Small sets in Mann pairs

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.

1 Introduction

This note is a natural extension of the work in [6]. In that reference, expansions $\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, P\rangle$ of an o-minimal structure ${{{\mathcal {M}}}}$ 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 ${{{\mathcal {M}}}}$ eliminates imaginaries. The tameness conditions were verified for dense pairs of real closed fields, for expansions of ${{\mathcal {M}}}$ by an independent set P, and for expansions of a real closed field ${{\mathcal {M}}}$ 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, ${{\mathcal {M}}}=\langle M, <, +, 0, \dots \rangle$ denotes an o-minimal expansion of an ordered group with a distinguished positive element 1. We denote by ${{\mathcal {L}}}$ its language, and by ${\text {dcl}}$ the usual definable closure operator in $\mathcal {M}$. An ‘${{\mathcal {L}}}$-definable’ set is a set definable in ${{\mathcal {M}}}$ with parameters. We write ‘${{\mathcal {L}}}_A$-definable’ to specify that those parameters come from $A\subseteq M$. It is well-known that ${{\mathcal {M}}}$ admits definable Skolem functions and eliminates imaginaries ( [4, Chapter 6]).

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

$$\begin{aligned} {{\mathcal {L}}}_{ind(D)}=\{R_{\phi (x)}(x): \phi (x)\in {{\mathcal {L}}}_D\}, \end{aligned}$$

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

$$\begin{aligned} P_{ind(D)}\models R_\phi (a) \,\,\Leftrightarrow \,\, \mathcal {M}\models \phi (a). \end{aligned}$$

If $Q\subseteq P^n$, by a trace on Q we mean a set of the form $Y\cap Q$, where Y is ${{\mathcal {L}}}$-definable. We call $Y\cap P^n$ a full trace. We call a set $Q\subseteq P^n$ fiber-dense if for every x in the projection $\pi (Q)$ of Q onto the first $n-1$ coordinates, the fiber $Q_x$ is dense in P.

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

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

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

(dcl)$_D$ Let $B, C\subseteq P$ and

$$\begin{aligned}A={\text {dcl}}(BD)\cap {\text {dcl}}(CD)\cap P.\end{aligned}$$

Then

$$\begin{aligned}{\text {dcl}}(AD)={\text {dcl}}(BD)\cap {\text {dcl}}(CD).\end{aligned}$$

(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 $P_{ind(D)}$ and fiber-dense. That is, there are ${{\mathcal {L}}}$-definable sets $Y_1, \dots , Y_l\subseteq M^n$, and sets $Q_1, \dots , Q_l\subseteq P^n$ that are $\emptyset$-definable in $P_{ind(D)}$ and fiber-dense, such that

$$\begin{aligned}X=\bigcup _i (Y_i \cap Q_i).\end{aligned}$$

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 Sect. 3. For now, let us state the main theorem.

Theorem 1.1

Assume (OP), (dcl)$_D$ and (ind)$_D$, and that D is ${\text {dcl}}$-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 ${{\mathcal {M}}}=\langle M, <, +, \cdot , 0, 1\rangle$ 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

$$\begin{aligned}a_1 x_1 + \dots +a_r x_r=1\end{aligned}$$

is called non-degenerate if for every non-empty $I\subseteq \{1, \dots , r\}$, $\sum _{i\in I} a_i q_i\ne 0$. 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 $G^r$.¹ Let us call such a pair $\langle {{{\mathcal {M}}}}, G\rangle$ 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,

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

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 $\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle$ is a Mann pair, such that for every prime p, $G^{[p]}$ has finite index in G. Let $D\subseteq M$ be ${\text {dcl}}$-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 $\widetilde{{\mathcal {M}}}$) fail ( [2]). Note also that the assumption of D being ${\text {dcl}}$-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 ${{\mathcal {L}}}_A$-definable function $h: M^m \rightarrow M^n$ such that $X \subseteq h(P^m)$. The recent work in [7] provides an analysis for all definable sets in terms of ‘${{\mathcal {L}}}$-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 ${\text {dcl}}$-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 $\tau :X\rightarrow P^l$. If A itself is ${\text {dcl}}$-independent over P, then the extra parameters from P can be omitted.

