Dimensions, matroids, and dense pairs of first-order structures Version 2.8

Antongiulio Fornasiero^{∗}

March 1, 2010

Abstract

A structure M is pregeometric if the algebraic closure is a pregeometry in all M′ elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique existential matroid.

Generalising previous results by L. van den Dries, we define dense elementary pairs of structures expanding a field and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We also extend the above result to dense tuples of structures.

Key words: Geometric structure; pregeometry; matroid; lovely pair; dense pair.

MSC2000: Primary 03Cxx; Secondary 03C64. Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Definable matroids . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Morley sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Local properties of dimension . . . . . . . . . . . . . . . . . . . . 20

4 Matroids from dimensions . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

∗University of Münster. <antongiulio.fornasiero@googlemail.com>

1. Introduction

6 Extension to imaginary elements . . . . . . . . . . . . . . . . . . . . 25 7 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 8 Dense pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.1 Small sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.2 Proof of Theorems 8.3 and 8.5 . . . . . . . . . . . . . . . . . . . 33 8.3 Additional facts . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.4 The small closure . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8.5 Elimination of imaginaries . . . . . . . . . . . . . . . . . . . . . 42

9 D-minimal topological structures . . . . . . . . . . . . . . . . . . . . 43 10 Cl-minimal structures . . . . . . . . . . . . . . . . . . . . . . . . . . 49 11 Connected groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 12 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 13 Dense tuples of structures . . . . . . . . . . . . . . . . . . . . . . . . 56

13.1 Dense tuples of topological structures . . . . . . . . . . . . . . . 57 14 The (pre)geometric case . . . . . . . . . . . . . . . . . . . . . . . . . 58

1 Introduction

A theory T is called pregeometric [HP94, Gagelman05] if, in every model 𝕂 of T, acl satisfies the Exchange Principle (and, therefore, acl is a pregeometry on 𝕂); if T is complete, it suffices to check that acl satisfies EP in one ω-saturated model of T. T is geometric if it is pregeometric and eliminates the quantifiers ∃^{∞}. We call a structure 𝕂 (pre)geometric if its theory is (pre)geometric (thus, 𝕂 is pregeometric iff there exists an ω-saturated elementary extension 𝕂′ of 𝕂 such that acl satisfies EP in 𝕂′).

Note that a pregeometric expansion of a field is geometric ([DMS08, 1.18], see also Lemma 3.46).

In the remainder of this introduction, all theories and all structures expand a field; in the body of the article we will sometimes state definitions and results without this assumption.

Geometric structures are ubiquitous in model theory: if 𝕂 is either ominimal, or strongly minimal, or a p-adic field, or a pseudo-finite field (or more generally a perfect PAC field, see [CDM92] and [HP94, 2.12]), then 𝕂 is geometric.

However, ultraproducts of geometric structures (even strongly minimal ones) are not geometric in general. We will show that there is a more general notion, structures with existential matroids, which instead is robust under taking ultraproducts. More in details, we consider structures 𝕂 with a matroid cl that satisfies some natural conditions (cl is an “existential matroid”).

A. Fornasiero  Dimensions, matroids, and dense pairs. V. 2.8

Under our hypothesis that 𝕂 is a field, then there is at most one existential matroid on 𝕂. An (almost) equivalent notion has already been studied by van den Dries [Dries89]: we will show that, if 𝕄 is a monster model, an existential matroid on 𝕄 induces a (unique) dimension function on the definable subset of 𝕄^{n}, satisfying the axioms in [Dries89], and conversely, any such dimension function, satisfying a slightly stronger version of the axioms, will be induced by a (unique) existential matroid. Moreover, a superstable group 𝕂 of U-rank a power of ω is naturally endowed by an existential matroid (van den Dries [Dries89, 2.25] noticed this already in the case when 𝕂 is a differential field of characteristic 0).

Given a geometric structure 𝕂, there is an abstract notion of dense subsets of 𝕂, which specialises to the usual topological notion in the case of o-minimal structures or of formally p-adic fields. More precisely, a subset X of 𝕂 is dense in 𝕂 if every infinite 𝕂-definable subset of 𝕂 intersect X [Macintyre75]. If T is a complete geometric theory, then the theory of dense elementary pairs of models of T is complete and consistent (the proof of this fact was already in [Dries98], but the result was stated there only for o-minimal structures).

We consider here the more general case when T is a complete theory and such that a monster model of T has an existential matroid. We show that there is a corresponding abstract notion of density in models of T. Given T as above, consider the theory of pairs ⟨𝕂, 𝕂′⟩, where 𝕂 ≺ 𝕂′ ⊧ T and 𝕂 is dense in 𝕂′; the theory of such pairs will not be complete in general, but we will show that it will become complete (and consistent) if we add the additional condition that 𝕂 is cl-closed in 𝕂′ (that is, cl(𝕂) ∩ 𝕂′ = 𝕂); we thus obtain the (complete) theory T^{d}. Moreover T^{d} also has an existential matroid. This allows us to iterate the above construction, and consider dense cl-closed pairs of models of T^{d}, which turn out to coincide with nested dense cl-closed triples of models of T; iterating many times, we can thus study nested dense cl-closed n-tuples of models of T.

Of particular interest are two cases of structures with an existential matroid: the cl-minimal case and the d-minimal one.

A structure 𝕂 (with an existential matroid) is cl-minimal if there is only one “generic” 1-type over every subset of 𝕂; the prototypes of such structures are given by strongly minimal structures and connected superstable groups of U-rank a power of ω. If T is the theory of 𝕂, we show that the condition that 𝕂 is dense in 𝕂′ is superfluous in the definition of T^{d}, and that T^{d} is also cl-minimal.

\underline{3. Matroids}

An first-order topological structure 𝕂 (expanding a topological field) is d-minimal if it is Hausdorff, it has an ω-saturated elementary extension 𝕂′ such that every definable (unary!) subset of 𝕂′ is the union of an open set and finitely many discrete sets, and it satisfies a version of Kuratowski-Ulam’s theorem for definable subset of 𝕂^{2} (the “d” stands for “discrete”). Examples of d-minimal structures are p-adic fields, o-minimal structures, and d-minimal structures in the sense of Miller. We show that a d-minimal structure has a (unique) existential matroid, and that the notion of density given by the matroid coincides with the topological one. Moreover, if T is the theory of a d-minimal structure, then T^{d} is the theory of dense elementary pairs of models of T (the condition that 𝕂 is a cl-closed subset of 𝕂′ is superfluous); hence, in the case when T is o-minimal, we recover van den Dries’ Theorem [Dries98]. However, if T is d-minimal, T^{d} will not be d-minimal. Moreover, while ultraproducts of o-minimal structures and of formally p-adic fields are d-minimal, ultraproducts of d-minimal structures are not d-minimal in general.

We show that if 𝕂 has an existential matroid, then 𝕂 is a perfect field: therefore, the theory exposed in this article does not apply to differential fields of finite characteristic, or to separably closed non-perfect fields.

2 Notations and conventions

Let T be a complete theory in some language L, with only infinite models. Let κ > |T | be a “big” cardinal. We work inside a κ-saturated and strongly κ-homogeneous model 𝕄 of T: we call 𝕄 a monster model of T.

A, B, and C, subsets of 𝕄 of cardinality less than κ; by ā, b̄, and c̄, finite tuples of elements of 𝕄; by a, b, and c, elements of 𝕄. As usual, we will write, for instance, ā ⊂ A to say that ā is a finite tuple of elements of A, and Ab̄ to denote the union of A with the set of elements in b̄.

Given a set X and m ≤ n ∈ N, denote by Π_{m}^{n} the projection from X^{n} onto the first m coordinates. Given Y ⊆ X^{n+m}, x̄ ∈ X^{n}, and z̄ ∈ X^{m}, denote the sections Yx̄ := {t̄∈ X^{m} : ⟨x̄, t̄⟩ ∈ Y } and Y z̄ := {t̄∈ X^{m} : ⟨t̄, z̄⟩ ∈ X}.

3 Matroids

sec:matroid

Let cl be a (finitary) closure operator on 𝕄: that is, cl : P(𝕄) → P(𝕄) satisfies, for every X ⊆ 𝕄:

extension X ⊆ cl(X);

monotonicity X ⊆ Y implies cl(X) ⊆ cl(Y);

idempotency cl(cl X) = cl(X);

finitariness cl(X) = ⋃{cl(A) : A ⊆ X & A finite}. cl is a (finitary) matroid (a.k.a. pregeometry) if moreover it satisfies the Exchange Principle: EP a ∈ cl(Xc) \ cl(X) implies c ∈ cl(Xa). Proviso. For the remainder of this section, cl is a finitary matroid on 𝕄.

As is well-known from matroid theory, cl defines notions of rank (which we denote by rk^{cl}), generators, independence, and basis.^{1} Definition 3.1. A set A generates C over B if cl(AB) = cl(CB). A subset A of 𝕄 is independent over B if, for every a ∈ A, a ∉ cl(Ba′ : a ≠ a′ ∈ A).

lem:Lascar Lemma 3.2 (Additivity of rank).
rk^{cl}(āb̄/C) = rk^{cl}(ā/b̄C) + rk^{cl}(b̄/C).

For the axioms of independence relations, we will use the nomenclature in [Adler05]. Definition 3.3. Given an infinite set X, a pre-independence relation^{2} on X is a the ternary relation | ⌣ on P(X) satisfying the following axioms: Monotonicity: If A | ⌣_{C} B, A′ ⊆ A, and B′ ⊆ B, then A′ | ⌣_{C} B′. Base Monotonicity: If D ⊆ C ⊆ B and A | ⌣_{D} B, then A | ⌣_{C} B. Transitivity: If D ⊆ C ⊆ B, B | ⌣_{C} A, and C | ⌣_{D} A, then B | ⌣_{D} A. Normality: If A | ⌣_{C} B, then AC | ⌣_{C} B. Finite Character: If A_{0} | ⌣_{C} B for every finite A_{0} ⊆ A, then A | ⌣_{C} B. | ⌣ is symmetric if moreover it satisfies the following axiom: Symmetry: A | ⌣_{C} B iff B | ⌣_{C} A.

1Sometimes in geometric model theory the “rank” is called “dimension” and/or the “dimension” (defined later) is called “rank”; however, since in many interesting cases (e.g.

algebraically closed fields, and o-minimal structures, with the acl matroid) what we call the dimension of a definable set induced by the matroid coincides with the usual notion of dimension given geometrically, our choice of nomenclature is clearly better.

2Pre-independence relations as defined here are slightly different than the ones defined in [Adler05]. However, as we will see later, if cl is definable, then | ⌣^{cl} is a pre-independence relation in Adler’s sense.

Definition 3.4. The pre-independence relation on 𝕄 induced by cl is the ternary relation | ⌣^{cl} on P(𝕄) defined by: X | ⌣^{cl}_{Y} Z if for every Z′ ⊂ Z, if Z′ is independent over Y, then Z′ remains independent over Y X. If X | ⌣^{cl}_{Y} Z, we say that X and Z are independent over Y (w.r.t. cl). Remark 3.5. If X | ⌣^{cl}_{Y} Z, then cl(XY) ∩ cl(ZY) = cl(Y). Lemma 3.6. | ⌣^{cl} is a symmetric pre-independence relation. Proof. The same given in [Adler05, Lemma 1.29]. Remark 3.7. | ⌣^{cl} also satisfies the following version of anti-reflexivity:

• A | ⌣^{cl}_{C} B iff cl(A) | ⌣^{cl}_{cl(C)} cl(B);

• a | ⌣^{cl}_{X} a iff a ∈ cl(X). Remark 3.8. X | ⌣^{cl}_{Y} Y. Lemma 3.9. T.f.a.e.:

1. X | ⌣^{cl}_{Y} Z;

2. ∀Z′ such that Y ⊆ Z′ ⊆ cl(Y Z), we have cl(XZ′) ∩ cl(Y Z) = cl(Z′);

3. there exists Z′ ⊆ Z which is a basis of Z/Y, such that Z′ remains independent over XY;

4. for every Z′ ⊆ Z which is a basis of Z/Y, Z′ remains independent over Y X;

5. if X′ ⊆ X is a basis of Y X/Y and Z′ ⊆ Z is a basis of Y Z/Y, then X′ and Z′ are disjoint, and X′Z′ is a basis of XZ over Y;

6. rk^{cl}(X/Y Z) = rk^{cl}(X/Y).

lem:ind-ternary Lemma 3.10. Let | ⌣ be a symmetric pre-independence relation on some infinite set X. Assume that ā | ⌣_{C} d̄and ād̄ | ⌣_{C} b̄. Then, ā | ⌣_{C} b̄d̄and d̄ | ⌣_{C} b̄ā. Proof. Cf. [Adler05, 1.9]. ā | ⌣_{C} b̄d̄ implies ā | ⌣_{Cb̄} d̄, which implies ā | ⌣_{Cb̄} b̄d̄, which, together with ā | ⌣_{C} d̄, implies ā | ⌣_{C} b̄d̄.

lem:ind-sequence Lemma 3.11. Let | ⌣ be a symmetric pre-independence relation on some infinite set X. Let ⟨I, ≤⟩ be a linearly ordered set, (ā_{i} : i ∈ I) be a sequence of tuples in X^{n}, and C ⊂ X. Then, t.f.a.e.:

1. For every i ∈ I, we have ā_{i} | ⌣_{C}(ā_{j} : j < i);

2. For every i ∈ I, we have ā_{i} | ⌣_{C}(ā_{j} : j ≠ i).

Proof. Assume, for contradiction, that (1) holds, but a_{i} |̸ ⌣_{C} (ā_{j} : j ≠ i), for some i ∈ I. Since | ⌣^{cl} satisfies finite character, w.l.o.g. I = {1, …, m} is finite. Let m′ such that i < m′ ≤ m is minimal with ā_{i} |̸ ⌣_{C} (ā_{j} : j ≤ m′ & j ≠ i); w.l.o.g., m = m′.

Let d̄:= (a_{j} : j ≠ i & j < m). By assumption, ā_{i} | ⌣_{C} d̄ and d̄ā_{i} | ⌣_{C} ā_{m}. Then, by Lemma 3.10, we have ā_{i} | ⌣_{C} d̄ā_{m}, absurd.

Definition 3.12. We say that a sequence (ā_{i} : i ∈ I) satisfying one of the above equivalent conditions is an independent sequence over C.

Remark 3.13. Let (a_{i} : i ∈ I) be a sequence of elements of 𝕄. There is a clash with the previous definition of independence; more precisely, let J := {i ∈ I : a_{i} ∉ cl(C)}; then, (a_{i} : i ∈ I) is an independent sequence over C according to | ⌣^{cl} iff all the a_{j} are pairwise distinct for j ∈ J, and the set {a_{j} : j ∈ J} is independent over C according to cl. Hopefully, this will not cause confusion.

3.1 Definable matroids

def:definable Definition 3.14. Let ϕ(x, ȳ) be an L-formula. We say that ϕ is x-narrow if, for every b̄ and every a, if 𝕄 ⊧ ϕ(a, b̄), then a ∈ cl(b̄). We say that cl is definable if, for every A,

cl(A) = ⋃{ϕ(𝕄, ā) : ϕ(x, ȳ) is x-narrow, ā ∈ A^{n}, n ∈ ℕ}.

Proviso. For the rest of the section, cl is a definable matroid.

Remark 3.15. For every A and every σ ∈ Aut(𝕄), σ(cl(A)) = cl(σ(A)).

lem:cl-definable Lemma 3.16. 1. | ⌣^{cl} satisfies the Invariance axiom: if A | ⌣^{cl}_{B} C and ⟨A′, B′, C′⟩ ≡ ⟨A, B, C⟩, then A′ | ⌣^{cl}_{B′} C′.

2. | ⌣^{cl} satisfies the Strong Finite Character axiom: if A |̸ ⌣^{cl}_{C} B, then there exist finite tuples ā ⊂ A, b̄ ⊂ B, and c̄ ⊂ C, and a formula ϕ(x̄, ȳ, z̄) without parameters, such that

• 𝕄 ⊧ ϕ(ā, b̄, c̄);

• ā′ |̸ ⌣^{cl}_{C} B for all ā′ satisfying 𝕄 ⊧ ϕ(ā′, b̄, c̄).

3. For every ā, B, and C, if tp(ā/BC) is finitely satisfied in B, then ā | ⌣^{cl}_{B} C.

3. Matroids  3.1. Definable matroids

4. | ⌣^{cl} satisfies the Local Character axiom: for every A, B there exists a subset C of B such that |C| ≤ |T | + |A| and A | ⌣^{cl}_{C} B.

Proof. (1) is obvious.

(2) Assume that A |̸ ⌣^{cl}_{C} B. Hence, there exists b̄ ∈ B^{n} independent over C, such that b̄ is not independent over AC. Hence, there exists ā ⊂ A and c̄ ⊂ C finite tuples, such that, w.l.o.g., b_{1} ∈ cl(c̄āb̃), where b̃ := ⟨b_{2}, …, b_{n}⟩. Let α(x, x̃, ȳ, z̄) be an x-narrow formula, such that 𝕄 ⊧ α(b_{1}, b̃, c̄, ā). If ā′ ⊂ 𝕄 satisfies α(b̄, c̄, ā′), then ā′ |̸ ⌣^{cl}_{C} B.

(3) and (4) follow as in [Adler05, 2.3–4]. Here is a direct proof of the Local Character axiom: let A and B be given. Let B′ ⊆ B be a basis of AB over A, A′ ⊆ A be a basis of A, and C ⊆ B be a basis of B over B′. Notice that CB′ is a basis of AB and A′B′ is a set of generators of AB; hence, by the Exchange Principle, |C| ≤ |A′| = rk^{cl}(A) ≤ |A|. Moreover, A | ⌣^{cl}_{C} B.

Definition 3.17. Let | ⌣ be a pre-independence relation on 𝕄. We say that | ⌣ is an independence relation on 𝕄 if it moreover satisfies Invariance, Local Character, and

Extension: If A | ⌣_{C} B and D ⊇ B, then there exists A′ ≡_{BC} A such that A′ | ⌣_{C} D.

We also define the following axiom:

Existence: For any A, B, and C, there exists A′ ≡_{C} A such that A′ | ⌣_{C} B.

The following result follows from [Adler05].

Corollary 3.18. If | ⌣^{cl} satisfies either the Extension or the Existence axiom, then it is an independence relation (and satisfies the Existence axiom).

Proof. See [Adler05, Thm. 2.5].

def:existence Definition 3.19. cl satisfies Existence if:

For every a, B, and C, if a ∉ cl B, then there exists a′ ≡_{B} a such that a′ ∉ cl(BC).

Denote by Aut(𝕄/B) the set of automorphisms of 𝕄 which fix B pointwise. Denote by Ξ(a/B) the set of conjugates of a over B: Ξ(a/C) := {a^{σ} : σ ∈ Aut(𝕄/B)}.

lem:cl-E Lemma 3.20. T.f.a.e.:

1. cl satisfies Existence.

2. For every a, B, and C, if Ξ(a/B) ⊆ cl(BC), then a ∈ cl(B).

3. For every a, b̄, and c̄, if a ∉ cl(b̄), then there exists a′ ≡b̄ a such that a′ ∉ cl(b̄c̄).

4. For every a, b̄, and c̄, and every x-narrow formula ϕ(x, ȳ, z̄), if 𝕄 ⊧ ϕ(a′, b̄, c̄) for every a′ ≡b̄ a, then a ∈ cl(b̄).

