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(I) Recover an [MATH] -definable map [MATH] , with [MATH] , that agrees with [MATH] on a definable set [MATH] with [MATH] (equivalently, [MATH] ). This is possible because [MATH] is strongly large. Indeed, in Lemma 2.5 we prove a generalization of (D6) for maps [MATH] with strongly large domain, which we can then apply... |
(II) Prove that [MATH] satisfies ‘group-like’ properties on an [MATH] -definable subset [MATH] of [MATH] with [MATH] , such as injectivity in each coordinate, and associativity. This is done using (I) and the fact that [MATH] is a group operation. Moreover, [MATH] |
(III) Show that [MATH] is an [MATH] -definable set with [MATH] . This is done using earlier work from Section for extracting [MATH] -definable sets. |
(IV) Construct a suitable definable embedding [MATH] . This is the heart of the whole proof. We first prove that [EQUATION] and then define [MATH] via (*). |
(V) Show that there is an [MATH] -definable set [MATH] with [MATH] and [MATH] , such that for every [MATH] [EQUATION] The proof combines all information for [MATH] [MATH] and [MATH] from Steps (II)-(IV). |
(VI) Apply a group chunk theorem to the quadruple [MATH] to conclude that [MATH] is definably isomorphic to an [MATH] -definable group. This group chunk theorem is proved in Section in a higher generality, where [MATH] and [MATH] are arbitrary structures satisfying only some of the dimension axioms (D1)-(D6) |
Example 1.3 Suppose [MATH] is an expansion of an ordered group M by a dense elementary substructure [MATH] . Let [MATH] and denote [MATH] . Consider the definable bijection [MATH] given by |
[EQUATION] and let [MATH] be the induced group structure on [MATH] . Namely, if we write [MATH] for the map [MATH] , then for every [MATH] |
[EQUATION] Then [MATH] is strongly large (even with [MATH] -definable domain), and it is definably isomorphic to the [MATH] -definable group [MATH] via [MATH] . We would like to recover [MATH] in an abstract way. This is done in Step (IV) below. However, we illustrate all steps from the general procedure. Let [MATH] |
(I) For every [MATH] , we have [EQUATION] So if we let for every [MATH] [MATH] , then [MATH] agrees with [MATH] exactly on the set |
[EQUATION] Since each [MATH] is co-small in [MATH] , it follows from (D3) that [MATH] (II) Let [MATH] (III) We have [MATH] (IV) We prove that here (*) actually yields exactly [MATH] . Namely, |
[EQUATION] To see this, one could perform a direct computation, or argue as follows (also in preparation for the sort of arguments that take place in general). Consider the following equalities: |
[EQUATION] The first equality holds for all those [MATH] such that [MATH] , and hence, by injectivity of [MATH] in the second coordinate, for co-small many [MATH] . The second equality holds for every [MATH] . The third equality holds for all [MATH] , again, co-small many. All together, [MATH] holds for co-small many [... |
(V) Let [MATH] (VI) The group chunk theorem here is not needed, as we actually have [MATH] , and [MATH] is the desirable definable isomorphism. |
Remark 1.4 The idea of recovering a group operation via (*) is a recast of a similar idea in . In Section 1.3 of that reference, the authors recover a definable isomorphism [MATH] between [MATH] and an ordered group [MATH] , satisfying additional properties, as follows: |
[EQUATION] In Example 1.3 , instead of letting [MATH] to be such a limit as [MATH] , we require the equation [MATH] to hold for co-small many [MATH] |
Remark 1.5 The first attempt to recover an [MATH] -definable domain in Step (III) of the general procedure would probably be to take [MATH] . Besides, this set is always contained in [MATH] , and hence the rest of the analysis (IV)-(VI) could be simplified. But [MATH] need not be [MATH] -definable; indeed, in Example 1... |
