]

whose universe is $P$ and, for every tuple $a\subseteq P$,

[TABLE]

If $Q\subseteq P^{n}$, by a trace on $Q$ we mean a set of the form $Y\cap Q$, where $Y$ is $\cal L$-definable. We call $Y\cap P^{n}$ a full trace.

For the rest of this paper we fix some $P\subseteq M$ and denote $\widetilde{\cal M}=\langle\cal M,P\rangle$. We let $\cal L(P)$ denote the language of $\widetilde{\cal M}$; namely, the language $\cal L$ augmented by a unary predicate symbol $P$. We denote by $\operatorname{dcl}_{\cal L(P)}$ the definable closure operator in $\widetilde{\cal M}$. Unless stated otherwise, by ‘($A$-)definable’ we mean ($A$-)definable in $\widetilde{\cal M}$, where $A\subseteq M$. We use the letter $D$ to denote an arbitrary, but not fixed, subset of $M$.

Tameness Conditions (for $\widetilde{\cal M}$ and $D$):

• (OP)

(Open definable sets are $\cal L$-definable.) For every set $A$ such that $A\setminus P$ is $\operatorname{dcl}$-independent over $P$, and for every $A$-definable set $V\subset M^{n}$, its topological closure $\overline{V}\subseteq M^{n}$ is $\cal L_{A}$-definable.

• (dcl)D

Let $B,C\subseteq P$ and

[TABLE]

Then

[TABLE]

• (ind)D

Let $X\subseteq P^{n}$ be definable in $P_{ind(D)}$. Then $X$ is a finite union of traces on sets which are $\emptyset$-definable in $P_{ind(D)}$. That is, there are $\cal L$-definable sets $Y_{1},\dots,Y_{l}\subseteq M^{n}$, and sets $Q_{1},\dots,Q_{l}\subseteq P^{k}$ that are $\emptyset$-definable in $P_{ind(D)}$, such that

[TABLE]

Conditions (OP) and (dcl)D are the same with those in [6], and are already known to hold for Mann pairs ([6, Remark 4.11]). Condition (ind)D is weaker than the corresponding one in [6], in three ways: (a) $X$ is now a finite union of traces (instead of a single trace), (b) the traces are on subsets of $P^{n}$ (instead of on the whole $P^{n}$), and (c) there is no control in parameters for the $Y_{i}$’s (although we achieve this in Corollary 3.5 below). These differences result in several non-trivial complications in the proof of our main theorem, which are handled in Section 3. For now, let us state the main theorem.

Theorem 1.1**.**

Assume (OP), (dcl)D and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Then $P_{ind(D)}$ eliminates imaginaries.

Condition (ind)D is modelled after the current literature on Mann pairs, which we now explain. Assume $\cal M=\langle M,<,+,\cdot,0,1\rangle$ is a real closed field, and $G$ a dense subgroup of $\langle M^{>0},\cdot\rangle$. For every $a_{1},\dots,a_{r}\in M$, a solution $(q_{1},\dots,q_{r})$ to the equation

[TABLE]

is called non-degenerate if for every non-empty $I\subseteq\{1,\dots,r\}$, $\sum_{i\in I}a_{i}q_{i}\neq 0$. We say that $G$ has the Mann property, if for every $a_{1},\dots,a_{r}\in M$, the above equation has only finitely many non-degenerate solutions $(q_{1},\dots,q_{r})$ in $G^{r}$.111 The original definition only involved equations with coefficients $a_{i}$ in the prime field of $\cal M$, but, by [5, Proposition 5.6], the two definitions are equivalent. Let us call such a pair $\langle{\cal M},G\rangle$ a Mann pair. Examples of Mann pairs include all multiplicative subgroups of $\langle\mathbb{R}_{>0},\cdot\rangle$ of finite rank ([8]), such as $2^{\mathbb{Q}}$ and $2^{\mathbb{Z}}3^{\mathbb{Z}}$. Van den Dries - Günaydin [5, Theorem 7.2] showed that in a Mann pair, where moreover $G$ is divisible (such as $2^{\mathbb{Q}}$), every definable set $X\subseteq G^{n}$ is a full trace; in particular, (ind)D from [6] holds. Without the divisibility assumption, however, this is no longer true. Consider for example $G=2^{\mathbb{Z}}3^{\mathbb{Z}}$ and let $X$ be the subgroup of $G$ consisting of all elements divisible by $2$. That is, $X=\{2^{2m}3^{2n}:m,n\in\mathbb{Z}\}$. This set is clearly dense and co-dense in $\mathbb{R}$, 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 $d\in\mathbb{N}$, the set $G^{[d]}$ of all elements of $G$ divisible by $d$,