5. For every formula (without parameters) ϕ(x, ȳ) and every x-narrow formula ψ(x, ȳ, z̄), if 𝕄 ⊧ ∀ȳ ∃z̄ ∀x (ϕ(x, ȳ) → ψ(x, ȳ, z̄)), then ϕ is x-narrow.

6. For every a and B, if rk^{cl}(Ξ(a/B) is finite, then a ∈ cl(B).

7. For every a and B, if rk^{cl}(Ξ(a/B) < κ, then a ∈ cl(B).

8. | ⌣^{cl} is an independence relation.

Remark 3.21. If cl satisfies Existence, then acl A ⊆ cl A.

lem:cl-Skolem-existence Lemma 3.22. Assume that cl(A) is an elementary substructure of 𝕄, for every A ⊂ 𝕄. Then, cl satisfies Existence, and therefore | ⌣^{cl} is an independence relation. Hence, if T has definable Skolem functions and cl extends acl, then cl is satisfies Existence.

Proof. Let Ξ(a/B) ⊆ cl(BC). We want to prove that a ∈ cl(B). Let B′ and C′ be elementary substructures of 𝕄, such that B ⊆ B′ ⊂ cl(B), B′C ⊆ C′ ⊂ cl(BC), |B′| < κ, and |C′| < κ (B′ and C′ exist by hypothesis on cl). By substituting B with B′ and C with C′, w.l.o.g. we can assume that B ⪯ C ≺ 𝕄. By saturation, there exist an x-narrow formula ϕ(x, ȳ, z̄), b̄ ⊂ B, and c̄ ⊂ C, such that Ξ(a/B) ⊆ ϕ(𝕄, b̄, c̄). Let p := tp(a/B), let q ∈ S_{1}(C) be a heir of p, and a′ be a realisation of q. Since ϕ(x, b̄, c̄) ∈ p, there exists b̄′ ∈ B such that ϕ(x, b̄, b̄′) ∈ q. Hence, a′ ∈ cl(B); since a′ ≡_{B} a, a ∈ cl(B).

Definition 3.23. The trivial matroid cl^{0} is given by cl^{0}(X) = X for every X ⊆ 𝕄. cl^{0} is a definable matroid and satisfies Existence. It induces the trivial pre-independence relation | ⌣^{0} , such that A | ⌣^{0} _{B} C for every A, B, and C. Notice that | ⌣^{0} is an independence relation.

def:existential Definition 3.24. We say that cl is an existential matroid if cl is a definable matroid, satisfies Existence, and is non-trivial (i.e., different from cl^{0}).

Examples 3.25. 1. Given n ∈ ℕ, the uniform matroid of rank n is defined as: cl^{n}(X) := X, if |X| < n, or 𝕄 if |X| ≥ n. cl^{n} is a definable matroid, but does not satisfy Existence in general (unless n = 0).

3. Matroids  3.2. Dimension

2. Define id(X) := X. id is a definable matroid, but does not satisfy Existence in general. The pre-independence relation induced by id is given by A | ⌣^{id}_{B} C iff A ∩ C ⊆ B.

Remark 3.26. Let 𝕄′ be another monster model of T. We can define an operator cl′ on 𝕄′ in the following way:

cl(X′) := ⋃{ϕ(𝕄′, ā) : ϕ(x, ȳ) x-narrow & ā′ ⊂ X′}.

Then, cl′ is a definable matroid. If cl satisfies existence, then cl′ also satisfies existence.

rem:general-closure Remark 3.27. Notice that the definitions of “definable” (3.14) and “existential” (3.24 and 3.19) make sense also for finitary closure operators (and not only for matroids).

However, we will not need such more general definitions.

Proviso. For the remainder of this section, cl is an existential matroid.

Summarising, we have: If cl is an existential matroid, then | ⌣^{cl} is an independence relation, satisfying the strong finite character axiom. In particular, if 𝕄 is a pregeometric structure, then | ⌣^{acl} is an independence relation.

3.2 Dimension

def:dimension Definition 3.28. Given a set V ⊆ 𝕄^{n}, definable with parameters A, the dimension of V (w.r.t. to the matroid cl) is given by

dim^{cl}(V) := max{rk^{cl}(b̄/A) : b̄ ∈ X},

with dim^{cl}(V) := −∞ iff V = ∅. More generally, the dimension of a partial type p with parameters A is given by

dim^{cl}(p) := max{rk^{cl}(b̄/A) : b̄ ⊧ p}.

The following lemma shows that the above notion is well-posed: in its proof, it is important that cl satisfies existence.

Lemma 3.29. Let V be a type-definable subset of 𝕄^{n}. Then, dim^{cl}(V) ≤ n, and dim^{cl}(V) does not depend on the choice of the parameters.

Remark 3.30. For every d ≤ n ∈ ℕ, the set of complete types in S_{n}(A) of dim^{cl} equal to d is closed (in the Stone topology). That is, dim^{cl} is continuous in the sense of [Poizat85, §17.b].

Lemma 3.31. Let p be a partial type over A. Then,

dim^{cl}(p) := min{dim^{cl}(V) : V is A-definable & V ∈ p}.

Moreover, if p is a complete type, then, for every b̄ ⊧ p, rk^{cl}(b̄/A) = dim^{cl}(p).

Proof. Let d := dim^{cl}(p), e := min{dim^{cl}(V) : V is A-definable & V ∈ p}, and b̄ ⊧ p, such that d = rk^{cl}(b̄/A). If V ∈ p, then b̄ ∈ V, and therefore

e ≥ dim^{cl}(V) ≥ rk^{cl}(b̄/A) = d.

For the opposite inequality, first assume that p is a complete type. W.l.o.g. b̃:= ⟨b_{1}, …, b_{d}⟩ are cl-independent over A, and therefore b_{i} ∈ cl(Ab̃) for every i = d + 1, …, n. For every i ≤ n, ϕ_{i}(x, ȳ, z̄) be an x-narrow formula such that 𝕄 ⊧ ϕ(b_{i}, b̃, ā) (where ā ⊂ A), ϕ(x̄, ȳ, z̄) := ⋂_{i=1}^{n} ϕ_{i}(x_{i}, x_{1}, …, x_{d}, z̄), and V := ϕ(𝕄^{n}, 𝕄^{d}, ā). Then, for every b̄′ ∈ V, rk^{cl}(b̄′/A) ≤ d, and therefore dim^{cl}(V) ≤ d. Moreover, b̄ ∈ V, hence V ∈ p, and therefore e ≤ d.

The general case when p is a partial type follows from the complete case, the fact that the set of complete types extending p is a closed (and hence compact) subset of S_{n}(A), and the previous remark.

Remark 3.32. dim^{cl}(𝕄^{n}) = n. Moreover, dim^{cl} is monotone: if U ⊆ V ⊆ 𝕄^{n}, then dim^{cl}(U) ≤ dim^{cl}(V).

Definition 3.33. Given p ∈ S_{n}(B), q ∈ S_{n}(C), with B ⊆ C, we say that q is a non-forking extension of p (w.r.t. cl), and write p ⊑ q, if q extends p and dim^{cl}(q) = dim^{cl}(p). We write q | ⌣^{cl}_{B} C if q ↾_{B}⊑ q.

Remark 3.34. Let B ⊆ C and q ∈ S_{n}(C). Then, q | ⌣^{cl}_{B} C iff, for some (for all) ā realising q, ā | ⌣^{cl}_{B} C.

Remark 3.35. Let p ∈ S_{n}(B) and B ⊆ C. Then, for every q ∈ S_{n}(C) extending p, dim^{cl}(q) ≤ dim^{cl}(p). Moreover, there exists q ∈ S_{n}(C) which is a non-forking extension of p.

lem:ind-heir Lemma 3.36. Let | ⌣^{f} be Shelah’s forking relation on 𝕄. Then, for every A, B, and C subsets of 𝕄, if A | ⌣^{f} _{B} C, then A | ⌣^{cl}_{B} C. In particular, if 𝕂 ≺ 𝕄, 𝕂 ⊆ C, and q ∈ S_{n}(C), and q is either a heir or a coheir of q ↾_{𝕂}, then q | ⌣^{cl}_{𝕂} C.

Proof. The fact that | ⌣^{f} implies | ⌣^{cl} is a particular case of [Adler05, Remark 1.27]. For the case when q is a heir of p := q ↾_{𝕂}, see also [Adler05, Remark 2.3].

Corollary 3.37. Assume that T is super-simple and p ∈ S_{n}(A) for some A ⊆ 𝕄. Then, SU(p) ≥ dim^{cl}(p), where SU is the SU-rank (see [Wagner00]).

rem:forking-closed Remark 3.38. Given B ⊇ A, let N_{n}(B/A) be the set of all n-types over B that do not fork over A. N_{n}(B, A) is closed in S_{n}(B). The same is true for any independence relation | ⌣, instead of | ⌣^{cl}.

Lemma 3.39. For every complete type p, dim^{cl}(p) is the maximum of the cardinalities n of chains of complete types p = q_{0} ⊂ q_{1} ⊂ … ⊂ q_{n}, such that each q_{i+1} is a forking extension of q_{i}.

Proof. Let A be the set of parameters of p, and b̄ ⊧ p. Let d := dim^{cl}(p); w.l.o.g., b̃ := ⟨b_{1}, …, b_{d}⟩ are independent over A. For every i ≤ n let A_{i} := Ab_{1} … b_{i}, and q_{i} := tp(b̄/A_{i}). Then, p = q_{0} ⊂ ⋯ ⊂ q_{d}, and each q_{i+1} is a forking extension of q_{i}.

Conversely, assume that p = q_{0} ⊂ ⋯ ⊂ q_{n}, and each q_{i+1} is a forking extension of q_{i}, and A_{i} be the set of parameters of q_{i}. Claim 1. For every i ≤ n, dim^{cl}(q_{n−1}) ≥ i; in particular, dim^{cl}(p) ≥ n.

By induction on i. The case i = 0 is clear. Assume that we have proved the claim for i, we want to show that it holds for i + 1. Since q_{i} is a forking extension of q_{i+1}, dim^{cl}(q_{i}) > dim^{cl}(q_{i+1}), and we are done.

lem:dim-rk Lemma 3.40. Let V ⊆ 𝕄^{n} be non-empty and definable with parameters ā. Then, either dim^{cl}(V) = 0 = rk^{cl}(V/ā), or dim^{cl}(V) > 0 and rk^{cl}(V) ≥ κ.

Lemma 3.41. A formula ϕ(x, ȳ) is x-narrow iff, for every b̄, dim^{cl}(ϕ(𝕄, b̄)) = 0.

lem:type-def Lemma 3.42. Let ϕ(x, ȳ) be a formula without parameters, and ā ∈ 𝕄^{n}. Then, dim^{cl}(ϕ(𝕄, ā)) = 0 iff there exists an x-narrow formula ψ(x, ȳ) such that ∀x (ϕ(x, ā) → ψ(x, ā)). Therefore, define

Γ_{ϕ}(ȳ) := {¬θ(ȳ) : θ(ȳ) formula without parameters s.t.
∀ā (θ(ā) → dim^{cl}(ϕ(𝕄, ā)) = 0)},
U_{ϕ}^{1} := {ā ∈ 𝕄^{n} : dim^{cl}(ϕ(𝕄, ā)) = 1}.

Then, U_{ϕ}^{1} = {ā ∈ 𝕄^{n} : 𝕄 ⊧ Γ_{ϕ}(ā)}, and in particular U_{ϕ}^{1} is type-definable (over the empty set).

More generally, let k ≤ n, x̄ := ⟨x_{1}, …, x_{n}⟩, and ϕ(x̄, ȳ) be a formula without parameters. Define

U_{ϕ}^{≥k} := {ā ∈ 𝕄^{m} : dim^{cl}(ϕ(𝕄^{n}, ā)) ≥ k}.

Then, U_{ϕ}^{≥k} is type-definable over the empty set. lem:cl-function Lemma 3.43 (Fibre-wise dimension inequalities). U ⊆ 𝕄^{m_{1}}, V ⊆ 𝕄^{m_{2}},

and F : U → V be definable, with parameters C. Let X ⊆ U and Y ⊆ V be type-definable, such that F (X) ⊆ Y. Define f := F ↾ X : X → Y. For every b̄ ∈ Y, let Xb̄ := f^{−1}(b̄) ⊆ X, and m := dim^{cl}(Y).

1. If, for every b̄ ∈ Y, dim^{cl}(Xb̄) ≤ n, then dim^{cl}(X) ≤ m + n.

2. If f is surjective and, for every b̄ ∈ Y, dim^{cl}(Xb̄) ≥ n, then dim^{cl}(X) ≥ m + n.

3. If f is surjective, then dim^{cl}(X) ≥ m.

4. If f is injective, then dim^{cl}(X) ≤ m.

5. If f is bijective, then dim^{cl}(X) = m. Proof. 1) Assume, for contradiction, that dim^{cl}(X) > m + n. Let ā ∈ X such that rk^{cl}(ā/C) > m + n, and b̄ := F (ā). Since ā ∈ Xb̄, and Xb̄ is type-definable with parameters Cb̄, rk^{cl}(ā/b̄C) ≤ n. Hence, by Lemma 3.2, rk^{cl}(ā/C) ≤ rk^{cl}(āb̄/C) ≤ m + n, absurd.

2) Let b̄ ∈ Y such that dim^{cl}(b̄/C) = m. Let ā ∈ Xb̄ such that dim^{cl}(ā/b̄C) ≥ n. Then, by Lemma 3.2, rk^{cl}(āb̄/C) ≥ m + n. However, since ā = F (b̄), ā ⊂ cl(b̄C), and therefore rk^{cl}(b̄/C) = rk^{cl}(āb̄/C) ≥ m + n.

(3) follows from (2) applied to n = 0. The other assertions are clear.

lem:cl-equality Lemma 3.44. Let cl′ be another existential matroid on 𝕄. T.f.a.e.: 1. cl ⊆ cl′;

2. rk^{cl} ≥ rk^{cl′};

3. dim^{cl} ≥ dim^{cl′} on definable sets;

4. dim^{cl} ≥ dim^{cl′} on complete types;

5. for every definable set X ⊆ 𝕄, if dim^{cl′}(X) = 0, then dim^{cl}(X) = 0. T.f.a.e.:

1. cl = cl′;

2. rk^{cl} = rk^{cl′};

3. dim^{cl} = dim^{cl′} on definable sets;

4. dim^{cl} = dim^{cl′} on complete types; 5. for every definable set X ⊆ 𝕄, dim^{cl}(X) = 0 iff dim^{cl′}(X) = 0.

We will show that, for many interesting theories, there is at most one existential matroid.

Define T_{R∤0} to be the theory of rings without zero divisors, in the language of rings L_{R} := (0, 1, +, ·).

Definition 3.45. If 𝕂 expands a ring without zero divisors, define F : 𝕂^{4} → 𝕂 the function, definable without parameters in the language L_{R},

⟨x_{1}, x_{2}, y_{1}, y_{2}⟩ ↦ \begin{cases}t & if y_{1} ≠ y_{2} & t · (y_{1} − y_{2}) = x_{1} − x_{2}; \\ 0 & otherwise. \end{cases}

Notice that F is well-defined, because in a ring without zero divisors, if y_{1} ≠ y_{2}, then, for every x, there exists at most one t such that t·(y_{1}−y_{2}) = x.

lem:cl-ring-function Lemma 3.46 ([DMS08, 1.18]). Assume that T expands T_{R∤0}. Let A ⊆ 𝕄 be definable. Then, dim^{cl}(A) = 1 iff 𝕄 = F (A^{4}).

Proof. Same as [DMS08, 1.18]. Assume for contradiction that dim^{cl}(A) = 1, but there exists c ∈ 𝕄 \ F (A^{4}). Since c ∉ F (A^{4}), the function ⟨x_{1}, x_{2}⟩ ↦ c · x_{1} + x_{2} : A^{2} → 𝕄 is injective. Hence, by Lemma 3.43, dim^{cl}(𝕄) ≥ dim^{cl}(A^{2}) = 2, absurd.

Conversely, by Lemma 3.43 again, if f(A^{4}) = 𝕄, then dim(A) = 1.

thm:cl-unique Theorem 3.47. If T expands T_{R∤0}, then cl is the only existential matroid on 𝕄. If S is a definable subfield of 𝕄 of dimension 1, then S = 𝕄.

Proof. Let A ⊆ 𝕄 be definable. By the previous lemma, dim(A) = 1 iff F (A^{4}) = 𝕄. Since the same holds for any existential matroid cl′ on 𝕄, we conclude that, for every definable set A ⊆ 𝕄, dim^{cl}(A) = 0 iff dim^{cl′}(A) = 0, and hence dim^{cl} = dim^{cl′}.

Given S a subfield of 𝕄, F (S^{4}) = S. Hence, if dim^{cl}(S) = 1, then S = 𝕄.

Example 3.48. In the above theorem, we cannot drop the hypothesis that T expands T_{R∤0}. In fact, let 𝕄_{0} be an infinite connected graph, such that 𝕄_{0} is a monster model, and acl is a matroid in 𝕄_{0} (e.g., 𝕄_{0} equal to a monster model of the theory of random graphs). Let 𝕄 be the disjoint union of κ copies of 𝕄_{0}: notice that 𝕄 is a monster model. For every a ∈ 𝕄, let cl(a) be the connected component of 𝕄 containing a (it is a copy of 𝕄_{0}), and cl(A) := ⋃_{a∈A} cl(a). Then, acl and cl are two different existential matroids on 𝕄.

Example 3.49. In Lemma 3.46 and Theorem 3.47 we cannot even relax the hypothesis to “T expands the theory of a vector space”. In fact, let 𝔽 be an ordered field, considered as a vector space over itself, in the language ⟨0, 1, +, <, λ_{c}⟩_{c∈𝔽}, and let T be its theory. Let T^{d} be the theory of dense pairs of models of T. [DMS08, 5.8] show that T^{d} has elimination of quantifiers, and acl is a matroid on T^{d}. However, as the reader can verify, the small closure Scl is another existential matroid on T^{d} (cf. §8.4), and it is different from acl.

Corollary 3.50. If 𝕄 expands a field, then 𝕄 must be a perfect field. In particular, the theory of separably closed and non-algebraically closed fields, and the theory of differentially closed fields of finite characteristic do not admit an existential matroid.

Proof. Cf. [Dries89, 1.6]. If 𝕄 is not perfect, then 𝕄^{p} is a proper definable subfield of 𝕄, where p := char(𝕄), and therefore dim^{cl}(𝕄^{p}) = 0. However, the map x ↦ x^{p} is a bijection from 𝕄 to 𝕄^{p}; therefore, dim^{cl}(𝕄) = 0, absurd.

Corollary 3.51. Let cl′ be a non-trivial definable matroid on some monster model 𝕄′. Assume that 𝕄′ expands a model of T_{R∤0}. Then, t.f.a.e.:

1. cl′ is an existential matroid;

2. for every formula (without quantifiers) ϕ(x, ȳ), ϕ is x-narrow iff, for y every b̄, F^{4}(ϕ(𝕄′, b̄) ≠ 𝕄′.

Proof. (1 ⇒ 2) is clear.

(2 ⇒ 1) follows from Lemma 3.20-5.

Lemma 3.52. Let 𝕂 be a ring without zero divisors definable in 𝕄, of dimension n ≥ 1. Let 𝔽 ⊆ 𝕂 be a definable subring such that 𝔽 is a skew field. If dim^{cl}(𝕂) = n, then 𝕂 = 𝔽.