Acknowledgements. I wish to thank Ya’acov Peterzil for pointing out the relevant literature and discussing the proof of the group chunk theorem in Section . The relevant discussions took place during the trimester in model theory, combinatorics and valued fields, 2018, at the Institut Henri Poincaré. I also thank Alfre... |
2. Preliminaries In this section, we fix some notation, prove some basic facts, analyze strongly large sets, and show how the pairs [MATH] from |
fit to the current setting. 2.1. Notation The topological closure of a set [MATH] is denoted by [MATH] If [MATH] , we call [MATH] dense in [MATH] if [MATH] . We call [MATH] co-dense in [MATH] if [MATH] is dense in [MATH] . Given a set [MATH] and [MATH] , we write [MATH] for |
[EQUATION] We write [MATH] for the projection onto the first [MATH] coordinates, unless stated otherwise. If [MATH] , we sometimes write [MATH] for [MATH] A tuple of elements is denoted just by one element, and we write [MATH] if [MATH] is a tuple with coordinates from [MATH] . Our use of the notions of a [MATH] -cell,... |
2.2. Basic facts Fact 2.1 Let [MATH] be a [MATH] -cell and [MATH] a definable set with [MATH] . Then [MATH] is dense in [MATH] Proof. |
Since [MATH] is a [MATH] -cell, a relatively open subset [MATH] of [MATH] has dimension [MATH] . Since [MATH] , we obtain [MATH] . In particular, [MATH] |
Fact 2.2 Let [MATH] be an [MATH] -definable set of dimension [MATH] , and [MATH] [MATH] -cell. If a definable set is dense in [MATH] , then so is it in [MATH] |
Proof. 16 , Lemma 2.6] The following lemma will be used in the proof of Lemma 5.4 . It generalizes 17 , Proposition 4.19] Lemma 2.3 |
Let [MATH] be an [MATH] -definable map, and [MATH] a definable set with [MATH] . Assume that [MATH] is injective. Then there is an [MATH] -definable set [MATH] such that [MATH] and [MATH] is injective. |
Proof. Assume [MATH] . Denote [EQUATION] We claim that [MATH] . Assume not, and let [MATH] be a [MATH] -cell. Then [MATH] . Now, by 17 , Fact 2.9] [MATH] has dimension [MATH] . In particular, [MATH] is in L-definable bijection with a subset of [MATH] . Hence [MATH] is in definable bijection with a subset of [MATH] , co... |
Now, by uniform finiteness in o-minimal structures, one can easily find an [MATH] -definable set [MATH] of dimension [MATH] , such that [MATH] is injective, and |
an [MATH] -definable map [MATH] , such that for every [MATH] [EQUATION] Observe that then [MATH] is injective, since if [MATH] and [MATH] , then [MATH] which implies [MATH] . Moreover, since [MATH] is injective, |
[EQUATION] But [MATH] , and hence by injectivity of [MATH] , the set on the right also has dimension [MATH] , contradicting [MATH] |
Question 2.4 In Lemma 2.3 , can it moreover be [MATH] 2.3. Strongly large sets Here we prove some statements about strongly large sets. We also introduce the notion of a full set. The first lemma extends property (D6) to functions [MATH] whose domain is any strongly large set, instead of just [MATH] |
Lemma 2.5 Let [MATH] be strongly large of dimension [MATH] . Then every definable map [MATH] agrees with an [MATH] -definable map [MATH] outside a definable set [MATH] of dimension [MATH] |
Proof. By working with the coordinate functions of [MATH] , we may assume that [MATH] . Indeed, if we find a suitable set [MATH] for the [MATH] -th coordinate [MATH] , then [MATH] works for [MATH] , by (D2) |
We may assume [MATH] . Indeed, [MATH] is a finite union of cells. If [MATH] is one of the cells and [MATH] , we can disregard it. Otherwise, [MATH] , and hence it is enough to work with one of these. After projecting [MATH] onto suitable coordinates, we may assume that [MATH] |
Define [MATH] as [MATH] , if [MATH] , and [MATH] , otherwise. This map [MATH] is definable, and hence, by (D6) , it agrees with an [MATH] -definable map [MATH] outside a set [MATH] of dimension [MATH] . Then [MATH] agrees with [MATH] outside [MATH] |
The above lemma supports the intuition that strongly large sets behave like [MATH] -definable sets. We strengthen the notion of being strongly large as follows. |