[TABLE]

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 then used in [1] to prove that every definable set $X\subseteq P^{n}$ is a finite union of traces on $\emptyset$-definable subsets of $P^{n}$ (Fact 3.10 below). Note this mild assumption is still satisfied by all multiplicative subgroups of $\langle\mathbb{R}_{>0},\cdot\rangle$ of finite rank (as noted in [9]).

Corollary 1.2**.**

Assume $\widetilde{\cal M}=\langle{\cal M},G\rangle$ is a Mann pair, such that for every prime $p$, $G^{[p]}$ has finite index in $G$. Let $D\subseteq M$ be $\operatorname{dcl}$-independent over $P$. Then (OP), (dcl)D and (ind)D hold. In particular, $P_{ind(D)}$ eliminates imaginaries.

Observe that Corollary 1.2 stands in contrast to the current literature, as it is known that in Mann pairs both existence of definable Skolem functions and elimination of imaginaries (for $\widetilde{\cal M}$) fail ([2]). Note also that the assumption of $D$ being $\operatorname{dcl}$-independent over $P$ is necessary; namely, without it, $P_{ind(D)}$ need not eliminate imaginaries ([6, Example 5.1]).

Theorem 1.1 has the following important consequence. Recall from [3] that a set $X\subseteq M^{n}$ is called $P$-bound over $A$ if there is an $\cal L_{A}$-definable function $h:M^{m}\to M^{n}$ such that $X\subseteq h(P^{m})$. The recent work in [7] provides an analysis for all definable sets in terms of ‘$\cal L$-definable-like’ and $P$-bound sets. Using Theorem 1.1, we further reduce the study of $P$-bound sets to that of definable subsets of $P^{l}$.

Corollary 1.3**.**

Assume (OP), (dcl)D and (ind)D hold for every $D\subseteq M$ which is $\operatorname{dcl}$-independent over $P$. Let $X\subseteq M^{n}$ be an $A$-definable set. If $X$ is $P$-bound over $A$, then there is an $A\cup P$-definable injective map $\tau:X\to P^{l}$. If $A$ itself is $\operatorname{dcl}$-independent over $P$, then the extra parameters from $P$ can be omitted.

Note that the assumption of Corollary 1.3 holds for $\widetilde{\cal M}$ as in Corollary 1.2. Note also that allowing parameters from $P$ is standard practice when studying definability in this context; see for example [7, Lemma 2.5 and Corollary 3.26].

Structure of the paper. In Section 2, we fix notation and recall some basic facts. In Section 3, we prove our results.

2. Preliminaries

We assume familiarity with the basics of o-minimality and pregeometries, as can be found, for example, in [4] or [10]. Recall that $\cal M=\langle M,<,+,0,\dots\rangle$ is our fixed o-minimal expansion of an ordered group with a distinguished positive element $1$ and $\operatorname{dcl}$ denotes the usual definable closure operator. We denote the corresponding dimension by $\dim$. If $A,B$ are two sets, we often write $AB$ for $A\cup B$. We denote by $\Gamma(f)$ the graph of a function $f$. If $T\subseteq M^{m}\times M^{n}$ and $x\in M^{n}$, we write $T_{x}$ for

[TABLE]

The topological closure of a set $Y\subseteq M^{n}$ is denoted by $\overline{Y}$ and its frontier $\overline{Y}\setminus Y$ by $\operatorname{fr}(Y)$. If $X\subseteq Y$, the relative interior of $X$ in $Y$ is denoted by $\operatorname{int}_{Y}(X)$. It is not hard to see that

[TABLE]

Fact 2.1**.**

Let $X\subseteq Y\subseteq M^{n}$ be two $\cal L$-definable sets. Then

[TABLE]

Proof.