Proof. Assume, for contradiction, that there exists c ∈ 𝕂 \ 𝔽. Define h : 𝔽×𝔽 → 𝕂, h(x, y) := x+cy. Since c ∉ 𝔽 and 𝔽 is a skew field, h is injective. Thus, 2n = dim(𝔽^{2}) ≤ dim(𝕂) = n, a contradiction.

Corollary 3.53. Let 𝕂 ⊆ 𝕄^{n} be a definable field, such that dim^{cl}(𝕂) ≥ 1. Then, 𝕂 is perfect.

The assumption that dim^{cl}(𝕂) ≥ 1 is necessary: non-perfect definable fields of dimension 0 can exist.

Proof. Let p := char 𝕂, and ϕ : 𝕂 → 𝕂 be the Frobenius automorphism ϕ(x) = x^{p}. Since ϕ is injective, dim^{cl}(𝕂^{p}) = dim^{cl}(𝕂), and therefore 𝕂^{p} = 𝕂.

ex:group Example 3.54. Let λ be an ordinal, which is an ordinal power of ω (e.g., λ = 1, λ = ω, …). Let 𝔾 be a monster model of a superstable group, such that U(𝔾) = λ, where U is Lascar’s rank. For every a and B, define a ∈ cl(B) iff U(a/B) < λ. Then, cl is an existential matroid. If X is a definable subset of 𝔾, then dim^{cl}(X) = 1 iff X is generic, that is finitely many left translates of X cover 𝔾.

Proof. See [Poizat87].

Example 3.55. Let 𝕂 be a monster differentially closed field, and p ≥ 0 be its characteristic. If p = 0, then 𝕂 is superstable, and U(𝕂) = ω; hence, by the previous example, there exists a (unique) existential matroid cl on 𝕂. It is easy to see that, if A is a differential subfield of 𝕂 and b ∈ 𝕂, then b ∈ cl(A) iff b is differential-algebraic over A (that is, iff b, db, d^{2}b, … are algebraically dependent over A); see [Wood76] and [Dries89, 2.25]. On the other hand, if p > 0, then there is no existential matroid on 𝕂, because 𝕂 is not perfect.

Definition 3.56. Let X ⊆ 𝕂^{n} any Y ⊆ 𝕂^{m} be definable. Let f : X ⇝ Y be a definable application (i.e., a multi-valued partial function), with graph F. For every x ∈ X, let f(x) := {y ∈ Y : ⟨x, y⟩ ∈ F } ⊆ Y. Such an application f is a Z-application if, for every x ∈ X, dim^{cl}(f(x)) ≤ 0.

rem:Zapplication Remark 3.57. Let A ⊆ 𝕂, and b ∈ 𝕂. Then, b ∈ cl(A) iff there exists an ∅- definable Z-application f : 𝕂^{n} ⇝ 𝕂 and ā ∈ A, such that b ∈ f(ā). Moreover, if c̄ ∈ 𝕂^{n}, then b ∈ cl(Ac̄) iff there exists an A-definable Z-application f : 𝕂^{n} → 𝕂, such that b ∈ f(c̄).

Definition 3.58. We say that dim^{cl} is definable if, for every X definable subset of 𝕄^{m} × 𝕄^{n}, the set {ā ∈ 𝕄^{m} : dim^{cl}(Xā) = d} is definable.

Lemma 3.59. T.f.a.e.:

1. dim^{cl} is definable;

2. for every X definable subset of 𝕄^{m} × 𝕄, the set X^{1,1} := {ā ∈ 𝕄^{m} : dim^{cl}(Xā) = 1} is also definable;

3. for every k ≤ n, every m, and every X definable subset of 𝕄^{m} × 𝕄^{n}, the set X^{n,k} := {ā ∈ 𝕄^{m} : dim^{cl}(Xā) = k} is also definable, with the same parameters as X.

Proof. (3 ⇒ 1 ⇒ 2) is obvious.

(2 ⇒ 1) is obvious. We will prove by induction on n that, for every Y definable subset of 𝕂^{n} × 𝕂^{m}, the set Y^{n,≥k} := {ā ∈ 𝕄^{m} : dim^{cl}(Xā) ≥ k} is definable. The case k = 0 is clear. The case k = 1 follows from the assumption and the observation that, for every Z definable subset of 𝕂^{n}, dim^{cl}(Z) ≥ 1 iff dim^{cl}(θ(Z)) ≥ 1 for some θ projection from 𝕂^{n} to a coordinate axis. The inductive step follows from the fact that

X^{n,≥k} = (Π_{n+m−1}^{n+m}(X))^{n−1,≥k} ∪ (X^{n+m−1,≥1})^{n−1,≥k−1}.

(1 ⇒ 3) Let X ⊆ 𝕂^{n+m} be definable with parameters A. Then, X^{n,k} is 𝕄definable, by assumption. Moreover, by Lemma 3.42, X^{n,k} is type-definable over A, and therefore invariant under automorphisms that fix A pointwise. Hence, by Beth’s definability theorem, X^{n,k} is definable over A.

Remark 3.60. If T expands T_{R∤0}, then dim^{cl} is definable.

3.3 Morley sequences

Most of the results of this subsection remain true for an arbitrary independence relation | ⌣ instead of | ⌣^{cl}.

Definition 3.61. Let C ⊆ B, p(x̄) ∈ S_{n}(B), and ⟨I, ≤⟩ be a linear order. A Morley sequence over C indexed by I in p is a sequence (ā_{i} : i ∈ I) of tuples in 𝕄^{n}, such that (ā_{i} : i ∈ I) is order-indiscernibles over B and independent over C, and every ā_{i} realises p(x̄). A Morley sequence over C is a Morley sequence over C in some p ∈ S_{n}(C). A Morley sequence in p is a Morley sequence over B in p.

lem:Morley-seq Lemma 3.62. Let ⟨I, ≤⟩ be a linear order, with |I| < κ. Let p(x̄) ∈ S_{n}(C). Then, there exists a Morley sequence over C indexed by I in p(x̄). If moreover b̄ | ⌣^{cl}_{C} d̄, then there exists a Morley sequence (ā_{i} : i ∈ I) over C indexed by I in p(x̄), such that (b̄ā_{i} : i ∈ I) is order-indiscernibles over Cd̄ and, for every i ∈ I, b̄ā_{i} | ⌣^{cl}_{C} d̄(ā_{j} : i ≠ j ∈ I).

Proof. Let (x̄_{i} : i ∈ I) be a sequence of n-tuples of variables. Consider the following set of C-formulae:

Γ_{1}(x̄_{i} : i ∈ I) := ⋀_{i∈I} p(x̄_{i}) & ⋀_{i∈I} x̄_{i} | ⌣^{cl}_{C}(x̄_{j} : j < i).

First, notice that, by Remark 3.38 Γ_{1} is a set of formulae.

3. Matroids  3.3. Morley sequences

Consider the following set of C-formulae:

Γ_{2}(x̄_{i} : i ∈ I) := Γ_{1}(x̄_{i} : i ∈ I) & (x̄_{i} : i ∈ I) is an order-indiscernible sequence of over C.

By [Adler05, 1.12], Γ_{2} is consistent.

We give an alternative proof of the above fact, which does not use Erdös-Rado. Claim 2. Γ_{1} is consistent.

It is enough to prove that Γ_{1} is finitely satisfiable; hence, w.l.o.g. I = {0, …, m} is finite. Let ā_{0} be any realisation of p(x̄). Let ā_{1} ≡_{C} ā_{0} such that ā_{i} | ⌣^{cl}_{C} ā_{0}, …, let ā_{m} ≡_{C} ā_{0} such that ā_{m} | ⌣^{cl}_{C} ā_{0} … ā_{m−1}.

By Ramsey’s Theorem, Γ_{2} is also consistent.

Since |I| < κ, there exists a realisation (ā_{i} : i ∈ I) of Γ_{2}. Then, by Lemma 3.11 (ā_{i} : i ∈ I) is a Morley sequence in p(x̄) over C.

If moreover b̄ and d̄ satisfy b̄ | ⌣^{cl}_{C} d̄, let q(x̄, ȳ, z̄) be the extension of p(x̄) to S^{∗}(Cb̄d̄) satisfying ȳ = b̄ and z̄ = d̄. Let (ā_{i}b̄d̄ : i ∈ I) be a Morley sequence in q(x̄, ȳ, z̄). By Lemma 3.10, for every i ∈ I we have b̄ā_{i} | ⌣^{cl}_{C} d̄(ā_{j} : i ≠ j ∈ I).

Lemma 3.63. A type p ∈ S_{n}(A) is stationary if, for every B ⊇ A, there exists a unique q ∈ S_{n}(B) such that p ⊑ q.

Remark 3.64. Let p ∈ S_{n}(A). If dim^{cl}(p) = 0, then p is stationary iff p is realised in dcl(A).

Hence, unlike the stable case, if cl ≠ acl, then there are types over models which are not stationary.

lem:indipendent-morley-seq Lemma 3.65. Let C ⊇ B, and q ∈ S_{n}(C) such that q | ⌣^{cl}_{B} C. Let (ā_{i} : i ∈ I) be a sequence of realisations of q independent over C. Then, (ā_{i} : i ∈ I) is also independent over B. If moreover q is stationary, then

1. (ā_{i} : i ∈ I) is a totally indiscernible set over C, and in particular it is a Morley sequence for q over B.

2. If (ā′ : i ∈ I) is another sequence of realisations of q independent over C, then (ā_{i} : i ∈ I) ≡_{C} (ā′_{i} : i ∈ I).

Proof. Standard proof. More precisely, for every i ∈ I, let d̄_{i} := (a_{j} : i ≠ j ∈ I). By assumption, ā_{i} | ⌣^{cl}_{C} d̄_{i}, and, since q | ⌣^{cl}_{B} C, ā_{i} | ⌣^{cl}_{B} C, and therefore ā_{i} | ⌣^{cl}_{B} d̄_{i}, proving that (ā_{i} : i ∈ I) is independent over B.

Let us prove Statement (2). By compactness, w.l.o.g. I = {1, …, m} is finite. Assume, for contradiction, that (ā : i ≤ m) ≢_{C} (ā′ : i ≤ m); by induction on m, we can assume that (ā_{i} : i ≤ m − 1) ≡_{C} (ā′_{i} : i ≤ m − 1), and therefore, w.l.o.g., that ā_{i} = ā′_{i} for i = 1, …, m − 1. However, since q is stationary, ā_{m} ≡_{C} ā′_{m}, ā_{m} | ⌣^{cl}_{C}(ā_{i} : i ≤ m − 1), and ā′_{m} | ⌣^{cl}_{C}(ā_{i} : i ≤ m − 1), we have that ā_{m} ≡_{C(ā_{i}:i≤m−1)} ā′_{m}, absurd.

Finally, it remains to prove that the set (ā_{i} : i ∈ I) is totally indiscernible over C. If σ is any permutation of I, then (ā_{σ(i)} : i ∈ I) is also a sequence of realisations of q independent over C, and therefore, by Statement (2), (ā_{σ(i)} : i ∈ I) ≡_{C} (ā_{i} : i ∈ I).

Corollary 3.66. Assume that there is a definable linear ordering on 𝕄. Then, p ∈ S_{n}(A) is stationary iff p is realised in dcl(A). Hence, if dim^{cl}(p) > 0, every non-forking extension of p is not stationary.

Contrast the above situation to the case of stable theories, where instead every type has at least one stationary non-forking extension.

Proof. Assume that p is stationary, but, for contradiction, that dim^{cl}(p) > 0. Then, there is a Morley sequence in p with at least two elements ā_{0} and ā_{1}. Since dim^{cl}(p) > 0, ā_{0} ≠ ā_{1}. By Lemma 3.65, ā_{0} and ā_{1} are indiscernibles over A, absurd.

cor:ind-morley Corollary 3.67. Let B ⊆ C and q ∈ S_{n}(C). Then, t.f.a.e.:

1. q | ⌣^{cl}_{B} C;

2. there exists an infinite sequence of realisations of q that are independent over B;

3. every sequence (ā_{i} : i ∈ I) of realisations of q that are independent over C are independent also over B;

4. there exists an infinite Morley sequence in q over B.

Proof. Cf. [Adler05, 1.12–13].

(1 ⇒ 3) Let (ā_{i} : i ∈ I) be a sequence of realisations of q independent over C. For every i ∈ I, let d̄_{i} := (ā_{j} : i ≠ j ∈ I). Since ā_{i} | ⌣_{C} d̄_{i} and ā_{i} | ⌣_{B} C,

we have ā_{i} | ⌣_{B} d̄_{i}.

(3 ⇒ 4) Let (ā_{i} : i ∈ I) be an infinite Morley sequence in q over C: such sequence exists by Lemma 3.62 (or by [Adler05, 1.12]). Then, (ā_{i} : i ∈ I) is independent also over B, and hence a Morley sequence for q over B.

3. Matroids  3.4. Local properties of dimension

(4 ⇒ 2) is obvious.

(2 ⇒ 1) Choose λ < κ a regular cardinal large enough. Let (ā′_{i} : i < ω) be a sequence of realisations of q independent over C. By saturation, there exists (ā_{i} : i < λ) a sequence of realisations of q independent over C. By Local Character, and since λ is regular, there exists α < λ such that ā_{α} | ⌣^{cl}_{Bd̄}C, where d̄:= (ā_{i} : i < α). Since moreover ā_{α} | ⌣^{cl}_{B} d̄, we have ā_{α} | ⌣^{cl}_{B} C, and therefore q | ⌣^{cl}_{B} C.

3.4 Local properties of dimension

In this subsection, we will show that the dimension of a set can be checked locally: what this means precisely will be clear in §9, where the results given here will be applied to a “concrete” situation.

Definition 3.68. A quasi-ordered set ⟨I, ≤⟩ is a directed set if every pairs of elements of I has an upper bound.

lem:order-ind Lemma 3.69. Let ⟨I, ≤⟩ be a directed set, definable in 𝕄 with parameters c̄. Then, for every ā ∈ I and d̄ ⊂ 𝕄 there exists b̄ ∈ I such that b̄ ≥ ā and d̄ā | ⌣^{cl}c̄ b̄.

Proof. Fix ā ∈ I and d̄⊂ 𝕄, and assume, for contradiction, that every b̄ ≥ ā satisfies d̄ā |̸ ⌣^{cl}c̄ b̄.

W.l.o.g., c̄ = ∅. Let λ be a large enough cardinal; at the price of increasing κ if necessary, we may assume that λ < κ. By Lemma 3.62, there exists a Morley sequence (d̄′ā′_{i} : i < λ) in tp(d̄ā/∅) over ∅. Consider the following set of formulae over {ā′_{i} : i < λ}:

Λ(x̄) := {x̄ ∈ I, x̄ ≥ ā′_{i} : i < λ}.

Since ⟨I, ≤⟩ is a directed set, Λ is consistent: let b̄ ∈ I be a realisation of Λ. By Erdös-Rado’s Theorem, there exists a Morley sequence (d̄_{i}ā_{i} : i < ω) in tp(d̄ā/∅) over ∅, such that all the d̄_{i}ā_{i} satisfy the same type q(x̄, ȳ) over b̄. Therefore, by Corollary 3.67, q | ⌣^{cl} b̄, and in particular ā_{0} | ⌣^{cl} b̄. Since ā_{0} ≡ ā, there exists b̄′ ≥ ā such that ā ≡ b̄′, a contradiction.

lem:neighbourhood-abstract Lemma 3.70. Let X ⊆ 𝕄^{n} be definable with parameters c̄ and (Ut̄)_{t̄∈I} be a family of subsets of 𝕄^{n}, such that each Ut̄ is definable with parameters t̄c̄. Let d ≤ n, and assume that, for every ā ∈ X there exists b̄ ∈ I such that ā ∈ Ub̄, ā | ⌣^{M}c̄ b̄, and dim^{cl}(X ∩ Ub̄) ≤ d. Then, dim^{cl}(X) ≤ d.

Proof. Assume, for contradiction, that dim^{cl}(X) > d; let ā ∈ X such that rk^{cl}(ā/c̄) > d. Choose b̄ as in the hypothesis of the lemma; then, rk^{cl}(ā/b̄c̄) > d, absurd.

lem:cl-U Lemma 3.71. Let I ⊆ 𝕄^{n} be definable and < be a definable linear ordering on I. Let (Xb̄)_{b̄∈I} be a definable increasing family of subsets of 𝕂^{m} and X := ⋃_{b̄∈I} Xb̄. Let d ≤ m, and assume that, for every b̄ ∈ I, dim^{cl}(Xb̄) ≤ d. Then, dim^{cl}(X) ≤ d.

Proof. Let c̄ be the parameters used to define I, <, and (Xb̄)_{b̄∈I}. Let ā ∈ X such that rk^{cl}(ā/c̄) = dim^{cl}(X). Let b̄ ∈ I such that ā ∈ Xb̄. Choose ā′, b̄′ ⊂ 𝕄 such that ā′b̄′ ≡c̄ āb̄ and ā′b̄′ | ⌣^{cl}c̄ āb̄. W.l.o.g., b̄′ ≥ b̄; hence, ā ∈ Xb̄′ and

d ≥ dim^{cl}(Xb̄′) ≥ rk^{cl}(ā/c̄b̄′) = rk^{cl}(ā/c̄) = dim^{cl}(X).

We can extend the above lemma to directed families.

Lemma 3.72. Let ⟨I, ≤⟩ be a definable directed set. Let (Xb̄)_{b̄∈I} be a definable increasing family of subsets of 𝕄^{m} and X := ⋃_{b̄∈I} Xb̄. Let d ≤ m, and assume that, for every b̄ ∈ I, dim^{cl}(Xb̄) ≤ d. Then, dim^{cl}(X) ≤ d.

Proof. W.l.o.g., ⟨I, ≤⟩ and the family (Xb̄)_{b̄∈I} are definable without parameters. Let ā ∈ X such that rk^{cl}(ā) = dim^{cl}(X), and let b̄_{0} ∈ I such that a ∈ Xb̄_{0}. By the Lemma 3.69, there exists b̄ ∈ I such that b̄ ≥ b̄_{0} and āb̄_{0} | ⌣^{cl} b̄. Hence, ā ∈ Xb̄ and ā | ⌣^{cl} b̄, and therefore

d ≥ dim^{cl}(Xb̄) ≥ rk(ā/b̄) = rk(ā) = dim^{cl}(X).

Remark 3.73. The above lemma is not true if (Xb̄)_{b̄∈I} be a definable decreasing family of subsets of 𝕄^{m}, instead of increasing. For instance, let 𝕂 be a real closed field, \overline{=} acl, I := 𝕂^{<0} × 𝕂 ∪ \{⟨0, 0⟩\}; define ⟨x, y⟩ ≤ ⟨x′, y′⟩ if x ≤ x′ and y = y′, or x = 0. Let I_{b_{1},b_{2}} := {⟨x, y⟩ ∈ I : ⟨x, y⟩ ≥ ⟨b_{1}, b_{2}⟩}. Then, ⟨I, ≤⟩ is a directed set, dim^{acl}(I) = 2, but dim^{acl}(Ib̄) ≤ 1 for every b̄ ∈ I.

4 Matroids from dimensions

Van den Dries in [Dries89] gave a definition of dimension for definable sets; we will show that his approach is almost equivalent to ours. Let 𝕂 be a first order structure.

4. Matroids from dimensions

Definition 4.1. A dimension function on 𝕂 is a function d from definable sets in 𝕂 to {−∞} ∪ ℕ, such that, for all m ∈ ℕ and S, S_{1} and S_{2} definable subsets of 𝕂^{m}, we have:

(D1m 1) d(S) = −∞ iff S = ∅, d({a}) = 0 for every a ∈ 𝕂, d(𝕂) = 1.