Note that the assumption of Corollary 1.3 holds for $\widetilde{{\mathcal {M}}}$ 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 Sect. 2, we fix notation and recall some basic facts. In Sect. 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 ${{\mathcal {M}}}=\langle M, <, +, 0, \dots \rangle$ is our fixed o-minimal expansion of an ordered group with a distinguished positive element 1 and ${\text {dcl}}$ 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 the fiber

$$\begin{aligned} \{ b \in M^m \ : \ (b,x) \in T\}. \end{aligned}$$

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

$$\begin{aligned} {{\,\mathrm{int}\,}}_Y(X)= \{x\in X: \text {there is open} B\subseteq M^n \text {containing} x \text {with} B\cap Y\subseteq X \}.\end{aligned}$$

We will need the following fact.

Fact 2.1

( [4, Ch. 4 (1.3)]) Let $X\subseteq Y\subseteq M^n$ be two ${{\mathcal {L}}}$-definable sets. Then

$$\begin{aligned}\dim (X{\setminus }{{\,\mathrm{int}\,}}_Y(X)) < \dim Y.\end{aligned}$$

2.1 Elimination of imaginaries

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

$$\begin{aligned}E(x,y) \,\,\Leftrightarrow \,\,f(x)=f(y).\end{aligned}$$

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

Fact 2.2

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

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

Then ${{\mathcal {N}}}$ eliminates imaginaries.

2.2 The induced structure

Recall from the introduction that

$$\begin{aligned}P_{ind(D)}=\langle P, \{R\cap P^l: R\subseteq M^l {{\mathcal {L}}}_{D}-\text {definable}, l\in {{\mathbb {N}}}\}\rangle . \end{aligned}$$

Remark 2.3

For $A\subseteq P$, we have:

• (1) if $Q\subseteq P^n$ is A-definable in $P_{ind(D)}$, and $Y\subseteq M^n$ is ${{\mathcal {L}}}_{AD}$-definable, then $Q\cap Y$ is A-definable in $P_{ind(D)}$. Indeed, $Q\cap Y=Q\cap (Y\cap P^n)$.

• (2) 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 ${{\mathcal {N}}}=P_{ind(D)}$ (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 ${{\mathcal {M}}}$. 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 $A\subseteq P$, ${\text {dcl}}_{\mathcal {L}(P)}(A)={\text {dcl}}(A)$.

Proof

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

Lemma 3.2

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

Proof

We work by induction on $k=\dim X$. If $X=\emptyset$, the statement is obvious. Assume $k\ge 0$. By (OP), ${\overline{X}}$ is $\mathcal {L}_C$-definable. By o-minimality, $\dim {{\,\mathrm{fr}\,}}(X)<k$. Since ${{\,\mathrm{fr}\,}}(X)={\overline{X}}{\setminus }X$ is both ${{\mathcal {L}}}$-definable and C-definable, by inductive hypothesis, it is ${{\mathcal {L}}}_C$-definable. So $X={\overline{X}} {\setminus }{{\,\mathrm{fr}\,}}(X)$ is ${{\mathcal {L}}}_C$-definable. $\square$

Lemma 3.3

Let $C\subseteq M$ and

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

where $Z_1, \dots , Z_m\subseteq M^n$ are ${{\mathcal {L}}}_{C}$-definable sets, and $R_1, \dots , R_m\subseteq P ^n$ are $\emptyset$-definable in $P_{ind(D)}$ and fiber-dense. Then

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

for some ${{\mathcal {L}}}_{C}$-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 $P_{ind(D)}$ and fiber-dense.

Proof

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

$$\begin{aligned}Q_\sigma = \bigcup _{i\in \sigma } R_i\end{aligned}$$

and

$$\begin{aligned}Y_\sigma = \left( \bigcap _{i\in \sigma } Z_i\right) {\setminus }\left( \bigcup _{j\not \in \sigma } Z_j\right) .\end{aligned}$$

Clearly, $Q_\sigma$ is fiber-dense. It is also easy to check that for any two distinct $\sigma , \tau \subseteq \{1, \dots , m\}$, we have $Y_\sigma \cap Y_\tau =\emptyset$, and that

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

as required. $\square$

Now, the key technical lemma.