If $\dim X<\dim Y$, we are done. Assume $\dim X=\dim Y$ and, towards a contradiction, that the inequality fails. Then there is a set $V\subseteq X$ with $\dim V=\dim Y$, such that $V\cap\operatorname{int}_{Y}(X)=\emptyset$. By cell decomposition, it is not hard to find open $B\subseteq M^{n}$ such that $\emptyset\neq B\cap Y\subseteq V\subseteq X$, and hence $V$ contains elements in $\operatorname{int}_{Y}(X)$, a contradiction. ∎

2.1. Elimination of imaginaries

We recall that a structure $\cal N$ eliminates imaginaries if for every $\emptyset$-definable equivalence relation $E$ on $N^{n}$, there is a $\emptyset$-definable map $f:N^{n}\to N^{l}$ such that for every $x,y\in N^{n}$,

[TABLE]

In the order setting, we have the following criterion (extracted from [10, Section 3]; for a proof see [6, Fact 2.2]).

Fact 2.2**.**

Let $\cal N$ be a sufficiently saturated structure with two distinct constants in its language. Suppose the following property holds.

• (*)

Let $B,C\subseteq N$ and $A=\operatorname{dcl}_{\cal N}(B)\cap\operatorname{dcl}_{\cal N}(C)$. If $X\subseteq N^{n}$ is $B$-definable and $C$-definable, then $X$ is $A$-definable.

Then $\cal N$ eliminates imaginaries.

2.2. The induced structure

Recall from the introduction that

[TABLE]

Remark 2.3*.*

For $A\subseteq P$, we have:

1. (1)

if $Q\subseteq P^{n}$ is $A$-definable in $P_{ind(D)}$, and $Y\subseteq M^{n}$ is $\cal L_{AD}$-definable, then $Q\cap Y$ is $A$-definable in $P_{ind(D)}$. Indeed, $Q\cap Y=Q\cap(Y\cap P^{n})$. 2. (2)

in general, if $Q\subseteq P^{n}$ is $A$-definable in $P_{ind(D)}$, then it is $AD$-definable. The converse will be true for Mann pairs, by Corollary 3.11 below.

3. Proofs of the results

In this section we prove elimination of imaginaries for $P_{ind(D)}$ under our assumptions (Theorem 1.1) and deduce Corollaries 1.2 and 1.3 from it. Our goal is to establish () from Fact 2.2 for $\cal N=P_{ind(D)}$ (Lemma 3.8 below). As in [6], the strategy is to reduce the proof of () to [10, Proposition 2.3], which is an assertion of (*) for $\cal M$. This reduction takes place in the proof of Lemma 3.8 below, and requires the key Lemma 3.4. The analogous key lemma in [6] (namely, [6, Lemma 3.1]) cannot help us here, because its assumptions are not met in the proof of Lemma 3.8. Furthermore, the proof of Lemma 3.4 requires an entirely new technique.

We begin with some preliminary observations.

Fact 3.1**.**

Assume (OP). Then for every $A\subseteq P$, $\operatorname{dcl}_{\cal L(P)}(A)=\operatorname{dcl}(A)$.

Proof.

Take $x\in\operatorname{dcl}_{\cal L(P)}(A)$. That is, the set $\{x\}$ is $A$-definable in $\widetilde{\cal M}$. By (OP), we have that $\overline{\{x\}}$ is $\cal L_{A}$-definable. But $\overline{\{x\}}=\{x\}$. ∎

Lemma 3.2**.**

Assume (OP). Let $X\subseteq M^{n}$ be an $\cal L$-definable set which is also $C$-definable, for some $C\subseteq M$ with $C\setminus P$ $\operatorname{dcl}$-independent over $P$. Then $X$ is $\cal L_{C}$-definable.

Proof.

We work by induction on $k=\dim X$. For $k=0$, $X$ is finite, and hence every element of it is in $\operatorname{dcl}_{\cal L(P)}(C)$. By Fact 3.1, it is in $\operatorname{dcl}(C)$. Now assume $k\geq 0$. By (OP), $\overline{X}$ is $\cal L_{C}$-definable. By o-minimality, $\dim\operatorname{fr}(X)<k$. Since $\operatorname{fr}(X)=\overline{X}\setminus X$ is both $\cal L$-definable and $C$-definable, by inductive hypothesis, it is $\cal L_{C}$-definable. So $X=\overline{X}\setminus\operatorname{fr}(X)$ is $\cal L_{C}$-definable. ∎