(D2m 2) d(S_{1} ∪ S_{2}) = max(d(S_{1}), d(S_{2})).

(D3m 3) d(S^{σ}) = d(S) for every permutation σ of the coordinates of 𝕂^{m}.

(D4m 4) Let U be a definable subset of 𝕂^{m+1}, and, for i = 0, 1, let U(i) := {x ∈ 𝕂^{m} : d(U_{x}) = i}. Then, U(i) is definable with the same parameters as U, and d(U ∩ π^{−1}(U(i))) = d(U(i)) + i, i = 0, 1,

where π := Π_{m}^{m+1}.

Notice that the axiom (Dim 4) is slightly stronger that the original axiom in [Dries89]; however, after expanding 𝕂 by at most |T | many constants, the situation in [Dries89] can be reduced to ours.

Definition 4.2. Given a dimension function d on 𝕂, for every A ⊂ 𝕂 and b ∈ 𝕂 we define b ∈ cl^{d}(A) iff there exists X ⊆ 𝕂 definable with parameters in A, such that d(X) = 0 and b ∈ X.

Theorem 4.3. cl^{d} (more precisely, the extension of cl^{d} to a monster model) is an existential matroid with definable dimension. The dimension induced by cl^{d} is precisely d.

Conversely, if cl is an existential matroid with definable dimension, then dim^{cl} is a dimension function, and cl^{dim^{cl}} = cl.

Proof. The only non-trivial facts are that, if d is a dimension function, then cl^{d} is definable, satisfies EP and the existence axiom.

Definability) Let a ∈ cl(B). Let X ⊆ 𝕂 be B-definable such that d(X) = 0 and a ∈ X. Let ϕ(x, b̄) be the B-formula defining X. By (Dim 4), w.l.o.g. d(ϕ(𝕂, ȳ) ≤ 0 for every ȳ.^{3} Hence, ϕ(x, ȳ) is an x-narrow formula.

EP) Let a ∈ cl(Bc) \ cl(B). Assume, for contradiction, that c ∉ cl(Ba). Let X ⊆ 𝕂^{2} be B-definable, such that a ∈ X_{c} and d(X_{c}) = 0. Let X′ := X ∩ π^{−1}(X(0)), where π := Π_{1}^{2}. By assumption, ⟨c, a⟩ ∈ X′ and, by (Dim 4), dim(X′) ≤ 1; w.l.o.g., X = X′.

Let Z := {u ∈ 𝕂 : d(X^{u}) = 1}. Since c ∈ X^{a} and c ∉ cl(Ba), a ∈ Z. Since a ∉ cl(B), d(Z) = 1. Hence, by (Dim 4) and (Dim 3), d(X) = 2, absurd.

Existence) Immediate from Lemma 3.20-5.

3Here it is important that in (Dim 4) we asked that the parameters of U(i) are the same as the parameters of U.

5 Expansions

Remember that 𝕄 is a monster model of a complete L-theory T. We are interested in the behaviour of definable matroids under expansions of 𝕄. In this section we assume that cl is a definable closure operator on the monster model 𝕄.

def:cl-induced Definition 5.1. Given X ⊆ 𝕄, let the restriction cl^{X} : P(X) → P(X) and the relativisation cl_{X} : P(𝕄) → P(𝕄) of cl be defined as cl^{X}(Y) := cl(Y)∩X and cl_{X}(Y) := cl(XY).

Lemma 5.2. Given X ⊆ 𝕄, cl^{X} is a closure operator on X and cl_{X} is a closure operator on 𝕄. If moreover cl is a matroid, then both cl^{X} and cl_{X} are matroids, A | ⌣^{^{cl}^{X}}_{B} C iff A | ⌣^{cl}_{XB} C, and | ⌣^{^{cl}^{X}} is the restriction of | ⌣^{cl} to the subsets of X.

Remark 5.3. Given B ⊂ 𝕄 (with |B| < κ), let 𝕄_{B} be the expansion of 𝕄 with all constants from B.

1. if cl is definable, then cl_{B} is also definable (see Remark 3.27).

2. if cl is a matroid, then cl_{B} is also a matroid;

3. if cl is definable and satisfies Existence, then cl_{B} satisfies Existence too;

4. if cl is an existential matroid, then cl_{B} is also an existential matroid, and dim^{cl} and dim^{cl^{B}} coincide (the definable sets of 𝕄 and of 𝕄_{B} are the same).

Example 5.4. In the above Remark, it is not true that, if cl is a definable matroid, and cl_{B} satisfies Existence, then cl satisfies Existence. For instance, let B be any non-empty subset of 𝕄 (of cardinality less than κ), and cl = cl^{1}; then, cl_{B} = cl^{0} satisfies Existence, but cl does not.

lem:cl-pair Lemma 5.5. Let X ⊆ 𝕄. Let 𝕄′ be the expansion of 𝕄 with a predicate P for X. Assume that 𝕄′ is a monster model of T (X), and denote by cl′_{X} the closure operator cl′_{X}(Y) := cl(XY) on 𝕄′ (cl′_{X} coincides with cl_{X}).

1. If cl is definable, then cl′_{X} is definable on 𝕄′;

2. if cl is a matroid, then cl′_{X} is a matroid;

Proof. Let D ⊆ X such that |D| < κ and cl(X) = cl(D).

5. Expansions

1. b ∈ cl′_{X}(A) iff b ∈ cl(AX) iff 𝕄 ⊧ ϕ(b, ā, c̄) for some x-narrow formula ϕ(x, ȳ, z̄) and some c̄ ∈ X^{n}. Define ψ(x, ȳ) := ∃z̄ (P (z̄) & ϕ(x, ȳ, z̄)). Notice that ψ is an L(P)-formula, and that, for every ā′ ⊂ 𝕄, ψ(𝕄′, ā′) ⊆ cl′_{X}(ā′).

2. Trivial.

lem:cl-small-relative Lemma 5.6. Let 𝕄, X and 𝕄′ be as in the above lemma. Let ⟨𝔹, Y ⟩ ≺ ⟨𝕄, X⟩; assume moreover that cl is a definable closure operator on 𝕄. Then, for every A ⊆ 𝔹, cl_{Y} (A) ∩ 𝔹 = cl_{X}(A) ∩ 𝔹.

Hence, in the above situation, inside 𝔹 we do not need to distinguish between cl_{X} and cl_{Y}.

lem:elementary-pair-independent Lemma 5.7. Let cl be a definable matroid (not necessarily existential) and X, Y, X^{∗}, and Y^{∗} be elementary substructures of 𝕄, such that X ⊆ X^{∗} ∩ Y and X^{∗}∪Y ⊆ Y^{∗}. Let L^{2} be the expansion of L with a new unary predicate P, and consider ⟨Y, X⟩ and ⟨Y^{∗}, X^{∗}⟩ as L^{2}-structures. Assume that (Y, X) ⪯ (Y^{∗}, X^{∗}). Then, X^{∗} | ⌣^{cl}_{X} Y.

Proof. Let x̄^{∗} ⊂ X^{∗}; it suffices to prove that x̄^{∗} | ⌣^{cl}_{X} Y. However, tp_{L}(x^{∗}/Y) is finitely satisfied in X, and we are done.

Assume that 𝕄 expands a ring without 0 divisors. Let 𝕄′ be an expansion of 𝕄 to a larger language L′; assume that 𝕄′ is also a monster model and that cl′ is an existential matroid on 𝕄′. We have seen that in this case cl′ is the unique existential matroid on 𝕄′, and that, for every X definable subset of 𝕄′, dim′(X) = 0 iff F (X^{4}) ≠ 𝕄′ (where dim′ is the dimension induced by cl′). It is clear that cl′, in general, is not definable in 𝕄. However, the dimension function dim′ is definable in 𝕄: Hence, we can restrict the dimension function dim′ to the sets definable in 𝕄 (with parameters), and get a function dim.

lem:cl-expansion Lemma 5.8. Let 𝕄, 𝕄′, dim′ and dim be as above. Then, dim is a dimension function on 𝕄 (i.e., it satisfies the axioms in Definition 3.28). The matroid cl induces by dim is characterized by For every A and b, b ∈ cl(A) iff there exists X ⊆ 𝕄, definable in 𝕄 with parameters A, such that dim′(X) = 0.

Corollary 5.9. Assume that 𝕄 expands a ring without zero divisors. Let 𝕄′ be an expansion of 𝕄. If 𝕄′ is geometric, then 𝕄 is also geometric.

Compare the above corollary with [Adler05, Corollary 2.38 and Example 2.40].

6 Extension to imaginary elements

sec:imaginary

Again, 𝕄 is a monster model of a complete theory T, and cl is an existential matroid on 𝕄. Let 𝕄^{eq} be the set of imaginary elements, and T^{eq} be the theory of 𝕄^{eq}. Our aim is to extend to 𝕄^{eq} the matroid cl to a closure operator cl^{eq} and the rank rk^{cl} to a “rank function” r̃k (see Definition 6.6).

Notice that acl^{eq} is a closure operator on 𝕄^{eq} extending acl; however, if cl = acl, then in general cl^{eq} ≠ acl^{eq}; hence, when cl = acl , we will have to pay attention not to confuse the two possible extensions of cl to 𝕄^{eq} (cf. Remark 6.11).

On the other hand, by dcl^{eq} we will always denote the usual extension of dcl to imaginary element: a ∈ dcl(b) if Ξ(a/B) = {a}.

We will start with the definition of a ∈ cl^{eq}(B) when a is real and B is imaginary.

def:matt-1 Definition 6.1. Let B be a set of imaginary elements (of cardinality less than κ), and a be a real element. We say that a ∈ cl^{eq}(B) iff Ξ(a/B) has finite rk^{cl}.

Remark 6.2. If a and B are real, then a ∈ cl^{eq}(B) iff a ∈ cl(B).

Remark 6.3. If a is real a B is imaginary, then a ∈ cl^{eq}(B) iff rk^{cl}(Ξ(a/B)) < κ.

Proof. Assume that rk^{cl}(Ξ(a/B)) ≥ ℵ_{0}; we want to show that rk^{cl}(Ξ(a/B)) ≥ κ. For every λ < κ, consider the type in λ variables p(x̄) over B, saying that, for every i < λ, x_{i} ≡_{B} a, and the x_{i} are cl-independent. By assumption, p(x̄) is consistent, and hence satisfied in 𝕄 by λ-tuple (a_{i})_{i<λ}; thus, the a_{i} are λ-independent elements in Ξ(a/B). Therefore, rk^{cl}(Ξ(a/B)) > λ for every λ < κ; since κ is a limit cardinal, we are done.

Recall that 𝕄 has geometric elimination of imaginaries if every for imaginary tuple ā there exists a real tuple b̄ such that ā and b̄ are inter-algebraic. If 𝕄 had geometric elimination of imaginaries, we could define ā ∈ cl^{eq}(B) iff there exists a real tuple c̄ such that ā ∈ acl^{eq}(c̄) and c̄ ⊂ cl^{eq}(B). In the general case, we need a more involved definition and some preliminary lemmata.

Lemma 6.4 (Exchange Principle [Gagelman05, 3.1]). cl^{eq} satisfies the Exchange Principle for real points over imaginary parameters. That is, for a and b real elements and C imaginary, if a ∈ cl^{eq}(bC) \ cl^{eq}(C), then b ∈ cl^{eq}(aC).

6. Extension to imaginary elements

Proof. Let a, b, and C as in the hypothesis, and assume, for contradiction, that b ∉ cl^{eq}(aC). Let B be a real set (of cardinality less than κ), such that Ξ(a/bC) ⊆ cl(B). By enlarging B, w.l.o.g. we can assume that C ⊆ dcl^{eq}(B). Since a ∉ cl^{eq}(C), Ξ(a/C) ⊈ cl(B), and therefore there exists σ ∈ Aut(𝕄/C) such that a′ := a^{σ} ∉ cl(B). Since b ∉ cl^{eq}(aC), we have b^{σ} ∉ cl^{eq}(a′C), and therefore there exists b′ ≡_{a′C} b^{σ} such that b′ ∉ cl(a′B).

Notice that a′b′ ≡_{C} ab; let μ ∈ Aut(𝕄/C) such that a′ = a^{μ} and b′ = b^{μ}, and define B′ := B^{μ}. Since Ξ(a/bC) ⊆ cl(B), we have Ξ(a′/b′C) ⊆ cl(B′). Moreover, since C ⊆ dcl^{eq}(B′), we have Ξ(a′/b′B′) ⊆ Ξ(a′/b′C) ⊆ cl(B′). Thus, a′ ∈ cl(b′B) \ cl(B); hence, since a′, b′, and B are real, b′ ∈ cl(a′B), absurd.

It is relatively easy to also prove the following:

Lemma 6.5 (Transitivity). cl^{eq} is transitive for real sets over imaginary parameters: that is, if A is a imaginary,b̄ is a tuple of reals, and c is real, such that b̄ ⊆ cl^{eq}(A) and c ∈ cl^{eq}(Ab̄), then c ∈ cl^{eq}(A).

def:rank Definition 6.6. Let X be a set and rk be a function from finite subsets of X to ℕ. Let ā and b̄ vary among the finite tuples of X and B, C among the subsets of X. rk is a rank function if it satisfies the following conditions:

Finite character: for all ā and B there exists B′ ⊆ B finite, such that rk(ā/B′) = rk(ā/B).

Additivity: for every ā, b̄, and B, rk(āb̄/B) = rk(ā/b̄C) + rk(b̄/C).

Transitivity: for all B ⊆ C and all a, rk(ā/C) ≤ rk(ā/B).

cor:rk-real-im Corollary 6.7 ([Gagelman05]). • For any real tuple b̄ and imaginary set A, any two maximally A-cl^{eq}-independent sub-tuples of b̄ have the same cardinality; thus, we may use the notation rk^{cl}(−/−) accordingly, as long as the first argument is real.

• The function rk^{cl} defined above coincides with the usual one when both arguments are real.

• The function rk^{cl} has Finite Character, is Additive and Transitive, as long as the first argument is real.

• rk^{cl} satisfies Extension (for real first argument): if ā is real and B and C are imaginary sets, then there exists ā′ ≡_{C} ā such that rk^{cl}(ā′/BC) = rk^{cl}(ā/B).

• If ā is real, then ā ∈ cl^{eq}(C) iff rk^{cl}(ā/C) = 0.

Definition 6.8. Let A and ā be imaginary. Choose b̄ real, such that ā ⊆ acl^{eq}(Ab̄). Define rk^{cl}(ā/A) := rk^{cl}(b̄/A) − rk^{cl}(b̄/āA).

Lemma 6.9 ([Gagelman05, 3.3]). The above definition does not depend on the choice of b̄. Moreover, rk^{cl}(ā/A) is a natural number, and coincides with the one given in Corollary 6.7 when ā is real. Finally, rk^{cl} is a rank function on 𝕄^{eq}.

Proof. Let’s prove that rk^{cl} does no depend on the choice of b̄. Let b̄ and b̄′ be real tuples, such t hat ā ⊆ acl^{eq}(Ab̄) and ā ⊆ acl^{eq}(Ab̄′). W.l.o.g., b̄ ⊆ b̄′ and hence b̄′ = ⟨b̄, b̄′′⟩. We must prove that rk^{cl}(b̄b̄′′/A) − rk^{cl}(b̄b̄′′/āA) = rk^{cl}(b̄/A) − rk^{cl}(b̄/āA). The above is equivalent to rk^{cl}(b̄b̄′′/A) − rk^{cl}(b̄/A) = rk^{cl}(b̄b̄′′/āA) − rk^{cl}(b̄/āA). But the left hand side is equal to rk^{cl}(b̄b̄′′/b̄A), while the right hand side is equal to rk^{cl}(b̄b̄′′/āb̄A). Since ā ∈ acl^{eq}(b̄A), the two sides are equal.

Let’s prove additivity. Let ā and ā′ be imaginary tuples and C be a set of imaginary elements. We want to prove that rk^{cl}(āā′/C) = rk^{cl}(ā/ā′C) + rk^{cl}(ā′/C). Let b̄ and b̄′ be real tuples, such that ā ⊂ acl^{eq}(Cb̄) and ā′ ⊂ acl^{eq}(Cb̄′). We have to show that rk^{cl}(b̄b̄′/C)−rk^{cl}(b̄b̄′/āā′C) = rk^{cl}(b̄/ā′C)− rk^{cl}(b̄/āā′C) + rk^{cl}(b̄′/C) − rk^{cl}(b̄′/ā′C), that is

− rk^{cl}(b̄b̄′/C) + rk^{cl}(b̄b̄′/āā′C) + rk^{cl}(b̄/ā′C) − rk^{cl}(b̄/āā′C)+

+ rk^{cl}(b̄′/C) − rk^{cl}(b̄′/ā′C) = 0.

The above is equivalent to

− rk^{cl}(b̄/b̄′C) + rk^{cl}(b̄′/b̄āā′C) + rk^{cl}(b̄/ā′C) − rk^{cl}(b̄′/ā′C) = 0. (1) eq:rank-1

Let C′ := Cā′. Since ā ⊂ acl^{eq}(Cb̄) and ā′ ⊂ acl^{eq}(Cb̄′), (1) is equivalent to

− rk^{cl}(b̄/b̄′C′) + rk^{cl}(b̄′/b̄C′) + rk^{cl}(b̄/C′) − rk^{cl}(b̄′/C′) = 0.

Finally, rk^{cl}(b̄′/b̄C′) + rk^{cl}(b̄/C′) = rk^{cl}(b̄b̄′/C′) = rk^{cl}(b̄/b̄′C′) + rk^{cl}(b̄′/C′), and we are done.

Finally, we define a ∈ cl^{eq}(B) iff rk^{cl}(ā/B) = 0, where a and B can be either real of imaginaries.

Lemma 6.10. The operator cl^{eq} defined above is a closure operator, coincides with cl for real elements, and extends the operator defined in 6.1.

rem:acleq Remark 6.11. Assume that 𝕄 is pregeometric structure and cl = acl. Given b̄ a real or imaginary tuple, we have acl^{eq}(b̄) ⊆ cl^{eq}(b̄) and cl^{eq}(b̄)∩𝕄 = \underline{7. Density}

acl^{eq}(b̄) ∩ 𝕄. However, it is not true in general that cl^{eq} = acl^{eq}: more precisely, cl^{eq} = acl^{eq} iff 𝕄 is surgical [Gagelman05]. For instance, if 𝕄 is a model of the theory of p-adic fields, then 𝕄 is geometric but not surgical; the imaginary sort Γ corresponding to the value group has dimension 0 but it is infinite; therefore, Γ ⊂ cl^{eq}(∅) \ acl^{eq}(∅).

7 Density

sec:density

Again, 𝕄 is a monster model of a complete theory T, and cl is an existential matroid on 𝕄.

def:dense Definition 7.1. Let 𝕂 ⪯ 𝕄, and X ⊆ 𝕂. We say that X is dense in 𝕂 if, for every 𝕂-definable subset U of 𝕂, if dim^{cl}(U) = 1, then U ∩ X ≠ ∅. We define cl_{𝕂}(X) := cl(X) ∩ 𝕂, and we say that X is cl-closed in 𝕂 if cl_{𝕂}(X) = X.

Examples 7.2. 1. If 𝕂 is geometric, then X is dense in 𝕂 iff X intersects every infinite definable subset of 𝕂.