Definition 2.6 A definable set [MATH] is called full if [MATH] By (D5) , every [MATH] -definable set is full. By (D2) , a full set is strongly large. The converse is not true; for example, let [MATH] be the disjoint union of an open interval and an infinite small set. Some natural examples of full sets come from the se... |
, see Fact 2.12 below. In Section , we will use the following consequence of Lemma 2.5 Corollary 2.7 Every strongly large set is a union of a full set and a set of smaller dimension. |
Proof. Let [MATH] be a strongly large set of dimension [MATH] . As in the proof of Lemma 2.5 , we may assume that [MATH] . Let [MATH] be a characteristic function for [MATH] ; namely, fix two elements [MATH] and let [MATH] , if [MATH] , and [MATH] , otherwise. By (D6) [MATH] agrees with an [MATH] -definable map [MATH] ... |
[EQUATION] and the latter set has dimension [MATH] , we obtain that [MATH] is full. Since also [EQUATION] we are done. The above conclusion may fail if we do not assume that the given set is strongly large. For example, consider any infinite small set. Also, in Corollary 6.8 below, we prove a partial converse of the ab... |
In Section , we will also need the following. Lemma 2.8 A finite union of full sets is full. Proof. Let [MATH] , where each [MATH] is a full set. By o-minimality, the union of the closures of finitely many [MATH] -definable sets equals the closure of their unions. It follows easily that [MATH] Therefore, |
[EQUATION] and hence [MATH] 2.4. The setting of In we studied pairs [MATH] , where [MATH] , such that three tameness conditions hold. Following |
, let us call a definable set [MATH] large if there is an [MATH] -definable map [MATH] such that [MATH] contains an open interval. Otherwise, it is called small . The three conditions in |
(see Section 2 there for more details) are: (I) [MATH] is small, (II) [MATH] is near-model complete, and (III) every open definable set is L-definable. In 17 , Section 2.2] , the following examples were shown to fall into this category: (a) dense pairs, (b) expansions of the real field by a multiplicative subgroup with... |
For the rest of this section, let [MATH] satisfy conditions (I)-(III) above. In , a suitable notion of dimension was introduced, which we describe next. |
Definition 2.9 supercone [MATH] [MATH] , and its shell [MATH] are defined recursively as follows: [MATH] is a supercone, and [MATH] |
A definable set [MATH] is a supercone if [MATH] is a supercone and there are [MATH] -definable continuous maps [MATH] with [MATH] , such that for every [MATH] [MATH] is contained in [MATH] and it is co-small in it. We let [MATH] |
Note that, [MATH] is the unique open cell in [MATH] such that [MATH] Definition 2.10 (Large dimension Let [MATH] be definable. If [MATH] , the large dimension of [MATH] is the maximum [MATH] such that [MATH] contains a set of the form [MATH] , where [MATH] is a supercone and [MATH] is an [MATH] -definable continuous in... |
The large dimension was used in to prove a cone decomposition theorem for all definable sets, in analogy with the cell decomposition theorem known for o-minimal structures. A consequence of this theorem was that the large dimension satisfies all properties (D1)-(D6) of the current paper. More precisely, these propertie... |
In what follows, the dimension [MATH] denotes the large dimension. Fact 2.11 Let [MATH] be a definable set of dimension [MATH] , and [MATH] a finite set. Then there is [MATH] which is [MATH] -independent over [MATH] |
Proof. By 17 , Theorem 5.7(1)] [MATH] contains a supercone, and hence we may assume that [MATH] is a supercone. Consider the operator [MATH] that maps [MATH] to [MATH] . By 17 , Section 6] [MATH] defines a pregeometry and the corresponding [MATH] -dimension for definable sets agrees with [MATH] . It is then easy to see... |
In the next fact, we draw a connection to the full sets from the last subsection. Fact 2.12 let [MATH] be as in Definition 2.10 . Namely, [MATH] is a supercone and [MATH] is an [MATH] -definable continuous injective map. Then [MATH] is a full set. |