Lemma 3.3**.**

Let $C\subseteq M$ and

[TABLE]

where $Z_{1},\dots,Z_{m}\subseteq M^{n}$ are $\cal L_{C}$-definable sets, and $R_{1},\dots,R_{m}\subseteq P^{n}$ are $\emptyset$-definable in $P_{ind(D)}$. Then

[TABLE]

for some $\cal L_{C}$-definable disjoint sets $Y_{1},\dots,Y_{l}\subseteq M^{n}$, and sets $Q_{1},\dots,Q_{l}\subseteq P^{n}$ which are $\emptyset$-definable in $P_{ind(D)}$.

Proof.

For $\sigma\subseteq\{1,\dots,m\}$, let

[TABLE]

and

[TABLE]

It is then easy to check that for any two distinct $\sigma,\tau\subseteq\{1,\dots,m\}$, we have $Y_{\sigma}\cap Y_{\tau}=\emptyset$, and that

[TABLE]

as required. ∎

Now, the key technical lemma.

Lemma 3.4**.**

Assume (OP) and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Let $B,C\subseteq P$ and $X\subseteq P^{n}$ be $B$-definable and $C$-definable in $P_{ind(D)}$. Then there are $W_{1},\dots,W_{l}\subseteq M^{n}$, that are both $\cal L_{BD}$-definable and $\cal L_{CD}$-definable, and sets $S_{1},\dots,S_{k}\subseteq P^{n}$, that are $\emptyset$-definable in $P_{ind(D)}$, such that

[TABLE]

Proof.

First note that $X$ is both $BD$-definable and $CD$-definable in $\langle\cal M,P\rangle$. Since $B,C\subseteq P$, by (OP) it follows that $\overline{X}$ is $\cal L_{BD}$-definable and $\cal L_{CD}$-definable.

We perform induction on the dimension of $\overline{X}$. For $\dim\overline{X}=0$, $X$ is finite and $X=\overline{X}=P^{n}\cap\overline{X}$, as needed. Suppose now that $\dim X=k>0$. By (ind)D and Lemma 3.3, there are $\cal L$-definable disjoint sets $Z_{1},\dots,Z_{m}\subseteq M^{n}$, and sets $R_{1},\dots,R_{l}\subseteq P^{n}$, each $\emptyset$-definable in $P_{ind(D)}$, such that

[TABLE]

For every $i$, define

[TABLE]

Let $T=\bigcup_{i}T_{i}$. It is immediate from the definition, that each $T_{i}$, and hence $T$, is relatively open in $\overline{X}$. Therefore, by (OP), it is $\cal L$-definable. On the other hand, each $T_{i}$ is $BD$-definable and $CD$-definable, because $X$ is, and $R_{i}$ is $D$-definable. Hence, by Lemma 3.2, each $T_{i}$, and hence $T$, is $\cal L_{BD}$-definable and $\cal L_{CD}$-definable.

Claim. $\dim\overline{X\setminus\bigcup_{i}(T_{i}\cap R_{i})}<k$.

Proof.

Observe first that $X\subseteq\bigcup_{i}Z_{i}$, and hence it suffices to show that for each $i$,

[TABLE]

We may write

[TABLE]

By Fact 2.1, it suffices to show that $\operatorname{int}_{\overline{X}}(Z_{i}\cap X)\subseteq(T_{i}\cap R_{i})$. Clearly, $\operatorname{int}_{\overline{X}}(Z_{i}\cap X)\subseteq\operatorname{int}_{\overline{X}}(Z_{i})\cap X$, and hence it suffices to show:

[TABLE]

Let $x\in\operatorname{int}_{\overline{X}}(Z_{i})\cap X$. Since $x\in\operatorname{int}_{\overline{X}}(Z_{i})$, there is a relatively open $V\subseteq\overline{X}$ containing $x$, with $V\subseteq Z_{i}$, and hence $V\cap R_{i}\subseteq Z_{i}\cap R_{i}\subseteq X$. Therefore $x\in T_{i}$. Since $x\in X\cap Z_{i}$ and the $Z_{j}$’s are disjoint, we must also have $x\in R_{i}$. Hence $x\in T_{i}\cap R_{i}$, as needed. ∎