2. If 𝕂 is strongly minimal, then X is dense in 𝕂 iff X is infinite.

3. If 𝕂 is o-minimal and densely ordered, then X is dense in 𝕂 in the sense of the above definition iff X is topologically dense in 𝕂 (this is our motivation for the choice of the term “dense”). See also §9 for a generalisation of this example.

Remark 7.3. If X ⊂ 𝕂 is dense (in 𝕂), and a ∈ X, then X \ {a} is also dense.

Proof. If U ⊆ 𝕂 is definable and of dimension 1, then U \{a} is also definable and of dimension 1.

lem:density-closure Lemma 7.4. Let X ⊆ 𝕂 ⪯ 𝕄. If X is cl-closed and dense in 𝕂, then X ⪯ 𝕂.

Proof. Robinson’s test. Let A ⊆ 𝕂 be definable, with parameters from A: we must show that A ∩ X ≠ ∅. If dim^{cl}(A) = 1, this is true because X is dense in 𝕂. If dim^{cl}(A) = 0, this is true because X is cl-closed in 𝕂.

lem:Td-consistent-GCH Lemma 7.5. Let 𝕂 ⪯ 𝕄 be a saturated model of cardinality λ > |T |. Then, there exists X ⊂ 𝕂 such that X is a cl-basis of 𝕂 and X is dense in 𝕂. Moreover, there exists 𝔽 ≺ 𝕂 such that 𝔽 is cl-closed and dense in 𝕂 and 𝔽 is not equal to 𝕂.

Proof. Let (A_{i})_{i<λ} be an enumeration of all subsets of 𝕂 which are definable (with parameters from 𝕂) and of dimension 1. Build a cl-independent sequence (a_{i})_{i<λ} inductively: for every μ < λ, we make so that (a_{i})_{i<μ} is cl-independent, and, for every i < μ there exists j < μ such that a_{j} ∈ A_{i}. Let i_{μ} be the smallest index such that A_{i_{μ}} does not contain any a_{i} for i < μ.

Claim 3. i_{μ} exists.

Otherwise, 𝕂 would have a basis of cardinality less than λ, contradicting the saturation hypothesis.

Claim 4. There exists a_{μ} ∈ A_{μ} such that a_{μ} is cl-independent from (a_{i})_{i<μ}.

Otherwise, rk^{cl}(A_{μ}) < λ, absurd.

Define a_{μ} as in the above claim.

By construction, X′ := {a_{i} : i < λ} is cl-independent and dense in 𝕂; we can complete it to a cl-basis X, which is also dense.^{4}

Choose a ∈ X, let Y := X \ {a}, and 𝔽 := cl(Y). Since X is dense, Y is also dense, and therefore 𝔽 is dense in 𝕂. Moreover, since X is a cl-basis, a ∉ 𝔽. Finally, by Lemma 7.4, 𝔽 ≺ 𝕂.

The proof of the above lemma shows the following stronger results.

Corollary 7.6. Let 𝕂 be as in Lemma 7.5. Let c ∈ 𝕂 \ cl ∅. Then, there exists 𝔽 ≺ 𝕂 cl-closed and dense in 𝕂, such that c ∉ 𝔽.

Given 𝕂 ⊧ T, and X, Y subsets of 𝕂, we say that X is dense in 𝕂 w.r.t. Y if for every subset U of 𝕂 definable with parameters from X, if dim^{cl}(U) = 1, then U ∩ X ≠ ∅.

lem:Td-consistent Lemma 7.7. There exists 𝔽 ≺ 𝕂 ⊧ T such that 𝔽 is a proper dense and cl-closed subset of 𝕂.

Proof. If T has a saturated model of cardinality > |T |, we can apply Lemma 7.5. Otherwise, let 𝕂_{0} ≺ 𝕂_{1} ≺ … be an elementary chain of models of T, such that, for every n ∈ ℕ, 𝕂_{n+1} is (|𝕂_{n}|+|T |)^{+}-saturated, and let 𝕂 := ⋃_{n∈ℕ} 𝕂_{n}. Proceeding as in Lemma 7.5, for every n ∈ ℕ we build a cl-independent set A_{n} of elements in 𝕂_{n+1}, such that A_{n} ⊆ A_{n+1} and A_{n} is dense in 𝕂_{n+1} w.r.t. 𝕂_{n}. Let A := ⋃_{n} A_{n}. Then, A is a cl-independent set of elements in 𝕂, which is also dense in 𝕂. Conclude as in Lemma 7.5.

4Is it true or not that X′ is already a basis?

8. Dense pairs

8 Dense pairs

sec:dense-pairs

Let 𝔹 be a real closed field and 𝔸 a proper dense subfield of 𝔸, such that 𝔸 is also real closed. We call ⟨𝔹, 𝔸⟩ a dense pair of real closed fields, and we consider its theory, in the language of ordered fields expanded with a predicate for a (dense) subfield. Robinson [Robinson74] proved that the theory of dense pairs of real closed fields is complete. Van den Dries [Dries98] extended the results to o-minimal theories: if T is a complete o-minimal theory expanding the theory of (densely) ordered Abelian groups, then the theory of dense elementary pairs of models of T is complete. Macintyre [Macintyre75] introduced an abstract notion of density, in the context of geometric theories, which for o-minimal theories specialises to the usual topological notion, and proved various results; more recent work has been done in the context of so called “lovely pairs” of geometric or simple structures: see for instance [Berenstein07, BPV03].

In § 7 we also proposed an abstract notion of density, which for geometric theories specialises to the one given by Macintyre (and, independently, by others): see Remark 14.9. However, it is not true in general that the theory of dense pairs of models of T is complete (unless T is geometric): the main result of this section is that if T expands the theory of integral domains, and we add the additional condition that 𝔸 is cl-closed in 𝔹, we obtain a complete theory, which we denote by T^{d}. (if T is geometric, the condition is trivially true). We will also show that T^{d} admits an existential matroid (§8.4). For the proofs we will follow closely [Dries98].

𝕄 is a monster model of a complete theory T, and cl is an existential matroid on 𝕄. For this section, we will write dim instead of dim^{cl}, rk instead of rk^{cl}, and | ⌣ instead of | ⌣^{cl}.

Definition 8.1. Let L^{2} be the expansion of L by a new unary predicate P. Let T^{2} be the L^{2}-expansion of T, whose models are the pairs ⟨𝕂, 𝔽⟩, with 𝔽 ≺ 𝕂, 𝔽 ≠ 𝕂, and 𝔽 cl-closed in 𝕂.

Assume that dim is definable. Let T^{d} be the L^{2}-expansion of T saying that 𝔽 is cl-closed and dense in 𝕂 (we need definability of dim to express in a first-order way that 𝔽 is dense in 𝕂).

Notice that, by Lemma 7.4, T^{d} extends T^{2}. Notice that if cl = acl, then T^{2} is the theory of pairs ⟨𝕂, 𝔽⟩, with 𝔽 ≺ 𝕂 ⊧ T; however, if cl ≠ acl, then there exists 𝔽 ≺ 𝕄 with 𝔽 not cl-closed in 𝕄.

Lemma 8.2. T^{d} is consistent.

Proof. By Lemma 7.7.

Proviso. For the remainder of this section, we assume that T expands the theory of integral domains (and therefore dim is definable) and that ⟨𝕂, 𝔽⟩ ⊧ T^{d}.

thm:Td-complete Theorem 8.3. T^{d} is complete.

Definition 8.4. An L^{2} formula ϕ(x̄) is basic if it is of the form

∃ȳ(U(ȳ) & ψ(x̄, ȳ)),

where ψ is an L-formula,^{5}

thm:Td-eq Theorem 8.5. Each L^{2}-formula ψ(x̄) is equivalent, modulo T^{d}, to a Boolean combination of basic formulae, with the same parameters as ψ.

Theorems 8.3 and 8.5 will be proved in §8.2.

8.1 Small sets

In this subsection we will assume that ⟨𝕂, 𝔸⟩ ⊧ T^{2}.

def:small Definition 8.6. A subset X of 𝕂 is 𝔸-small if X ⊆ f(𝔸^{n}), for some Zapplication f : 𝕂^{n} ⇝ 𝕂 which is definable in 𝕂.

Definition 8.7. Let X ⊆ 𝕂^{n}. We say that X is weakly dense in 𝕂 if, for every definable U ⊆ 𝕂^{n}, if X ⊆ U, then dim(U) = n.

For instance, if cl = acl, then X is a weakly dense subset of 𝕂 iff X is infinite.

Remark 8.8. If X is a weakly dense subset of 𝕂, then X^{n} is a weakly dense subset of 𝕂^{n}.

lem:weakly-dense Lemma 8.9. If 𝕂 ⊧ T and 𝕂′ ⪯ 𝕂, then 𝕂′ is weakly dense in 𝕂.

Proof. W.l.o.g., the pair ⟨𝕂, 𝕂′⟩ is ω-saturated. Assume, for contradiction, that U ⊂ 𝕂 is definable, with parameters b̄ ∈ 𝕂^{n}, dim(U) = 0, and 𝕂′ ⊆ U. By saturation, rk(𝕂′) is infinite; let c̄ ∈ 𝕂^{′n+1} be independent elements. However, c̄ ∈ U, and therefore c̄ ⊂ cl(b̄), absurd.

The following result is the most delicate one.

5Basic formulae were called “special” in [Dries98].

8. Dense pairs  8.1. Small sets

lem:infinite-dimension Lemma 8.10 ([Dries98, 1.1]). Let f : 𝕂^{n+1} ⇝ 𝕂 be a Z-application 𝔸definable in 𝕂, and let b_{0} ∈ 𝕂 \ 𝔸. For every x ∈ 𝕂 and ȳ = ⟨y_{0}, …, y_{n}⟩ ∈ 𝕂^{n+1} let p(ȳ, x) := y_{0} + y_{1}x + ⋯ + y_{n}x^{n}, Then, there exists ā𝔸^{n+1} such that

p(ā, b_{0}) ∉ f(𝔸^{n} × {b}).

Proof. Otherwise there is for each ā ∈ 𝔸^{n+1} a tuple c̄ ∈ 𝔸^{n} such that p(ā, b_{0}) ∈ f(c̄, b_{0}). W.l.o.g., f is definable without parameters. For each ȳ ∈ 𝕂^{n+1} and z̄ ∈ 𝕂^{n} let D(ȳ, z̄) := {x ∈ 𝕂 : p(ȳ, x) ∈ f(z̄, x}. Define W := {⟨ȳ, z̄⟩ := dim(D(ȳ, z̄) = 1)}, and Y := Π_{n+1}^{2n+1}(W). Since b_{0} ∉ 𝔸 and 𝔸 is cl-closed, 𝔸^{n+1} ⊆ Y. Since Y is definable, Lemma 8.9 implies that dim(Y) = n + 1; therefore, dim(W) ≥ n + 1. Let Z := {z̄ ∈ 𝕂^{n} : dim({ȳ : ⟨ȳ, z̄⟩ ∈ W }) ≥ 1}. Since dim(W) ≥ n + 1 and dim(𝕂^{n}) = n, we have that Z is non-empty.

Choose c̄ ∈ Z. Let ā ∈ 𝕂^{n+1} such that ⟨ā, c̄⟩ ∈ W and rk(ā/c̄) ≥ 1. By definition of W, dim(D(ā, c̄) = 1; choose b ∈ D(ā, c̄) such that rk(b/c̄ā) = 1. Define d := p(ā, b); remember that d ∈ f(c̄, b), and therefore d ∈ cl(c̄b). Let ā′ ∈ 𝕂^{n+1} such that ā′ ≡_{c̄bd} ā and ā′ | ⌣_{c̄bd} ā. Since d ∈ cl(c̄, b), we haveā′ | ⌣_{c̄b} ā. Moreover, p(ā′, b) = d; therefore, p(ā − ā′, b) = 0.

If ā ≠ ā′, this implies that b is algebraic over ā − ā′, and therefore b ∈ cl(āā′), contradicting the fact that b ∉ \overline{(}āc̄) and ā′ | ⌣_{c̄b} ā.

If instead ā = ā′, then ā′ | ⌣_{c̄b} ā implies that ā ⊂ cl(c̄b), contradicting the facts that b ∉ cl(c̄ā) and rk(ā/c̄) ≥ 1.

Notice that the hypothesis of the above lemma can be weakened to: 𝕂 ⊧ T, 𝔸 is a proper cl-closed and weakly dense subset of 𝕂.

rem:pair-small Remark 8.11 ([Dries98, 1.3]). Each 𝔸-small subset of 𝕂 is a proper subset of 𝕂.

Proof. Same as [Dries98, Corollary 1.3].

rem:pair-small-union Remark 8.12. A finite union of 𝔸-small subsets of 𝕂 is also 𝔸-small.

Lemma 8.13. Let B ⊆ 𝕂 be a proper cl-closed subset. Then, B is co-dense in 𝕂; that is, 𝕂 \ B is dense in 𝕂.

Proof. Since B is cl-closed, F (B^{4}) ⊆ B. Assume, for contradiction, that there exists U definable in 𝕂, such that dim(U) = 1 and U ⊆ B. Then, F (U^{4}) = 𝕂, and therefore F (B^{4}) = 𝕂. However, since 𝕂 is cl-closed, F (B^{4}) ⊆ B contradicting the assumption that B ≠ 𝕂.

The hypothesis in the above lemma can be weakened to: B proper subset of 𝕂 ⊧ T, and F (B^{4}) ⊆ B. cor:dense-cardinality Lemma 8.14 ([Dries98, Lemma 1.5]). If the pair ⟨𝕂, 𝔸⟩ is λ-saturated, where

λ is an infinite cardinal and λ > |T |, then dim(𝕂/𝔸) ≥ λ. Hence, if |X| < λ, then cl(𝔸X) is co-dense in 𝕂.

Proof. Same as [Dries98, Lemma 1.5]. Let E be a generating set for 𝕂/𝔸, and suppose that |E| < λ. Let Γ(v) be the set of L^{2}-formulae of the form

∀y_{1} … ∀y_{n}(U(ȳ) → v ∉ f(ȳ, e_{1}, …, e_{p})),

where f(ȳ, z̄) is a Z-application ∅-definable in 𝕂, and e_{1}, …, e_{p} are in E. By Remarks 8.11 and 8.12, Γ(v) is a consistent set of formulae, with less than λ many parameters. By saturation, there exists b ∈ 𝕂 realising the partial type Γ(v). Thus b ∉ cl(𝔸E), absurd.

Notice that in the original [Dries98, Lemma 1.5], if T expands RCF, then van den Dries’ assumption that A is dense in B is superfluous; density is used if however T expands only the theory of ordered Abelian groups.

8.2 Proof of Theorems 8.3 and 8.5

sec:Td-proof

The proof is similar to the one in [Boxall09].

Definition 8.15. Let ⟨𝔹, 𝔸⟩ ⊧ T^{2} and C ⊆ 𝔹. Let c̄ be a tuple of elements from 𝔹^{eq}; the P -type of c̄, denoted by P - tp(c̄), is the information which tells us which members of c̄ are in 𝔸 (notice: the elements in c̄ are real or iminaginary, but only real elements can be in 𝔸). We say that c̄ is P -independent if c̄ | ⌣_{𝔸∩c̄}𝔸 (where, again, only the real element of c̄ can be in 𝔸 ∩ c̄).

We will use a superscript 1 to denote model-theoretic notions for L, and a superscript 2 to denote those notions for L^{2}: for instance, we will write a ≡_{C}^{1} a′ if the L-type of a and a′ over C is the same, and a ≡_{C}^{2} a′ the L^{2}-type of a and a′ over C is the same.

Both theorems are an immediate consequence of the following proposition.

prop:back-and-forth Proposition 8.16. Let ⟨𝔹, 𝔸⟩ and ⟨𝔹′, 𝔸′⟩ be models of T^{d}. Let c̄ be a (possibly infinite) P -independent tuple in 𝔹^{eq}, and c̄′ be an P -independent tuple in (𝔹′)^{eq} of the same length and the same sorts. If c̄ ≡^{1} c̄′ and P - tp(c̄) = P - tp(c̄′), then c̄ ≡^{2} c̄′.

Proof. Back-and-forth argument. Let λ < κ be a cardinal strictly greater than |T | and the length of c̄. W.l.o.g., we can assume that both ⟨𝔹, 𝔸⟩ and

8. Dense pairs  8.2. Proof of Theorems 8.3 and 8.5

⟨𝔹′, 𝔸′⟩ are λ-saturated. Let ē (resp. ē′) be the subtuple of c̄ (resp. of d̄′) of non-real elements. Let

Γ := {f : c̃ → c̃′ : c̄ ⊂ c̃ ⊂ 𝔹^{eq}, c̄′ ⊂ c̃′ ⊂ (𝔹′)^{eq},

c̃ & c̃′ of the same length less than λ and of the same sorts, with all non-real elements of c̃ in ē, f is a bijection,

c̃ & c̃′ are P -independent, c̃ ≡^{1} c̃′, P - tp(c̃) = P - tp(c̃′)}.

We want to prove that Γ has the back-and-forth property. So, let f : c̃ → c̃′ be in Γ, and d ∈ 𝔹\c̄; we want to find g ∈ Γ such that g extends f and d is in the domain of g. W.l.o.g., c̃ = c̄ and c̃′ = c̄′. Let ā := c̄∩ 𝔸, and ā′ := c̄′ ∩ 𝔸′. Notice that f(ā) = ā′ and that 𝔸 ∩ cl(c̄) = 𝔸 ∩ cl(ā), and similarly for c̄′. We distinguish some cases. Case 1. d ∈ 𝔸 ∩ cl(c̄) = 𝔸 ∩ cl(ā). Notice that c̄d | ⌣_{ād} 𝔸, and therefore c̄d is P -independent. There is a x-narrow formula ϕ(x, ȳ) such that 𝔹 ⊧ ϕ(d, ā). Choose d′ ∈ 𝔸′ such that c̄d ≡^{1} c̄′d′; therefore, d′ ∈ cl(ā′), and thus c̄′d′ is also P -independent and has the same P -type as c̄d. Thus, we can extend f to c̄d setting g(d) = d′. Case 2. d ∈ 𝔸 \ cl(c̄) = 𝔸 \ cl(ā). Since c̄ | ⌣ ā 𝔸 and c ∈ 𝔸, we have c̄ | ⌣_{ād} 𝔸, and therefore c̄d is P -independent. Let q(x) := tp^{1}(d/c̄), and q′ := f(q) ∈ S_{1}^{1}(c̄′). Notice that q | ⌣ ā c̄ (because d | ⌣ ā c̄), and therefore q′ | ⌣_{ā′} c̄′. Since 𝔸′ is dense in 𝔹′ and ⟨𝔹′, 𝔸′⟩ is λ-saturated, there exists d′ ∈ 𝔸′ realizing q′. It is now easy to see that c̄′d′ is P -independent, and that we can extend f to c̄d setting g(d) = d′. Case 3. d ∈ cl(c̄𝔸) \ 𝔸. Let ā_{0} ∈ 𝔸^{n} such that d ∈ cl(b̄ā_{0}) (ā_{0} exists because cl is finitary). By applying n times cases 1 or 2, we can extend f to f′ ∈ Γ such that ā_{0} is a subset of the domain of f′. By substituting f with f′, we are reduced to the case that d ∈ cl(c̄) \ 𝔸. Hence, c̄ | ⌣ ā 𝔸 and d ∈ cl(c̄): therefore, c̄d | ⌣ ā 𝔸, and hence c̄d is P -independent. Let d′ ∈ 𝔹′ such that d′c̄′ ≡^{1} dc̄. For the same reason as above, c̄′d′ is also P -independent. It remains to show that c̄d and c̄′d′ have the same P -type, that is that d′ ∉ 𝔸′. If, for contradiction, d′ ∈ 𝔸′ ∩ cl(c̄), then d′ ∈ cl(ā); therefore, there would be a x-narrow-formula witnessing it, and thus d ∈ cl(ā) ⊆ 𝔸, absurd. Case 4. d ∉ cl(c̄𝔸). Let ā_{0} ⊂ 𝔸 be of cardinality less than λ such that d | ⌣_{ā_{0}ā} 𝔸 (ā_{0} exists because | ⌣ satisfies Local Character). By applying cases 2 and 3 sufficiently many times, we can extend f to f′ ∈ Γ such that ā_{0} is contained in the domain of f′; thus, w.l.o.g., d | ⌣ ā P. Let d′ ∈ Am′ such that d′c̄′ ≡^{1} dc̄; moreover, by Lemma 8.14 we can also assume that d′ | ⌣_{ā′} 𝔸′. We need only to show that d′ ∉ 𝔸′. Assume, for contradiction, that d′ ∈ 𝔸′ and d′ | ⌣_{ā′} 𝔸′; then, d′ | ⌣_{ā′} d′, thus d′ ∈ cl(ā′), and hence d ∈ cl(ā), absurd.