Lemma 3.4

Assume (OP) and (ind)$_D$, and that D is ${\text {dcl}}$-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 $\mathcal {L}_{BD}$-definable and ${{\mathcal {L}}}_{CD}$-definable, and sets $S_1, \dots , S_k\subseteq P^n$, that are $\emptyset$-definable in $P_{ind(D)}$ and fiber-dense, such that

$$\begin{aligned}X=\bigcup _{i=1}^l W_i\cap S_i.\end{aligned}$$

Proof

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

We perform induction on the dimension of ${\overline{X}}$. For $X=\emptyset$, the statement is obvious. Suppose now that $\dim {\overline{X}}=k\ge 0$. By (ind)$_D$ and Lemma 3.3, there are $\mathcal {L}$-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 $P_{ind(D)}$ and fiber-dense, such that

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

For every i, define

$$\begin{aligned}T_i=\{ x\in {\overline{X}} : \text { there is relatively open} V\subseteq {\overline{X}} \text {around} x, \text {with} V\cap R_i\subseteq X\}.\end{aligned}$$

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 ${{\mathcal {L}}}$-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 ${{\mathcal {L}}}_{BD}$-definable and ${{\mathcal {L}}}_{CD}$-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,

$$\begin{aligned} \dim \big ((Z_i\cap X) {\setminus }(T_i \cap R_i)\big ) < k. \end{aligned}$$

We may write

$$\begin{aligned}(Z_i\cap X) {\setminus }(T_i \cap R_i)= \big ((Z_i\cap X) {\setminus }{{\,\mathrm{int}\,}}_{{\overline{X}}} (Z_i\cap X)\big ) \cup \big ({{\,\mathrm{int}\,}}_{\overline{X}} (Z_i\cap X) {\setminus }(T_i \cap R_i)\big ),\end{aligned}$$

By Fact 2.1, it suffices to show that ${{\,\mathrm{int}\,}}_{{\overline{X}}} (Z_i\cap X) \subseteq (T_i \cap R_i)$. Clearly, ${{\,\mathrm{int}\,}}_{{\overline{X}}} (Z_i\cap X) \subseteq {{\,\mathrm{int}\,}}_{{\overline{X}}} (Z_i)\cap X$, and hence it suffices to show:

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

Let $x\in {{\,\mathrm{int}\,}}_{{\overline{X}}} (Z_i)\cap X$. Since $x\in {{\,\mathrm{int}\,}}_{{\overline{X}}} (Z_i)$, 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. $\square$

By Remark 2.3(1), the set $X{\setminus }\bigcup _i (T_i \cap R_i)$ 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

$$\begin{aligned} X= \left( X{\setminus }\bigcup _i (T_i\cap R_i)\right) \cup \bigcup _i (T_i\cap R_i), \end{aligned}$$

and we are done. $\square$

Corollary 3.5

Assume (OP) and (ind)$_D$, and that D is ${\text {dcl}}$-independent over P. Let $A\subseteq P$ and $X\subseteq P^n$ be A-definable in $P_{ind(D)}$. Then there are ${{\mathcal {L}}}_{AD}$-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 $P_{ind(D)}$ and fiber-dense, such that

$$\begin{aligned}X=\bigcup _i (W_i \cap S_i).\end{aligned}$$

Proof

By Lemma 3.4 for $B=C=A$. $\square$

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 ${\text {dcl}}$-independent over P. Let $f: P^n\rightarrow P$ be an A-definable map in $P_{ind(D)}$. Then there are ${{\mathcal {L}}}_{AD}$-definable maps $F_1, \dots , F_l: M^n\rightarrow M^k$, such that for every $x\in P^n$, there is $i=1, \dots , l$, with $f(x)=F_i(x)$.