By Remark 2.3(1), the set $(X\cap T)\setminus\bigcup_{i}(T_{i}\cap R_{i})$ is both $B$-definable and $C$-definable in $P_{ind(D)}$. Hence, by inductive hypothesis and the claim, the conclusion holds for this set. Now, for each $i$, by definition of $T_{i}$, we have $T_{i}\cap R_{i}\subseteq X$. Hence

[TABLE]

and we are done.∎

Corollary 3.5**.**

Assume (OP) and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Let $A\subseteq P$ and $X\subseteq P^{n}$ be $A$-definable in $P_{ind(D)}$. Then there are $\cal L_{AD}$-definable sets $W_{1},\dots,W_{l}\subseteq M^{n}$, and sets $S_{1},\dots,S_{l}\subseteq P^{k}$ that are $\emptyset$-definable in $P_{ind(D)}$, such that

[TABLE]

Proof.

By Lemma 3.4 for $B=C=A$. ∎

Our next goal is to prove the promised Lemma 3.8. Denote by $cl_{D}$ the definable closure operator in $P_{ind(D)}$. We first prove that, under (OP) and (ind)D, $cl_{D}$ defines a pregeometry (Corollary 3.7).

Lemma 3.6**.**

Assume (OP) and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Let $f:P^{n}\to P$ be an $A$-definable map in $P_{ind(D)}$. Then there is an $\cal L_{AD}$-definable map $F:M^{n}\to M^{k}$ that extends $f$.

Proof.

By Corollary 3.5, there are finitely many $\cal L_{AD}$-definable sets $W_{1},\dots,W_{l}\subseteq M^{n+1}$ and $\emptyset$-definable sets $S_{1},\dots,S_{l}\subseteq P^{n+1}$, such that $\Gamma(f)=\bigcup_{i}W_{i}\cap S_{i}$. Fix $i$, and let $f_{i}$ be the map whose graph equals $W_{i}\cap S_{i}$. It clearly suffices to prove the lemma for $f_{i}$. By (OP) and o-minimality, each fiber $(S_{i})_{x}$ is dense in a finite union of open intervals and points. Hence, without loss of generality, we may assume that for every $x\in\pi(W_{i})\cap P^{n}$, the fiber $(W_{i})_{x}$ is a singleton. Denote by $\pi:M^{n+1}\to M^{n}$ the projection onto the first $n$ coordinates. The set

[TABLE]

is $\cal L_{AD}$-definable. So, $\pi(S_{i})\subseteq X_{i}$. Now let

[TABLE]

Then $W^{\prime}_{i}$ is $\cal L_{AD}$-definable, it is the graph of a function $F_{i}:M^{n}\to M$, and $\Gamma(f_{i})=W_{i}^{\prime}\cap S_{i}$, as required. ∎

Corollary 3.7**.**

Assume (OP) and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Then for every $A\subseteq P$, $\operatorname{cl}_{D}(A)=\operatorname{dcl}(AD)\cap P$. In particular, $cl_{D}$ defines a pregeometry.

Proof.

The inclusion $\supseteq$ is immediate from the definitions, whereas the inclusion $\subseteq$ is immediate from Lemma 3.6. Since $\operatorname{dcl}(-D)$ defines a pregeometry in $\cal M$, it follows easily that so does $cl_{D}(-)$ in $P_{ind(D)}$. ∎

Lemma 3.8**.**

Assume (OP), (dcl)D and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Let $B,C\subseteq P$ and $A=cl_{D}(B)\cap cl_{D}(C)$. If $X\subseteq P^{n}$ is $B$-definable and $C$-definable in $P_{ind(D)}$, then $X$ is $A$-definable in $P_{ind(D)}$.

Proof.

Let $X\subseteq P^{n}$ be $B$-definable and $C$-definable in $P_{ind(D)}$. By Lemma 3.4, there are $W_{1},\dots,W_{l}\subseteq M^{n}$, each both $\cal L_{BD}$-definable and $\cal L_{CD}$-definable, and $S_{1},\dots,S_{k}\subseteq P^{n}$, each $\emptyset$-definable in $P_{ind(D)}$, such that

[TABLE]

By [10, Proposition 2.3], each $W_{i}$ is L-definable over $\operatorname{dcl}(BD)\cap\operatorname{dcl}(CD)$. By (dcl)D, $W_{i}$ is L-definable over $\operatorname{dcl}(BD)\cap\operatorname{dcl}(CD)\cap PD$. Hence $X$ is definable over $\operatorname{dcl}(BD)\cap\operatorname{dcl}(CD)\cap P$ in $P_{ind(D)}$. But

[TABLE]

and hence $X$ is $A$-definable in $P_{ind(D)}$. ∎

We can now conclude our results.