8.3 Additional facts

Reasoning as in [Dries98, 2.6–2.9], from Theorems 8.3 and 8.5, and Proposition 8.16, we can deduce the following facts. We are still assuming that T expands an integral domain. We will say that A is T^{2}-algebraically closed if A is a subset of some pair ⟨𝔹, ℂ⟩ ⊧ T^{d} and A is algebraically closed w.r.t. the language L^{2}, and similarly for T^{2}-definably closed. We denote by acl^{1} the algebraic closure in T, and by acl^{2} the algebraic closure in T^{2}. We denote by ≡^{1} elementary equivalence w.r.t. the language L, and by We denote by ≡^{2} elementary equivalence w.r.t. the language L^{2}. Similarly, tp^{1}(b̄/X) will denote the L-type of b̄ over X, while tp^{2}(b̄/X) will denote the L^{2} type.

cor:vdd-26 Corollary 8.17 ([Dries98, 2.6]). Let ⟨𝔹, 𝔸⟩ be a model of T^{d}. Suppose Y ⊆ 𝔹^{n} is A_{0}-definable in ⟨𝔹, 𝔸⟩, for some A_{0} ⊂ 𝔸. Then Y ∩ 𝔸^{n} is A_{0}-definable in 𝔸.

cor:vdd-27 Corollary 8.18 ([Dries98, 2.7]). Let ⟨𝔹, 𝔸⟩ and ⟨𝔹′, 𝔸′⟩ be models of T^{d}, such that ⟨𝔹′, 𝔸′⟩ ⊆ ⟨𝔹, 𝔸⟩ and 𝔹′ and 𝔸 are cl-independent over 𝔸′. Then, ⟨𝔹′, 𝔸′⟩ ⪯ ⟨𝔹, 𝔸⟩. In particular, if 𝔸 ≺ 𝔹′ ⪯ 𝔹, with 𝔸 ≠ 𝔹′, then ⟨𝔹′, 𝔸⟩ ⪯ ⟨𝔹, 𝔸⟩.

cor:Td-2extensions Corollary 8.19 ([Dries98, 2.8]). Let A ⊆ B ⊂ 𝕄 be substructures. Assume that ⟨B, A⟩ have extensions ⟨𝔹_{1}, 𝔸_{1}⟩ ⊧ T^{d} and ⟨𝔹_{2}, 𝔸_{2}⟩ ⊧ T^{d}, such that B | ⌣_{A} 𝔸_{k} and B ∩ 𝔸_{k} = A, k = 1, 2. Then, ⟨𝔹_{1}, 𝔸_{1}⟩ ≡_{B}^{2} ⟨𝔹_{2}, 𝔸_{2}⟩, that is ⟨𝔹_{1}, 𝔸_{1}⟩ and ⟨𝔹_{2}, 𝔸_{2}⟩ satisfy the same L^{2}-formulae with parameters from B. More generally, for every ā_{1} ∈ (𝔸_{1})^{n} and ā_{2} ∈ (𝔸_{2})^{n}, if ā_{1} ≡_{B}^{1} ā_{2}, then ā_{1} ≡_{B}^{2} ā_{2}; that is, if ā_{1} and ā_{2} realise the same L-types over B in 𝔹_{1} and 𝔹_{2} respectively, then they realise the same L^{2}-type over B in ⟨𝔹_{1}, 𝔸_{1}⟩ and ⟨𝔹_{2}, 𝔸_{2}⟩ respectively.

Notice that the hypothesis of the above Corollary implies that A is cl-closed (but not necessarily dense) in B.

Proof. Let c̄_{k} := Bā_{k}. c̄_{1} and c̄_{2} have the same P -type, they are both P -in-dependent, and c̄_{1} ≡^{1} c̄_{2}; the conclusion follows from Proposition 8.16.

cor:Td-type-1 Corollary 8.20 ([Dries98, 2.9]). Let ⟨𝔹_{1}, 𝔸_{1}⟩ ⊧ T^{d} and ⟨𝔹_{2}, 𝔸_{2}⟩ ⊧ T^{d}, and let A be a common subset of 𝔸_{1} and 𝔸_{2}. Suppose that b_{1} ∈ 𝔹_{1} \ 𝔸_{1} and b_{2} ∈ 𝔹_{2} \ 𝔸_{2} realise the same L-types over A in 𝔹_{1} and 𝔹_{2} respectively, that is b_{1} ≡_{A}^{1} b_{2}. Then, they realise the same L^{2}-types over A in ⟨𝔹_{1}, 𝔸_{1}⟩ and ⟨𝔹_{2}, 𝔸_{2}⟩ respectively, that is b_{1} ≡_{A}^{2} b_{2}.

8. Dense pairs  8.3. Additional facts

Proof. Let c̄_{i} := b_{i}𝔸_{i}, i = 1, 2. By assumption, c̄_{1} ≡^{1} c̄_{2}, they have the same P -type, and they are both P -independent. The conclusion follows from Proposition 8.16.

For the remainder of this section, we will assume that ⟨𝔹, 𝔸⟩ is a model of T^{d}, and that λ is a cardinal number such that κ > λ > max(|T |, |𝔹|).

lem:thm-2 Lemma 8.21 ([Dries98, Theorem 2]). Let b̄ ⊂ 𝔹 be P -independent. Given a set Y ⊂ 𝔸^{n}, t.f.a.e.:

1. Y is T^{2}-definable over b̄;

2. Y = Z ∩ 𝔸^{n} for some set Z ⊆ 𝔹^{n} that is T -definable over b̄.

Proof. (1 ⇒ 2) is as in [Dries98, Theorem 2]. (2 ⇒ 1) is obvious.

Lemma 8.22 ([Dries98, 3.1]). 𝔸 is T^{2}-algebraically closed in ⟨𝔹, 𝔸⟩.

Proof. As in [Dries98, 3.1]: let b ∈ 𝔹 \ 𝔸. Let ⟨𝔹^{∗}, 𝔸^{∗}⟩ ⪰ ⟨𝔹, 𝔸⟩ be a monster model. Since cl is existential, and b ∉ cl(𝔸), there exists infinitely many distinct b′ ∈ 𝔹^{∗} such that b ≡_{𝔸}^{1} b′. By Corollary 8.20, b ≡_{𝔸}^{2} b′. Thus, b is not T^{2}-𝔸-algebraic in ⟨𝔹^{∗}, 𝔸^{∗}⟩, and therefore not T^{2}-𝔸-algebraic in ⟨𝔹, 𝔸⟩.

lem:3.2 Lemma 8.23 ([Dries98, 3.2]). Let A_{0} ⊆ 𝔸 be T -algebraically closed (T -definably closed). Then A_{0} is T^{2}-algebraically closed (resp. T^{2}-definably closed).

Proof. Assume A_{0} is T -algebraically closed. Let c ∈ acl^{2}(A_{0}), and C := {c_{1}, …, c_{n}} be the set of L^{2}-conjugates of c/A_{0}. By definition, C is A_{0}- definable in ⟨𝔹, 𝔸⟩, and, by the above Lemma, C ⊂ 𝔸. Hence, by Corollary 8.17, C is A_{0}-definable in 𝔸. The case when A_{0} is T -definably closed is similar.

lem:orbit-dense Lemma 8.24. Let ⟨𝔹, 𝔸⟩ be a κ-saturated model of T^{d}. Let D ⊂ 𝔹 such that |D| < λ, and c ∈ 𝔹 \ cl(D). Define C := {c′ ∈ 𝔹 : c′ ≡_{D}^{1} c} ∩ A. Then, |C| ≥ λ.

Proof. Consider the following partial L^{2}-type over D:

p(x_{i} : i < λ) := (⋀_{i} x_{i} ≡_{D}^{1} c) & (⋀_{i} U(x_{i})) & (⋀_{i<j} x_{i} ≠ x_{j}).

Claim 5. p is consistent.

If not, there exist d̄ ⊂ D, b̄ ⊂ 𝔹, ϕ(x, d̄) ∈ tp^{1}(c/D), such that ϕ(𝔹, d̄) \ 𝔸 = b̄. Let X := ϕ(𝔹, d̄) \ b̄: notice that X is definable in 𝔹, and X ⊆ 𝔸. Hence, since 𝔸 is co-dense in 𝔹, we conclude that dim(X) ≤ 0, and therefore dim(ϕ(𝔹, d̄)) ≤ 0. Thus, c ∈ cl(d̄) ⊆ cl(D), absurd.

The conclusion follows immediately from the claim. prop:3.3 Proposition 8.25 ([Dries98, 3.3]). Let b̄ ⊂ 𝔹 be P -independent. Then, dcl^{2}(b̄) =

dcl^{1}(b̄), and similarly for the algebraic closure. Let c ∈ 𝔹^{eq} (i.e., c is an imaginary element for the structure 𝔹). Then, c ∈ dcl^{2}(b̄) iff c ∈ dcl^{1}(b̄), and similarly for the algebraic closure.

Idea for proof. W.l.o.g., we can assume that ⟨𝔹, 𝔸⟩ is λ-saturated and that b̄ has finite length. So, let c be a 𝔹-imaginary such that c ∈ acl^{2}(b̄). We have to prove that c ∈ acl^{1}(b̄).

Let 𝔹_{1} := cl^{𝔹}(𝔸b̄); by Corollary 8.18, ⟨𝔹_{1}, 𝔸⟩ ⪯ ⟨𝔹, 𝔸⟩, and in particular 𝔹_{2} is T^{2}-algebraically closed in ⟨𝔹, 𝔸⟩, and therefore c ∈ 𝔹_{1}^{eq}. Let n ≥ 0 minimal and ā ∈ 𝔸^{n} such that c ∈ cl(b̄ā).

cl:33-2 Claim 6. c ∈ cl(b̄), i.e. n = 0.

If n > 0, by substituting b̄ with b̄a_{1}, …, a_{n−1}, and proceeding by induction on n, we can reduce to the case n = 1; let a := a_{1}. Consider the following partial L-type over b̄a:

q(x) := (x ≡1 ; b̄ a) & (x | ⌣_{b̄} a).

Since | ⌣ satisfies Existence, q is consistent. Let d ∈ 𝔹 be any realisation of q. Since d | ⌣ b̄ a, we conclude that either d ∉ cl(b̄a) or d ∈ cl(b̄). However, the latter cannot happen, since d ≡1 ; b̄ a ∉ cl(b̄); thus, d ∉ cl(b̄a), and therefore dim(q) = 1. Hence, since 𝔸 is dense in 𝔹 and ⟨𝔹, 𝔸⟩ is ω-saturated, there exists a′ ∈ 𝔸 satisfying q. Reasoning in the same way, we can show that there exists (a_{2}, a_{3}, a_{4}, …) a Morley sequence in q contained in 𝔸. By Corollary 8.19, a_{i} ≡2 ; b̄ a for every i. Let c_{1}, c_{2}, …, c_{m} be all the L^{2}-conjugates of c over b̄ (there are finitely many of them), and let ϕ(x, y, z̄) be an x-narrow L-formula without parameters such that 𝔹 ⊧ ϕ(c, a, b̄).

The L-formula (in y, with parameters in b̄c_{1}, …, c_{m}) ⋁_{i} ϕ(c_{i}, y, b̄) is equivalent to an L^{2}-formula in y with parameters b̄; hence, every a_{i} satisfies it (because a_{i} ≡2 ; b̄ a). Hence, w.l.o.g. c_{1} ∈ cl(b̄a_{2}) ∩ cl(b̄a_{3}) = cl(b̄) (because a_{2} | ⌣ b̄ a_{3}). Therefore, c ∈ cl(b̄).

It remains to show that c ∈ acl^{1}(b̄). Let c_{2} ∈ 𝔹^{eq} such that c_{2} ≡1 ; b̄ c. Since 𝔹 is ω-saturated, it suffices to prove that there are only finitely many such c_{2}. Since c ∈ acl^{2}(b̄), it suffices to prove that c_{2} ≡2 ; b̄ c. Let b̄_{1} := b̄c, b̄_{2} := b̄c_{2}, and ā := b̄ ∩ 𝔸. By assumption, b̄_{1} ≡^{1} b̄_{2}. By Claim 6, we have b̄_{1} ⊆ cl(b̄), and therefore, since b̄ | ⌣ ā 𝔸, b̄_{1} is P -independent. Claim 6 also implies that b̄_{2} ⊆ cl(b̄), and hence b̄_{2} is also P -independent. It remains to show that b̄_{1} and b̄_{2} have the same P -type. Assume e.g. that c ∈ 𝔸. Since b̄ | ⌣ ā 𝔸, we have that c ∈ cl(ā), and therefore c_{2} ∈ cl^{𝔹}(ā) ⊆ 𝔸.

The other assertions are proved in the same way.

8. Dense pairs  8.4. The small closure

8.4 The small closure

subsec:small-closure

We will are still assuming that T expands an integral domain. Let 𝕄^{∗} := ⟨𝔹^{∗}, 𝔸^{∗}⟩ be a κ-saturated and strongly κ-homogeneous monster model of T^{d}, and ⟨𝔹, 𝔸⟩ ≺ 𝕄^{∗}, with |𝔹| < κ. Notice that rk(𝔹^{∗}/𝔸^{∗}) ≥ κ.

Definition 8.26. For every X ⊆ 𝔹^{∗} we define the small closure of X as

Scl(X) := cl(X𝔸^{∗}).

For lovely pairs (e.g., dense pairs of o-minimal structures), the small closure was already defined in [Berenstein07].

Remark 8.27. Scl is a definable matroid (on 𝕄^{∗}).

Proof. Scl coincides with the operator cl_{𝔸^{∗}} in Lemma 5.5.

Notice that we can apply Lemma 5.6: Scl^{𝔹} = (cl^{𝔹})_{𝔸}: we can “compute” the small closure of a subset of 𝔹 inside 𝔹 using 𝔸 instead of 𝔸^{∗}.

We want to prove that Scl is existential; we will need a preliminary lemma.

Lemma 8.28. Let b ∈ 𝔹^{∗} \ 𝔸^{∗}. Define 𝕄_{b}^{∗} the expansion of 𝕄^{∗} with a constant for b, and Scl_{b}(X) := Scl(bX) = cl(X𝔸^{∗}b). Then, Scl_{b} is an existential matroid on 𝕄_{b}^{∗}.

Proof. That Scl_{b} is a definable matroid follows from Lemma 5.5, applied to Scl. Let X ⊆ 𝕄^{∗}, and Y := Scl_{b}(X). Claim 7. Y ≺ 𝕄^{∗} (as an L^{2}-structure).

By Lemma 7.4, Y is an elementary L-substructure of 𝔹^{∗}. By Theorems 8.3, and 8.5 and, again, Lemma 7.4, it suffices to show that 𝔸^{∗} is a cl-closed, dense, and proper subset of Y, which is trivially true.

The lemma then follows from the above claim and Lemma 3.22; nontriviality follows from the fact that Zrk(𝔹^{∗}/𝔸^{∗}) ≥ κ.

lem:scl Lemma 8.29. Scl is an existential matroid.

Proof. The only thing that needs proving is Existence. Define Ξ^{2}(a/C) as the set of conjugates of a over C in 𝕄. Assume that Ξ^{2}(a/C) ⊆ Scl(CD). We want to prove that a ∈ Scl(C). Choose b, b′ ∈ 𝔹^{∗} which are cl-independent over 𝔸^{∗}C. By applying the previous lemma to Scl_{b} and Scl_{b′}, we see that

a ∈ Scl_{b}(C) ∩ Scl_{b′}(C) = cl(𝔸^{∗}Cb) ∩ cl(𝔸^{∗}Cb′) = cl(𝔸^{∗}C) = Scl(C).

Hence, we can define the dimension induced by Scl, and denote it by Sdim. Notice that, by Theorem 3.47, Scl is the only existential matroid on T^{d}.

Lemma 8.30. Let X ⊆ (𝔹^{∗})^{n} be definable in 𝔹^{∗}. Then Sdim(X) = dim(X).

Proof. From cl ⊆ Scl follows immediately that Sdim(X) ≤ dim(X). For the opposite inequality, we proceed by induction on k := dim(X). Assume, for contradiction, that Sdim(X) < k. W.l.o.g., dim(Π_{k}^{n}(X)) = k; therefore, w.l.o.g. k = n. If k = 1, then Sdim(X) = 0, and therefore F^{4}(X) ≠ 𝔹^{∗}^{n}, contradicting dim(X) = 1. For the inductive step, assume k = n > 1, and let U := {a ∈ 𝔹^{n} : dim(X_{a}) = n − 1}. U is definable in 𝔹, and therefore, by inductive hypothesis, Sdim(U) = dim(U) = n − 1. By the case k = 1, for every a ∈ 𝕂^{n−1}, dim(X_{a}) = Sdim(X), and therefore Sdim(X_{a}) = 1 for every a ∈ U. Thus, Sdim(X) = n.

Definition 8.31. Let X ⊆ (𝔹^{∗})^{n} be definable in ⟨𝔹^{∗}, 𝔸^{∗}⟩. We say that X is small if Sdim(X) = 0. Let Y ⊆ 𝔹^{n} be definable in ⟨𝔹, 𝔸⟩. We say that Y is small if Sdim(Y^{∗}) = 0, where Y^{∗} is the interpretation of Y inside ⟨𝔹^{∗}, 𝔸^{∗}⟩.