Proof. We first note that [MATH] is a full set, by 17 , Corollary 4.28] . Now, let [MATH] . Observe that [EQUATION] Since [MATH] is [MATH] -definable, the first part of the last union has dimension [MATH] . Since also [MATH] is continuous and injective, the set |
[EQUATION] also has dimension [MATH] , as needed. 3. Local [MATH] -definability This section contains a key result (Corollary 3.8 ) which will be used in the proof of Lemma 5.7 below in order to extract an [MATH] -definable set from some given data. At first, one recovers only a ‘locally L-definable’ set, which we prov... |
3.1. Preliminaries on local [MATH] -definability Definition 3.1 Let [MATH] be a definable set, and [MATH] . We call [MATH] locally [MATH] -definable at [MATH] if there is an open box [MATH] containing [MATH] such that [MATH] is [MATH] -definable. We call [MATH] |
locally L-definable if it is locally L-definable at every point. The following fact follows easily from the definition. Fact 3.2 |
Suppose [MATH] is a [MATH] -cell and [MATH] a definable set. Then [MATH] is locally [MATH] -definable if and only if for every [MATH] , there is a [MATH] -cell [MATH] containing [MATH] |
Of course, an [MATH] -definable set is locally [MATH] -definable. We will also prove the converse. Lemma 3.3 Suppose [MATH] is locally [MATH] -definable and [MATH] is [MATH] -definable. Then [MATH] is locally [MATH] -definable. |
Proof. If [MATH] is an open box and [MATH] is [MATH] -definable, then so is [MATH] Lemma 3.4 A locally [MATH] -definable set is [MATH] -definable. |
Proof. Let [MATH] be locally [MATH] -definable and suppose that the closure [MATH] has dimension [MATH] . We work by induction on [MATH] . If [MATH] , then [MATH] is finite and hence [MATH] -definable. Suppose [MATH] . We first prove that |
[EQUATION] If not, there is a [MATH] -cell [MATH] in which [MATH] is dense. Since [MATH] is dense in [MATH] , by Fact 2.2 it is also dense in [MATH] . In particular, there is [MATH] . Since [MATH] is locally [MATH] -definable (Lemma 3.3 ), there is a [MATH] -cell [MATH] containing [MATH] such that [MATH] is [MATH] -def... |
By Lemma 3.3 , the set [MATH] is locally [MATH] -definable. Since its closure is contained in [MATH] , by inductive hypothesis we obtain that it is [MATH] -definable. Since |
[EQUATION] is also [MATH] -definable, we conclude that [MATH] is [MATH] -definable. Although it will not be used in this paper, we note that local [MATH] -definability at a point is a definable notion. |
Lemma 3.5 Let [MATH] be a definable set. Then the set [MATH] of points in [MATH] at which [MATH] is locally [MATH] -definable is [MATH] -definable. |
Proof. It is easy to see that for any point [MATH] , we have that [MATH] is locally [MATH] -definable at [MATH] if and only if there is a closed box [MATH] containing [MATH] such that [MATH] . Therefore, [MATH] is definable. By its definition, it is thus locally [MATH] -definable. By Lemma 3.4 , it is [MATH] -definable... |
3.2. Extracting local [MATH] -definability A simple and illustrative example of what follows is this. Let [MATH] be the set of non-algebraic real numbers, and [MATH] the usual addition. Then [MATH] is [MATH] -definable. The statements that follow generalize this observation. The extra complication in proving Corollary ... |
Lemma 3.6 Let [MATH] be two [MATH] -cells, and [MATH] definable sets with [MATH] , for [MATH] . Suppose [MATH] is an [MATH] -definable continuous map, injective in each coordinate, and let [MATH] . Then [MATH] contains an open subset of [MATH] that contains [MATH] |
Proof. The proof is inspired by an example in , page 5] . Let [MATH] , where [MATH] . We first claim that there is a definable set [MATH] of dimension [MATH] , such that [MATH] |
contains an open set [MATH] that contains [MATH] . Since [MATH] is [MATH] -definable, continuous and injective, by [MATH] contains an open set that contains [MATH] . By continuity of [MATH] , there is a [MATH] -cell [MATH] containing [MATH] , such that [MATH] contains an open set that contains [MATH] . We can thus set ... |