Proof

By Corollary 3.5, there are finitely many $\mathcal {L}_{AD}$-definable sets $W_1, \dots , W_l\subseteq M^{n+1}$ and sets $S_1, \dots , S_l\subseteq P^{n+1}$ which are $\emptyset$-definable in $P_{ind(D)}$ and fiber-dense, 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 suffices to prove that the graph of $f_i$ is contained in finitely many graphs of ${{\mathcal {L}}}_{AD}$-definable maps. To simplify the notation, we set $f=f_i$, $S=S_i$ and $W=W_i$. So $\Gamma (f)=W\cap S$, where $f:X\subseteq P^n\rightarrow P$ is A-definable, $W\subseteq M^{n+1}$ is ${{\mathcal {L}}}_{AD}$-definable, and $S\subseteq P^{n+1}$ is $\emptyset$-definable in $P_{ind(D)}$ and fiber-dense. By o-minimality, for every x in the projection $\pi (W)$ of W onto the first n coordinates, each fiber $W_x$ is a finite union of open intervals and points. Since for every $x\in X$, $(W\cap S)_x=\{f(x)\}$, and the fiber $S_x$ is dense in P, it follows that $W_x$ is finite, with only one of its elements belonging to P. Let

$$\begin{aligned}T=\{x\in \pi (W): W_x \text { is finite}\}.\end{aligned}$$

So, $X\subseteq T$. Moreover, T is ${{\mathcal {L}}}_{AD}$-definable. Now, by o-minimality $W\cap (T\times M)$ is a finite union of $\mathcal {L}_{AD}$-definable maps that contains $\Gamma (f)$, as needed. $\square$

Corollary 3.7

Assume (OP) and (ind)$_D$, and that D is ${\text {dcl}}$-independent over P. Then for every $A\subseteq P$, ${\text {cl}}_D(A)={\text {dcl}}(AD)\cap P$. 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 ${\text {dcl}}(-D)$ defines a pregeometry in ${{\mathcal {M}}}$, it follows easily that so does $cl_D(-)$ in $P_{ind(D)}$. $\square$

Lemma 3.8

Assume (OP), (dcl)$_D$ and (ind)$_D$, and that D is ${\text {dcl}}$-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 ${{\mathcal {L}}}_{BD}$-definable and ${{\mathcal {L}}}_{CD}$-definable, and $S_1, \dots , S_k\subseteq P^n$, each $\emptyset$-definable in $P_{ind(D)}$, such that

$$\begin{aligned}X=\bigcup _{i=1}^l W_i\cap S_i.\end{aligned}$$

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

$$\begin{aligned}{\text {dcl}}(BD)\cap {\text {dcl}}(CD)\cap P=cl_D(B) \cap cl_D(C)=A,\end{aligned}$$

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

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 ${\text {dcl}}$-independent over P. Let ${{\mathcal {M}}}'$ be the expansion of $\mathcal {M}$ with constants for all elements in P, and $\widetilde{{\mathcal {M}}}'=\langle {{\mathcal {M}}}', P\rangle$. Then (ind)$_D$ holds for $\widetilde{{\mathcal {M}}}'$ and D.

Proof

Denote by $P'_{ind(D)}$ the D-induced structure on P by ${{\mathcal {M}}}'$. Let $X\subseteq P^n$ be A-definable in $P'_{ind(D)}$. It follows that X is AP-definable in $P_{ind(D)}$. By Corollary 3.5, there are ${{\mathcal {L}}}_{APD}$-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

$$\begin{aligned}X=\bigcup _i (Y_i \cap Q_i).\end{aligned}$$

Such $Y_i$’s are ${{\mathcal {L}}}_{AD}$-definable in ${{\mathcal {M}}}$, and the $Q_i$’s are of course $\emptyset$-definable in $P'_{ind(D)}$, as required. $\square$

Proof of Corollary 1.3

The proof when A is ${\text {dcl}}$-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 $\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle$ 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 $\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle$ be a Mann pair, such that for every prime p, $G^{[p]}$ has finite index in G. Let $X\subseteq G^n$ a definable set. Then X is a finite union of traces on sets which are $\emptyset$-definable in $G_{ind(\emptyset )}$ and fiber-dense. 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 $\widetilde{{\mathcal {M}}}$). By Fact 3.1, $g\in {\text {dcl}}(\emptyset )$. By the above observation, $g(G^{[d]})^n$ is $\emptyset$-definable in $G_{ind(\emptyset )}$. It is also fiber-dense. $\square$

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 $\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle$ be a Mann pair, such that for every prime p, $G^{[p]}$ has finite index in G. Let $D\subseteq M$ be ${\text {dcl}}$-independent over P. 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 . $\square$