Notice that, if X ⊂ 𝔹^{n} is 𝔸-small (in the sense of Definition 8.6), then X is also small in the above sense. The next lemma shows that the converse is also true.

lem:pair-small Lemma 8.32. Let ⟨𝔹, 𝔸⟩ ⪯ ⟨𝔹^{∗}, 𝔸^{∗}⟩ and X ⊆ 𝔹^{n} be definable in ⟨𝔹, 𝔸⟩. Let X^{∗} be the interpretation of X inside ⟨𝔹^{∗}, 𝔸^{∗}⟩. Let c̄ ∈ 𝔹^{k} be the parameters of definition of X. T.f.a.e.:

1. X is small;

2. X^{∗} is small;

3. X^{∗} ⊆ Scl(b̄) for some finite tuple b̄ ⊂ 𝔹^{∗};

4. X^{∗} ⊆ Scl(c̄);

5. X^{∗} ⊆ cl(c̄𝔸^{∗});

6. X is 𝔸-small: that is, there exists a Z-application f^{∗} : (𝔹^{∗})^{m} ⇝ (𝔹^{∗})^{n}, definable in 𝔹^{∗} with parameters, such that f^{∗}(𝔸^{∗}^{m}) ⊇ X^{∗};

7. X^{∗} is 𝔸^{∗}-small: that is, there exists a Z-application f : 𝔹^{m} ⇝ 𝔹^{n}, definable in 𝔹 with parameters c̄, such that f^{∗}(𝔸^{∗}^{m}) ⊇ X^{∗}, where f^{∗} is the interpretation of f in 𝔹^{∗};

8. there exists a Z-application g : 𝔹^{m+k} ⇝ 𝔹^{n}, definable in 𝔹 without parameters, such that g^{∗}(𝔸^{∗}^{m} × {c̄}) ⊇ X^{∗};

9. there exists a Z-application f : 𝔹^{m} ⇝ 𝔹^{n}, definable in 𝔹 without parameters, such that f(𝔸^{m} × {c̄}) ⊇ X.

Proof. The only non-trivial implication is (5 ⇒ 7), which is proved by a compactness argument using Remark 3.57.

Conjecture 8.33 ([Dries98, 3.6]). Let f : 𝔸^{n} → 𝔸 be T^{2}-definable with parameters b̄. Let ā ⊂ 𝔸^{m} such that b̄ | ⌣ ā 𝔸 and dcl^{1}(b̄ā) ∩ 𝔸 = dcl^{1}(ā). Then, f is given piecewise by functions definable in 𝔸 with parameters ā.

Idea for proof. By compactness, it suffices to show that, given an elementary extension ⟨𝔹^{∗}, 𝔸^{∗}⟩ of ⟨𝔹, 𝔸⟩ and a point a^{∗} ∈ (𝔸^{∗})^{n}, we have f(a^{∗}) ∈ 𝔸′ := dcl^{1}(āa^{∗}). Let 𝔹′ := dcl^{1}(b̄āa^{∗}). Since b̄ | ⌣ ā 𝔸, we have 𝔹′ | ⌣_{𝔸′} 𝔸. Hence, by Proposition 8.25, 𝔹′ is dcl^{2}-closed, and therefore f(a^{∗}) ∈ 𝔹′. Hence,

f(a^{∗}) ∈ dcl^{1}(b̄āa^{∗}) ∩ 𝔸^{∗} = cl(b̄āa^{∗}) ∩ 𝔸^{∗} ∩ dcl^{1}(b̄āa^{∗}) = cl(āa^{∗}) ∩ dcl^{1}(b̄āa^{∗}).

It remains to show that cl(āa^{∗}) ∩ dcl^{1}(b̄āa^{∗}) = dcl^{1}(āa^{∗}).

Lemma 8.34 ([Boxall09, 6.1.3]). Let f : 𝔸^{n} → 𝔹 be T^{2}-definable with parameters b̄. Assume that b̄ is P -independent. Then, there exists g : 𝔹^{n} → 𝔹 which is T -definable with parameters b̄, and such that f = g ↾ 𝔸^{n}.

Proof. Let ⟨𝔹^{∗}, 𝔸^{∗}⟩ be an elementary extension of ⟨𝔹, 𝔸⟩ and a^{∗} ∈ (𝔸^{∗})^{n}. By Proposition 8.25, there exists a function g_{i} : 𝔹^{n} → 𝔹 which is T -definable with parameters b̄, such that f(a) = g_{i}(a). By compactness, finitely many g_{i} will suffice. The conclusion then follows from Lemma 8.21.

prop:3.5 Proposition 8.35 ([Dries98, 3.5]). Let b̄ ∈ 𝔹^{k} and ā ∈ 𝔹^{k′} such that b̄ | ⌣ ā 𝔸 and b̄ ∩ 𝔸 ⊆ ā. Let X ⊆ 𝔹^{n} be T -definable (possibly, inside an imaginary sort) with parameters b̄, such that dim(X) = d. Let Y ⊆ X be T^{2}-definable, with parameters b̄. Then, there exist S ⊂ X which is T^{2}-definable with parameters b̄, and Z ⊆ X which is T^{2}-definable with parameters b̄ā, such that Z Δ Y ⊆ S and Sdim(S) < d.

In particular, if dim(X) = 0, then every T^{2}-definable subset of X is already T -definable.

Proof. The proof is a variant of Beth’s definability theorem, using Proposition 8.16. W.l.o.g., ⟨𝔹, 𝔸⟩ is λ-saturated for some cardinal λ such that |T | < λ < κ.

Let W := {p ∈ S_{X}^{2}(āb̄) : Sdim(p) = d}. Notice that W is a closed subset of S_{X}^{2}(āb̄) (the Stone space of T^{2}-types over āb̄ containing the formula “x ∈ X”). Let θ : S_{X}^{2}(āb̄) → S_{X}^{1}(āb̄) be the restriction map: notice that θ is a continuous homomorphism, and therefore V := θ(W) is closed in S_{X}^{1}(āb̄). Let ρ := θ ↾ W. Claim 8. ρ is injective (and therefore ρ is a homeomorphism between W and V).

We have to prove that, for every c̄ and c̄′ ∈ X, if Srk(c̄/āb̄) = Srk(c̄′/āb̄) = d and c̄ ≡_{āb̄}^{1} c̄′, then c̄ ≡_{āb̄}^{2} c̄′. Let d̄:= āb̄c̄ and d̄′ := āb̄c̄′. By Proposition 8.16, it suffices to prove that d̄ and d̄′ are both P -independent and have the same P -type. Since Srk(c̄/āb̄) = d and c̄ ∈ X, we have that Srk(c̄/āb̄) = rk(c̄/āb̄), which is equivalent to c̄ | ⌣_{āb̄} 𝔸, and hence (since b̄ | ⌣ ā 𝔸) d̄ | ⌣ ā 𝔸, that is d̄ is P -independent, and similarly for d̄′. It remains to show that d̄ and d̄′ have the same P -type. Let d_{i} ∈ 𝔸; we have to prove that d′_{i} ∈ 𝔸. Since d̄ | ⌣ ā 𝔸, we have d_{i} ∈ cl(ā), and hence d′_{i} ∈ cl^{𝔹}(ā′) ⊆ 𝔸.

Let U := S_{Y}^{2} (āb̄) ∩ W; since Y is definable, U is clopen in W, and since ρ is a homeomorphism, ρ(U) is clopen in V. Hence, there exists Z subset of X, such that Z is T -definable over āb̄ and V ∩ S_{Z_{1}}^{1}(āb̄) = ρ(U). Claim 9. There exists S ⊂ X which is T_{2}-definable over b̄, such that Sdim(S) < d and Y Δ Z ⊆ S.

Assume not. Then, the following partial type over āb̄ is consistent:

Φ(x̄) := x̄ ∈ X & x̄ ∈ Y Δ Z & x̄ ∉ S,

where S varies among the subsets of X which are T^{2}-definable over b̄, with Sdim(S) < d. Let c̄ ∈ X be a realization of Φ and p := tp^{2}(c̄/āb̄) ∈ S_{X}^{2}(āb̄). By assumption, Sdim(c̄/āb̄) = d, and therefore p ∈ W. Hence, ρ(p) = tp^{1}(c̄/āb̄) ∈ V. Since ρ is injective, we have

ρ(p) ∈ ρ(S_{Y}^{2} (āb̄) ∩ W ) Δ ρ(S_{Z}^{2}(āb̄) ∩ W ) ⊆ S_{Z}^{1}(āb̄) Δ S_{Z}^{1}(āb̄) = ∅,

absurd.

In general, given b̄ ∈ 𝔹^{n}, it is always possible to find ā ∈ 𝔸^{n′} such that b̄ | ⌣ ā 𝔸. However, there are some examples when 𝔹 is o-minimal, but ā cannot be found inside dcl^{2}(b̄).^{6}

cor:3.4 Corollary 8.36 ([Dries98, 3.4]). Let b̄ and ā be as in the above Proposition. Let Γ be a T -definable set (possibly, in some imaginary sort) over b̄, and let the function f : 𝔹^{n} → Γ be T^{2}-definable with parameters b̄. Then, there exist S ⊆ 𝔹^{n}, which is T^{2}-definable over b̄ and with Sdim(S) < n, and f̂: 𝔹^{m} → Γ, which is T -definable over b̄ā, such that f agrees with f̂ outside S.

Proof. W.l.o.g., ⟨𝔹, 𝔸⟩ is κ-saturated. Claim 10. There exists a set S ⊂ 𝔹^{n} which is T^{2}-definable with parameters b̄, with Sdim(S) < n, and finitely many functions g_{1}, …, g_{k} : 𝔹^{n} → Γ that are T -definable with parameters b̄ā, such that f agrees off S with some of the g_{i}.

6Thanks to J. Ramakrishnan for pointing this out.

8. Dense pairs  8.5. Elimination of imaginaries

Assume that the claim does not hold. Hence, for every S and every g as in the claim, there exists c̄ ∈ 𝔹^{n} such that c̄ ∉ S and f(c̄) ≠ g(c̄). Thus, the following partial L^{2}-type over b̄ā is consistent:

p(x̄) := (x̄ ∈ 𝔹^{n} \ Scl(b̄)) & (f(x̄) ≠ g(x̄)),

where we let g : 𝔹^{n} → Γ vary among the functions that are T -definable with parameters b̄ā. Let c̄ be a realisation of p. Notice that the choice of ā and the fact that Srk(c̄/āb̄) = n imply that c̄b̄ā | ⌣ ā 𝔸. Hence, by Proposition 8.25, f(c̄) ∈ dcl^{1}(c̄b̄ā). Hence f(c̄) = g(c̄) for some function g : 𝔹^{n} → 𝔹 which is T -definable with parameters b̄ā, absurd.

The above Claim plus Proposition 8.35 imply the conclusion.

The above Corollary gives a way to find the parameters of definition of f̂ (and of S) starting from the parameters b̄ of f.

Example 8.37. In general, f̂ cannot be defined using only b̄ as parameters. Consider a_{1} and a_{2} in 𝔸 which are independent over the empty set, b_{1} ∈ 𝔹\𝔸, and b_{2} := a_{1} + b_{1}a_{2} ∈ 𝔹 \ 𝔸. Let ā := ⟨a_{1}, a_{2}⟩ and b̄ := ⟨b_{1}, b_{2}⟩. Notice that rk(āb̄) = 3, while Srk(āb̄) = 2. Let f be the constant function a_{1}. Then, f is T^{2}-definable over b̄:= ⟨b_{1}, b_{2}⟩, but is not T -definable over b̄.

question:dmin-scl-1 Question 8.38. Assume that T is d-minimal (see §9). Is it true that, for every X ⊆ 𝔹^{∗}, Scl(X) = acl(𝔸^{∗}X)?

Conjecture 8.39 (J. Ramakrishnan). Assume that T is o-minimal. Then, for every X ⊂ 𝔹,

acl^{2}(X) = acl^{1}(X ∪ (acl^{2}(X) ∩ 𝔸)).

8.5 Elimination of imaginaries

Let cl be an existential matroid on 𝕄. Remember that element e ∈ 𝕄^{eq} is an equivalence class X ⊆ 𝕄^{n} for some ∅-definable equivalence relation E on 𝕄^{n}. If c̄ ∈ X we say that c̄ represents e.

Definition 8.40. We say that 𝕄 has cl-elimination of imaginaries if, for every e ∈ 𝕄^{eq}, there exists c̄ representing e, such that c̄ ∈ cl^{eq}(e). Given b̄ ⊂ 𝕄, we say that 𝕄 has cl-elimination of imaginaries modulo b̄ if, for every e ∈ 𝕄^{eq}, there exists c̄ representing e, such that c̄ ∈ cl^{eq}(eb̄).

If 𝕂 ⪯ 𝕄 we say that 𝕂 has cl-elimination of imaginaries (modulo some b̄ ⊂ 𝕂) if 𝕄 has it.

Compare the above notion with weak elimination of imaginaries (see [CF04]). prop:cl-elimination Proposition 8.41. Let b̄ ⊂ 𝕄. Assume that cl(b̄) is dense in 𝕄. Then, 𝕄

has cl-elimination of imaginaries modulo b̄.

Corollary 8.42. Let 𝕄 be geometric. Assume that acl(∅) is acl-dense in 𝕄 (e.g., 𝕄 is an algebraically closed field). Then, 𝕄 has weak elimination of imaginaries. If moreover 𝕄 expands a field, then 𝕄 has elimination of imaginaries.

Corollary 8.43. Assume that 𝕄 expands an integral domain. Let ⟨𝔹, 𝔸⟩ ⊧ T^{d}. Let b ∈ 𝔹\𝔸. Then, ⟨𝔹, 𝔸⟩ has Scl-elimination of imaginaries modulo b.

Proof. Scl(b) is Scl-dense in ⟨𝔹, 𝔸⟩ for every b ∈ 𝔹 \ 𝔸.

In the situation of the above corollary, it is not true that ⟨𝔹, 𝔸⟩ has Scl-elimination of imaginaries (modulo ∅). For instance, let X := 𝔹 \ 𝔸. Then, X ∩ Scl^{eq}(⌜X⌝) = ∅.

Before proving the Proposition 8.41, we need some preliminaries. Let X ⊆ 𝕄^{n} be a subset definable with parameters b̄. Let 𝕄′ be the expansion of 𝕄 with a new predicate denoting X. Notice that 𝕄 and 𝕄′ have the same definable sets. However, cl is no longer an existential matroid on 𝕄′: for instance, if X = {b} is a singleton, and b ∉ cl(∅), then b ∈ acl_{𝕄′}(∅) \ cl(∅), and therefore cl is not existential on 𝕄′. Notice that | ⌣^{cl} satisfies all the axioms of a symmetric independence relation on 𝕄′, except the Extension axiom.

Let e := ⌜X⌝ ∈ 𝕄^{eq} be the canonical parameter for X. For every Z ⊆ 𝕄, define cl_{e}(Z) := cl^{eq}(eZ) ∩ 𝕄 (notice that, if e = ∅, then cl_{e} = cl).

Lemma 8.44. cl_{e} is an existential matroid on 𝕄′.

Proof. We only need to check that cl_{e} satisfies Existence. Let B and C be subsets of 𝕄 such that a ∉ cl_{e}(B), that is a ∉ cl^{eq}(eB). Let a′ ≡_{eB}^{𝕄} a, such that a′ | ⌣^{cl}_{e} BC. Then, a′ ≡_{B}^{𝕄′} a and a′ ∉ cl^{eq}(eBC) = cl_{e}(BC).

Proof of Proposition 8.41. W.l.o.g., b̄ = ∅. Let e ∈ 𝕄^{eq}. Let E be a ∅- definable equivalence relation on 𝕄 and X be an equivalence class of E, such that e = X. Let cl_{e} be defined as above. Since cl(∅) is dense in 𝕄 and cl ⊆ cl_{e}, 𝕂 := cl_{e}(∅) is cl_{e}-dense in 𝕄′. Hence, by Lemma 7.4, 𝕂 ⪯ 𝕄′. Thus, since X is ∅-definable in 𝕄′, there exists c̄ ∈ X ∩ 𝕂.

9 D-minimal topological structures

sec:dmin

9. D-minimal topological structures

In this section we will introduce d-minimal structures. They are topological structures whose definable sets are particularly simple from the topological point of view; they generalise o-minimal structures. We will show that for d-minimal structure the topology induces a canonical existential matroid, which we denote by Zcl. Moreover, the abstract notion of density introduced in §7 coincides with the usual topological notion. Finally, if T is a complete d-minimal theory expanding the theory of fields, then in T^{d} the condition that the smaller structure is cl-closed is superfluous. Our definition of dminimality extends an older definition by C. Miller [Miller05], that applied only to linearly ordered structures.

Let 𝕂 be a first-order topological structure in the sense of [Pillay87]. That is, 𝕂 is a structure with a topology, such that a basis of the topology is given by {Φ(𝕂, ā) : ā ∈ 𝕂^{m}} for a certain formula without parameters Φ(x, ȳ); fix such a formula Φ(x, ȳ), and denote Bā := Φ(𝕂, ā). Examples of topological structures are valued fields, or ordered structures. On 𝕂^{n} we put the product topology.^{7} Let 𝕄 ⪰ 𝕂 be a monster model. Given X ⊆ 𝕂^{n}, we ill denote by \overline{X} the topological closure of X inside 𝕂^{n}.

Definition 9.1. 𝕂 is d-minimal if:

1. it it is T_{1} (i.e., its points are closed);

2. it has no isolated points;

3. for every X ⊆ 𝕄 definable subset (with parameters in 𝕄), if X has empty interior, then X is a finite union of discrete sets.

4. for every X ⊂ 𝕂^{n} definable and discrete, Π_{1}^{n}(X) has empty interior;

5. given X ⊆ 𝕂^{2} and U ⊆ Π_{1}^{2}(X) definable sets, if U is open and nonempty, and X_{a} has non-empty interior for every a ∈ U, then X has non-empty interior.

Lemma 9.2. Assume that 𝕂 is d-minimal. Let Z ⊂ 𝕂^{2} be definable, such that Π_{1}^{2}(Z) has empty interior, and Z_{x} has empty interior for every x ∈ 𝕂. Then, θ(Z) has empty interior, where θ is the projection onto the second coordinate.

Proof. By assumption, w.l.o.g. Π_{1}^{2}(Z) is discrete and, for every x ∈ 𝕂, Z_{x} is also discrete. Therefore, Z is discrete, and hence θ(Z) has empty interior.

Definition 9.3. Given A ⊂ 𝕄 and b ∈ 𝕄, we say that b ∈ Zcl(A) if there exists X ⊂ 𝕄 A-definable such that b ∈ A and A has empty interior (or, equivalently, A is discrete).

7Allowing a different topology (e.g. Zariski topology) might be a better choice.

Examples 9.4.

1. p-adic fields and algebraically closed valued fields are d-minimal;

2. densely ordered o-minimal structures are d-minimal.

ex:dmin Example 9.5. A structure 𝕂 is definably complete if it expands a linear order ⟨K, <⟩, and every 𝕂-definable subset of K has a supremum in K ⊔ {±∞}. C. Miller defines a d-minimal structure as a definably complete structure 𝕂 such that, given 𝕂′ an ℵ_{0}-saturated elementary extension of 𝕂, every 𝕂′-definable subset of 𝕂′ is the union of an open set and finitely many discrete sets. In particular, o-minimal structures and ultra-products of ominimal structures are d-minimal in Miller’s sense. If 𝕂 expands a field and is a d-minimal structures in the sense of Miller, then 𝕂 is d-minimal in our sense (the proof will be given elsewhere). Conversely, any definably complete structure which is d-minimal in our sense is also d-minimal in Miller’s sense.

Proviso. For the remainder of this section, we assume that 𝕂 is d-minimal, and T is the theory of 𝕂.

Remark 9.6. If A and B are definable subsets of 𝕂 with empty interior, then A ∪ B has empty interior. Hence, for every C ⊆ 𝕂 definable, bd(C) has empty interior.

lem:open-orbit Lemma 9.7. If c ∉ Zcl(A), then Ξ(c/A) has non-empty interior.

Proof. Let X ⊆ 𝕄 be an A-definable set containing c. Since c ∉ Zcl(A), c ∈ X̊. Consider the partial type over cA

Γ(ȳ) := {Φ(c, ȳ) & Φ(𝕄, ȳ) ⊆ X},

where X varies among the A-definable sets containing c. By the above consideration, Γ is consistent; let b̄ ⊂ 𝕄 be a realisation of Γ. Claim 11. c ∈ Φ(𝕄, b̄) ⊆ Ξ(c/A).