Now let [MATH] and [MATH] be as above. We prove that actually [MATH] contains [MATH] . Assume towards a contradiction that there is [MATH] such that [MATH] . By the claim in the first paragraph, |
[EQUATION] By injectivity of [MATH] in the first coordinate, we obtain [EQUATION] which is a contradiction, because [MATH] , and hence its image under the definable map [MATH] cannot contain [MATH] (see, for example, 17 , Corollary 5.3] ). ∎ |
We next derive a version of the last lemma where the range of [MATH] is a [MATH] -cell in any [MATH] Lemma 3.7 Let [MATH] be three [MATH] -cells, and [MATH] definable sets with [MATH] , for [MATH] . Suppose [MATH] is an [MATH] -definable continuous map, injective in each coordinate, and let [MATH] . Then [MATH] contain... |
Proof. Suppose [MATH] , for [MATH] Let [MATH] be a coordinate projection which is injective on [MATH] . Then [MATH] is an [MATH] -definable continuous map that satisfies the conditions of Lemma 3.6 . So [MATH] contains an open box [MATH] of [MATH] that contains [MATH] . Then [MATH] is a [MATH] -cell that contains [MATH... |
It is not hard to see that the above lemma remains true if [MATH] is any [MATH] -definable set of dimension [MATH] , but we will not need this fact here. However, if [MATH] is of higher dimension, then the lemma fails: let [MATH] be the identity map and [MATH] contain no open [MATH] -definable set. |
We can now prove the exact statement that will be used in the proof of Lemma 5.7 Corollary 3.8 Let [MATH] be a [MATH] -cell, [MATH] a finite union of [MATH] -cells, and [MATH] a definable set with [MATH] . Suppose that [MATH] is an [MATH] -definable continuous map, which is injective in each coordinate. Then the set [M... |
Proof. By Lemma 3.4 , it suffices to show that [MATH] is locally [MATH] -definable. So let [MATH] , where [MATH] . Since [MATH] is a finite union of [MATH] -cells and [MATH] , it is easy to find [MATH] -cells [MATH] such that [MATH] . Since [MATH] , we have [MATH] and hence by Lemma 3.7 , for [MATH] , there is a [MATH]... |
[EQUATION] By Fact 3.2 [MATH] is locally [MATH] -definable. 4. A Weil’s group chunk theorem The goal of this section is to recover an [MATH] -definable group from an [MATH] -definable ‘group chunk’. Theorems of this spirit have already been considered in classical model theory. The current account borrows ideas from We... |
. The proof of Theorem 4.4 is based on discussions with Y. Peterzil. In this section, we work in a more general setting than in the rest of this paper. Let [MATH] and [MATH] be any two first-order structures, with [MATH] expanding M. Assume that there is a map [MATH] from the class of all definable sets in N to [MATH] ... |
(D1), (D2), (D3b), (D4) , and if the family [MATH] in (D3) is L-definable, then so are the sets [MATH] in (D3a) We refer to the second property above as (Ldef) . Note that we do not assume that M is o-minimal, nor that it admits a dimension function. But even for an o-minimal [MATH] , the current setting is much richer... |
), such as [MATH] , and also weakly o-minimal non-valuational structures ( ), such as [MATH] Remark 4.1 An example where N satisfies the above properties, but not (D3a) , is that of a semi-bounded o-minimal structure [MATH] ; namely, when M is a linear o-minimal structure, [MATH] is an o-minimal expansion of a real clo... |
), the results of this section appear to be new also in that setting. Under these assumptions, we recover a group which is interpretable in [MATH] . This will be enough for our purposes in this paper, in view of Fact 4.3 below. To avoid any ambiguities, let us recall the following definition from |
Definition 4.2 Let [MATH] be any structure. By a definable quotient (in R) we mean a quotient [MATH] of a definable set [MATH] by a definable equivalence relation [MATH] A map [MATH] between two definable quotients is called definable if the set |