Proof of Theorem 1.1.

By Fact 2.2 and Lemma 3.8. ∎

For the proof of Corollary 1.3, we additionally need the following lemma.

Lemma 3.9**.**

Assume (OP and (ind)D, and that $D$ is $\operatorname{dcl}$-independent over $P$. Let $\cal M^{\prime}$ be the expansion of $\cal M$ with constants for all elements in $P$, and $\widetilde{\cal M}^{\prime}=\langle\cal M^{\prime},P\rangle$. Then (ind)D holds for $\widetilde{\cal M}^{\prime}$ and $D$.

Proof.

Denote by $P^{\prime}_{ind(D)}$ the $D$-induced structure on $P$ by $\cal M^{\prime}$. Let $X\subseteq P^{n}$ be $A$-definable in $P^{\prime}_{ind(D)}$. It follows that $X$ is $AP$-definable in $P_{ind(D)}$. By Corollary 3.5, there are $\cal L_{APD}$-definable sets $Y_{1},\dots,Y_{l}\subseteq M^{n}$, and $Q_{1},\dots,Q_{l}\subseteq P^{k}$, which are $\emptyset$-definable in $P_{ind(D)}$, such that

[TABLE]

Such $Y_{i}$’s are $\cal L_{AD}$-definable in $\cal M$, and the $Q_{i}$’s are of course $\emptyset$-definable in $P^{\prime}_{ind(D)}$, as required. ∎

Proof of Corollary 1.3.

The proof when $A$ is $\operatorname{dcl}$-independent over $P$ is identical to that of [6, Theorem B]. The proof of the general case is identical to that of [6, Corollary 1.4], after replacing in [6, Lemma 3.4] the clause about (ind)D with Lemma 3.9 above. ∎

We finally turn to our targeted example of Mann pairs. The proof of Corollary 1.2 will be complete after we recall the fact below, which is extracted from [1]. First, observe that if $\widetilde{\cal M}=\langle{\cal M},G\rangle$ is a Mann pair, then for every $d\in\mathbb{N}$, $G^{[d]}$ is $\emptyset$-definable in $P_{ind(\emptyset)}$. Indeed, $G^{[d]}$ is the projection onto the first coordinate of the set $\{(x^{d},x):x\in M\}\cap G^{2}.$

Fact 3.10**.**

Let $\widetilde{\cal M}=\langle{\cal M},G\rangle$ be a Mann pair, such that for every prime $p$, $G^{[p]}$ has finite index in $G$. Let $X\subseteq P^{n}$ a definable set. Then $X$ is a finite union of traces on sets which are $\emptyset$-definable in $P_{ind(\emptyset)}$. That is, (ind)D holds.

Proof.

By [1, Corollary 57], $X$ is as a finite union of traces on sets of the form $g(G^{[d]})^{n}$, $d\in\mathbb{N}$. As pointed out in the proof of [1, Theorem 1], each such $g$ can be chosen to be $\emptyset$-definable (in $\widetilde{\cal M}$). By Fact 3.1, $g\in\operatorname{dcl}(\emptyset)$. By the above observation, $g(G^{[d]})^{n}$ is $\emptyset$-definable in $P_{ind(\emptyset)}$. ∎

Proof of Corollary 1.2.

By Fact 3.10, (ind)D hold. By [6], as explained in Remark 4.11 therein, (OP) and (dcl)D holds. By Theorem 1.1, we are done. ∎

A byproduct of our work is the following corollary.

Corollary 3.11**.**

Let $\widetilde{\cal M}$ and $D$ be as in Corollary 1.2. Let $X\subseteq P^{n}$ be $AD$-definable, with $A\subseteq P$. Then $X$ is $A$-definable in $P_{ind(D)}$. In particular, the conclusion of Corollary 3.5 holds.

Proof.

By Corollaries 1.2 and 3.5. ∎