[EQUATION] is definable (in R). An interpretable group [MATH] is a group whose universe is a definable quotient, and whose group operation is a definable map. |
Fact 4.3 20 , Theorem 1] If [MATH] is o-minimal, then every interpretable group is definably isomorphic to a definable group. We extend our terminology from the introduction to definable quotients: a quotient, map and group as in Definition 4.2 , is called ‘definable’ or ‘interpretable’ if [MATH] , and ‘ [MATH] -defina... |
Theorem 4.4 Let [MATH] be a definable group with [MATH] and [MATH] . Suppose that [MATH] and [MATH] are two [MATH] -definable sets, with [MATH] |
[MATH] is a definable injective map, with [MATH] , and [MATH] is an [MATH] -definable map, such that for every [MATH] [EQUATION] |
Then [MATH] is definably isomorphic to an [MATH] -interpretable group. If, moreover, M is o-minimal, then [MATH] is definably isomorphic to an [MATH] -definable group. |
Proof. We may assume that [MATH] and [MATH] . Indeed, one can form the disjoint union of [MATH] and [MATH] , and induce on it a definable group structure after identifying [MATH] with [MATH] , and [MATH] with itself. We then have [MATH] [MATH] , and [MATH] is an [MATH] -definable map, such that for every [MATH] |
[EQUATION] To simplify the notation, for [MATH] , we may write [MATH] for [MATH] . Moreover, we may assume that for every [MATH] [MATH] Indeed, by (Ldef) , the set [MATH] is L-definable. Moreover, since [MATH] , it follows that [MATH] . We may thus replace [MATH] by [MATH] . Finally, note that [MATH] is injective in ea... |
Claim 1. The following sets are [MATH] -definable: [MATH] [MATH] Proof of Claim 1. For [MATH] , it suffices by (Ldef) to prove that for every [MATH] [MATH] if and only if the set |
[EQUATION] has co-dimension [MATH] in [MATH] . To see this, it suffices to show that for every [MATH] , the set [EQUATION] has co-dimension [MATH] in [MATH] . Clearly, every [MATH] such that [MATH] is contained in the last set. But by injectivity of [MATH] in the second coordinate, there are co-dimension [MATH] many su... |
For [MATH] , one can see similarly that for every [MATH] [MATH] if and only if the set [EQUATION] has co-dimension [MATH] in [MATH] |
Let [MATH] be the following equivalence relation on [MATH] [EQUATION] By definability of the set [MATH] from Claim 1, the relation [MATH] is [MATH] -definable. Let [MATH] and denote by [MATH] the equivalence class of [MATH] . So [MATH] is an [MATH] -definable quotient. We aim to equip [MATH] with an [MATH] -interpretab... |
Claim 2. (1) For every [MATH] , there are [MATH] , such that [MATH] [MATH] and [MATH] (2) For every [MATH] , there are [MATH] , such that [MATH] [MATH] [MATH] and [MATH] |
Proof of Claim 2. We only prove (1), as the proof for (2) is similar. Consider the sets [EQUATION] and [EQUATION] Since [MATH] , the projections [MATH] and [MATH] on the first and last [MATH] coordinates, respectively, have co-dimension [MATH] in [MATH] . In particular, |
[EQUATION] Now take [MATH] in this set and let [MATH] [MATH] and [MATH] . By construction, [MATH] and they satisfy the equalities of the conclusion. |
Now, for every [MATH] , define the relation [EQUATION] By Claim 1, [MATH] is an [MATH] -definable relation. By Claim 2(1), for every [MATH] , there are [MATH] such that [MATH] . Moreover, if [MATH] [MATH] [MATH] and [MATH] , then [MATH] . Indeed, let [MATH] witnessing the first two relations. Then |
[EQUATION] We can thus define the following [MATH] -definable operation on [MATH] [EQUATION] Let [MATH] for some/any [MATH] such that [MATH] . Namely, take [MATH] , which exists since [MATH] |
Claim 3. [MATH] is an [MATH] -definable group. Proof of Claim 3. We already saw that [MATH] and [MATH] are [MATH] -definable. We prove associativity of [MATH] . Let [MATH] . Take [MATH] as in Claim 2(2). Then |
[EQUATION] It is also easy to check that [MATH] is the identity element, using Claim 2(1). Claim 4. [MATH] is definably isomorphic to [MATH] |
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