In fact, if c ∈ X, where X is A-definable, then, by definition, c ∈ Φ(𝕄, b̄) ⊆ X, and therefore any x′ ∈ Φ(𝕄, b̄) satisfies all the A-formulae satisfied by c.

Theorem 9.8. Zcl is an existential matroid.

Proof. Finite character, extension and monotonicity are obvious. The fact that Zcl is definable is also obvious. Idempotence) Let b̄ := ⟨b_{1}, …, b_{n}⟩, a ∈ Zcl(b̄c̄) and b̄ ⊂ Zcl(c̄). We must prove that a ∈ Zcl(c̄). Let ϕ(x, ȳ, z̄) and ψ_{i}(y, z̄) be formulae, i = 1, …, n, such that ϕ(𝕂, ȳ, z̄) and ψ_{i}(𝕂, z̄) are discrete for every ȳ and z̄, and 𝕂 ⊧ ϕ(a, b̄, c̄) and 𝕂 ⊧ ψ_{i}(b_{i}, c̄), i = 1, …, n. Let

Z := {⟨x, ȳ⟩ : 𝕂 ⊧ ϕ(x, ȳ, c̄) & ⋁_{i=1}^{n} ψ_{i}(y_{i}, z_{i})},

and W := Π_{1}^{n+1}Z. By hypothesis, Z is a discrete subset of 𝕂^{n+1}, and therefore, by Assumption (4), W has empty interior. Moreover, W is c̄-definable and a ∈ W, and hence a ∈ Zcl(c̄). Notice that weak d-minimality suffices. EP) Let a ∈ Zcl(bc̄) \ Zcl(c̄). We must prove that b ∈ Zcl(ac̄). Assume not. Let Z ⊂ 𝕂^{2} be c̄-definable, such that ⟨a, b⟩ ∈ Z and Z^{y} is discrete for every y ∈ 𝕂. Since b ∈ Z_{a} and b ∉ Zcl(ac̄), b ∈ int(Z_{a}); hence, w.l.o.g. Z_{x} is open for every x ∈ 𝕂. Let U := Π_{1}^{2}(Z). Since a ∈ U and a ∉ Zcl(c̄), a ∈ Ů. Hence, by Condition (5), Z has non-empty interior; but this contradict the fact Z^{y} is discrete for every y ∈ 𝕂. Existence follows from Lemma 9.7. Non-triviality) Consider the following partial type over the empty set:

Λ(x) := {x ∉ Y },

where Y varies among the discrete ∅-definable sets. Since 𝕂 has no isolated points, Λ is finitely satisfiable; if a ∈ 𝕄 is a realisation of Λ, then a ∉ Zcl(∅).

We will denote by Zrk and dim the rank and dimension on 𝕄 according to Zcl. We will say Z-closed instead of Zcl-closed.

Remark 9.9. Let X ⊆ 𝕂^{n} be definable. If X has non-empty interior, then dim(X) = n. If Π_{d}^{n}(X) has non-empty interior, then dim(X) ≥ d.

Conjecture 9.10. Let X ⊆ 𝕂^{n} be definable. Then, dim(X) = d iff, after a permutation of variables, Π_{d}^{n}(X) has non-empty interior.

Conjecture 9.11. For every X ⊆ 𝕂^{n} definable, dim(\overline{X}) = dim X.

However, it is not true that dim(∂X) < dim(X) if X is definable and non-empty.

Lemma 9.12. X is dense in 𝕂 according to definition 7.1 (w.r.t. Zcl) iff X is topologically dense in 𝕂.

Proof. Assume that X is dense in 𝕂 according to Zcl. Let A ⊆ 𝕂 be an open definable set; thus, dim(A) = 1, and therefore A∩X ≠ ∅. Conversely, if X is topologically dense in 𝕂, let A ⊆ 𝕂 be definable and of dimension 1. Thus, A has non-empty interior, and therefore A ∩ X ≠ ∅.

Proviso. For the remainder of this section, will assume that 𝕂 is d-minimal and expands an integral domain, that + and − are continuous (and therefore ⟨𝕂, +⟩ is a topological group), and that T is the theory of 𝕂.

Notice that an algebraically closed field with the Zariski topology is not a topological group, because + is not continuous. Notice also that, since we required that points are closed, 𝕂 is a regular topological space.

rem:dense-sum Remark 9.13. Let X ⊆ 𝕂 be dense (but not necessarily definable). Then, for every b ∈ 𝕂 and every V neighbourhood of 0, there exists a ∈ X such that b ∈ a + V.

Proof. Since − is continuous, there exists V′ neighbourhood of 0 such that V′ = −V′ and V′ ⊆ V. Since X is dense, there exists a ∈ X such that a ∈ b + V′. Hence, b ∈ a − V′ ⊆ a + V.

cor:dense-dminimal Corollary 9.14. T^{d} is complete. Besides, T^{d} is the theory of pairs ⟨𝕂, 𝔽⟩ such that 𝔽 ≺ 𝕂 ⊧ T and 𝔽 is a (topologically) dense proper subset of 𝕂.

Proof. We must show that if 𝔽 ⪯ 𝕂 is topologically dense in 𝕂, then 𝔽 is cl-closed in 𝕂. W.l.o.g., the pair ⟨𝕂, 𝔽⟩ is ω-saturated. Let b ∈ cl^{𝕂}(𝔽); we must prove that b ∈ 𝔽. Let Z ⊂ 𝕂 be 𝔽-definable and discrete, such that b ∈ Z. Let U′ be a definable neighbourhood of b, such that Z ∩ U′ = {b}. Define U := U′ − b; since 𝕂 is a topological group, U is a neighbourhood of 0 in 𝕂, and there exists V open neighbourhood of 0 definable in 𝕂, such that V = −V and V + V ⊆ U. Claim 12. There exists and 𝔽-definable open neighbourhood W of 0 such that W ⊆ V.

Suppose the claim is not true. Since 𝔽 is a regular space, there exists X definable open neighbourhood of 0 such that \overline{X} ⊆ V. Let X_{𝔽} := X ∩ 𝔽. X_{𝔽} is a neighbourhood of 0 in 𝔽; thus, since the topology has a definable basis, there exists W_{𝔽} ⊆ X_{𝔽} such that W_{𝔽} is 𝔽-definable and W_{𝔽} is an open neighbourhood of 0. Let W be the interpretation of W_{𝔽} in 𝕂. Since W is open and 𝔽 is dense in 𝕂, W_{𝔽} is dense in W; therefore, W ⊆ \overline{W_{𝔽}} ⊆ \overline{X} ⊆ V.

By Remark 9.13, there exists a ∈ 𝔽 such that b ∈ W′, where W′ := a+W. Claim 13. W′ ⊆ U′.

The claim is equivalent to a + W ⊆ b + U, that is W + (a − b) ⊆ U. By assumption, b ∈ a + W, and therefore a − b ∈ −W. Hence, W + (a − b) ⊆ W − W ⊆ V − V ⊆ U.

However, W′ is 𝔽-definable, and b ∈ W′ ∩ Z ⊆ V ∩ Z = {b}. Hence, b is 𝔽-definable, and therefore b ∈ 𝔽.

Denote | ⌣ := | ⌣^{Zcl}. Given ā := ⟨ā_{1}, …, ā_{n}⟩ ∈ 𝕄^{n×m} and b̄ ∈ 𝕄^{n}, denote

Bā + b̄:= (Bā_{1} + b_{1}) × ⋯ × (Bā_{n} + b_{n}) ⊆ 𝕄^{n}.

lem:neighbourhood-0 Lemma 9.15. Let d ∈ 𝕄, V be a definable neighbourhood of d, and C ⊂ 𝕄. Then, there exists ā ∈ 𝕄^{m} such that ā | ⌣_{d} C and d ∈ Bā ⊆ V.

Proof. Let X := {ā ∈ 𝕄^{n} : d ∈ Bā}. Let ≤ be the quasi-ordering on X given by reverse inclusion: ā ≤ ā′ if Bā ⊇ B_{}_{ā′}. Fix b̄ ∈ X such that Bb̄ ⊆ V. Since (X, ≤) is a directed set, by Lemma 3.69, there exists ā ∈ X such that ā | ⌣^{cl}_{d} C and Bā ⊆ Bb̄ ⊆ V.

lem:neighbourhood-1 Lemma 9.16. Let d̄∈ 𝕄^{n}, V be a definable neighbourhood of d̄, and C ⊂ 𝕄. Then, there exist ā ∈ 𝕄^{m×n} and b̄ ∈ 𝕄^{n} such that d̄ ∈ Bā + b̄ ⊆ V and āb̄ | ⌣ Cd̄.

Proof. Proceeding by induction on n, it suffices to treat the case n = 1. Let V_{0} := V − d; it is a definable neighbourhood of 0. Since 𝕄 is a topological group, there exists V_{1} definable and open, such that 0 ∈ V_{1}, V_{1} = −V_{1}, and V_{1} + V_{1} ⊆ V_{0}. By Lemma 9.15, there exists ā ∈ 𝕄^{m} such that ā | ⌣ Cd and 0 ∈ Bā ⊆ V_{1}. Let W := d − Bā. Since dim(W) = 1, there exists b ∈ W such that b ∉ Zcl(Cād). Claim 14. d ∈ Bā + b.

In fact, b ∈ −Bā + d, and therefore d − b ∈ Bā Claim 15. ab̄ | ⌣ Cd.

By construction, b | ⌣ Cād, and therefore b | ⌣ ā Cd, and hence āb | ⌣ ā Cd. Together with ā | ⌣ Cd, this implies the claim.

Corollary 9.17. Let X ⊆ 𝕄^{n} be a definable set, and k ∈ ℕ. Assume that, for every x̄ ∈ X, there exists Vx̄ definable neighbourhood of x̄, such that dim(Vx̄ ∩ X) ≤ k. Then, dim(X) ≤ k.

Proof. Let C be the set of parameters of X. By Lemma 9.16, for every x ∈ X there exist ā ∈ 𝕂^{n×m} and b̄ ∈ 𝕂^{n} such that āb̄ | ⌣ Cx and x̄ ∈ Bā + b̄ ⊆ Vx̄; notice that dim(X ∩ (Bā + b̄)) ≤ k Hence, by Lemma 3.70, dim(X) ≤ k.

We do not know if the above Corollary remains true if we drop the assumption that 𝕄 expands a group.

Corollary 9.18. Let C ⊂ 𝕄 and p ∈ S_{n}(C). Then, p is stationary iff p is realised in dcl(C).

Proof. Assume for contradiction, that p is stationary, but dim(p) > 0. Let ā_{0} and ā_{1} be realisations of p independent over C. Since dim(p) > 0, ā_{0} ≠ ā_{1}. Since 𝕄 is Hausdorff, Lemma 9.16 implies that there exists V open neighbourhood of ā_{0}, definable with parameters b̄, such that ā_{1} ∉ V and b̄ | ⌣ Cā_{0}ā_{1}. Hence, by Lemma 3.10, ā_{0} | ⌣_{Cb̄} ā_{1}. Since p is stationary, Lemma 3.65 implies ā_{0} ≡b̄ ā_{1}, contradicting the fact that ā_{0} ∈ V, while ā_{1} ∉ V.

10 Cl-minimal structures

Let 𝕄 be a monster model, T be the theory of 𝕄, and cl be an existential matroid on 𝕄. We denote by dim and rk the dimension and rank induced by cl.

Definition 10.1. p ∈ S_{n}(A) is a generic type if dim(p) = n. 𝕄 is cl-mini-mal if, for every A ⊂ 𝕄, there exists only one generic 1-type over A.

Lemma 10.2. For every 0 < n ∈ ℕ and A ⊂ 𝕄, there exists at least one generic type in S_{n}(A). If 𝕄 is cl-minimal, then for every n and A there exists exactly one generic type in S_{n}(A).

Lemma 10.3. If 𝕄 is cl-minimal, then dim is definable.

Proof. Notice that, given x̄ := ⟨x_{1}, …, x_{n}⟩ and a formula ϕ(x̄, ȳ), the set U_{ϕ}^{n} := {ā : dim(ϕ(𝕂, ā)) = n} is always type-definable (Lemma 3.42). By the above Lemma, 𝕂^{n} \ U_{ϕ}^{n} = U_{¬ϕ}^{n}, and therefore U_{ϕ}^{n} is both type-definable and ord-definable, and hence definable.

Remark 10.4. 𝕄 is cl-minimal iff, for every n > 0 and every X definable subset of 𝕂^{n}, exactly one among X and 𝕂^{n} \ X, has dimension n.

rem:clminimal-fixed Remark 10.5. If 𝕂 ⪯ 𝕄 and dim is definable, then 𝕂 is cl-minimal iff, for every X definable subset of 𝕂, either dim(X) = 0, or dim(𝕂 \ X) = 0; that is, we can check cl-minimality directly on 𝕂.

Examples 10.6. 1. 𝕄 is strongly minimal iff acl is a matroid and 𝕄 is acl-minimal.

11. Connected groups

2. Consider Example 3.54. In that context, a type is generic in our sense iff it is generic in the sense of stable groups. Hence, 𝔾 is cl-minimal iff it has only one generic type iff it is connected (in the sense of stable groups). lem:minimal-pair Lemma 10.7. Assume that T is cl-minimal. Then, T^{d} is also cl-minimal.

Moreover, T^{d} coincides with T^{2}.

Proof. Let ⟨𝔹^{∗}, 𝔸^{∗}⟩ be a monster model of T^{d}. Let C ⊂ 𝔹^{∗} with |C| < κ. Define 𝔸 := cl(𝔸^{∗}C), and q_{C}(x) the partial L^{2}-type over C given by

q_{C}(x) := x ∉ 𝔸.

It is clear that every generic 1-T^{d}-type over C expands q_{C}. Hence, it suffices to prove that q_{C} is complete. Let b and b′ ∈ 𝔹^{∗} satisfy q_{C}. By Corollary 8.18, ⟨𝔹^{∗}, 𝔸^{∗}⟩ ⪯ ⟨𝔹^{∗}, 𝔸⟩. By assumption, b and b′ are not in 𝔸; hence, since T is cl-minimal, they satisfy the same generic 1-T -type p_{𝔸}; thus, by Corollary 8.20, b ≡_{𝔸}^{2} b′.

11 Connected groups

Let 𝕄 be a monster model, and cl be an existential matroid on it. Denote dim := dim^{cl}, rk := rk^{cl}, and | ⌣ := | ⌣^{cl}.

Definition 11.1. Let X ⊆ 𝕄^{n} be definable (with parameters). Assume that m := dim(X) > 0. We say that X is connected if, for every Y definable subset of X, either dim(Y) < n, or dim(X \ Y) < n.

For instance, if 𝕄 is cl-minimal and X = 𝕄, then X is connected.

Remark 11.2. If X is connected, then, for every l ≥ 0, X^{l} is also connected.

Remark 11.3. Let X ⊆ 𝕄^{n} be definable, of dimension m > 0. Then, X is connected iff for every A ⊂ 𝕄 containing the parameters of definition of X, there exists exactly one n-type over A in X which is generic (i.e., of dimension m).

lem:semigroup Lemma 11.4. Let G ⊆ 𝕄^{n} be definable and connected. Assume that G is a semigroup with left cancellation. Assume moreover that G has either right cancellation or right identity. Then G is a group.

Cf. [Poizat87, 1.1].

Proof. Assume not. Let m := dim(G). W.l.o.g., G is definable without parameters. For every a ∈ G, let a · G := {a · x : a ∈ G}. Since G has left cancellation, we have dim(a · 𝕄) = m.

Let F := {a ∈ G : a · G = G}. Our aim is to prove that F = G.

It is easy to see that G is multiplicatively closed.

First, assume that G has a right identity element 1. The following claim is true for any abstract semigroup with left cancellation and right identity 1.

Claim 16. 1 is also the left identity.

In fact, for every a, b ∈ G, a · b = (a · 1) · b = a · (1 · b). Since we have left cancellation, we conclude that b = 1 · b for every b, and we are done.

Obviously, 1 ∈ F. For every a ∈ F, denote by a^{−1} the (unique) element of G such that a · a^{−1} = 1. Claim 17. a^{−1} · a = 1.

In fact, a · (a^{−1} · a) = 1 · a = a · 1, and the claim follows from left cancellation. Claim 18. F is a group.

We have already seen that F is multiplicatively closed and 1 ∈ F. Let a ∈ F. Then, for every g ∈ G, a^{−1} · (a · g) = g, and therefore a^{−1} ∈ F. Claim 19. dim(F) < m.

Assume, for contradiction, that dim(F) = m. Let a ∈ G \ F. Then, F ∩ (a · F) ≠ ∅; let u, v ∈ F such that u = a · v.

Since u ∈ F and F is a group, there exists w ∈ F such that v · w = 1; hence, u · w = a · 1 = a, and therefore a ∈ F, absurd.

Choose a, b ∈ G independent (over the empty set). Since dim(a · G) = dim(b · G) = m, we have a ∈ b · G and b ∈ a · G. Let u, v ∈ G such that b = a · u and a = b · v. Hence, a = a · u · v.

Since a · 1 = a · u · v, we have 1 = u · v. Hence, both u and v are in F. However, since dim(F) < m and b = a · u, we have rk(b/a) ≤ rk(u) < m, absurd.

If instead G has right cancellation, it suffices, by symmetry, to show that G has a left identity. Reasoning as above, we can show that there exists a and b in G such that a · b = a. We claim that b is a left identity. In fact, for every c ∈ G, we have a · b · c = a · c, and therefore b · c = c, and we are done.

Proviso. For the remainder of this section, ⟨G, ·⟩ is a definable connected group, of dimension m > 0, with identity 1.

If G is Abelian, we will also use + instead of · and 0 instead of 1.

Hence, if G expands a ring without zero divisors, then, by applying the above lemma to the multiplicative semigroup of G, we obtain that G is a division ring.

Remark 11.5. Let X ⊆ G be definable, such that X · X ⊆ X. Then, dim(X) = m iff X = G.

Proof. Assume that dim(X) = m. Let a ∈ G. Then, X ∩ (a · X^{−1}) ≠ ∅; choose u, v ∈ X such that u = a · v^{−1}. Hence, a = u · v ∈ X · X = X. lem:homo Lemma 11.6. Let f : G → G be a definable homomorphism. If dim(ker f) =

0, then f is surjective.

Cf. [Poizat87, 1.7].

Proof. Let H := f(G) and K := ker(f); notice that H < G and K < G. Moreover, by additivity of dimension, m = dim(H) + dim(K). Hence, if dim(K) = 0, then dim(H) = m, therefore H = G and f is surjective.

Example 11.7. ⟨ℤ, +⟩ cannot be cl-minimal, because the homomorphism x ↦ 2x has trivial kernel but is not surjective.

lem:quotient Lemma 11.8. Let H < G be definable, with dim(H) = k < m. Then, G/H is connected, and dim(G/H) = m − k.

Proof. That dim(G/H) = m − k is obvious. Let X ⊆ G/H be definable of dimension m−k. We must prove that dim(G/H\X) < m. Let π : G → G/H be the canonical projection, and Y := π^{−1}(X). Then, dim(Y) = m, and therefore dim(G \ Y) < m. Thus, dim(G/H \ X) = dim(π(Y)) < m − k.

Conjecture 11.9. If m = 1, then G is Abelian. Cf. Reineke’s Theorem [Poizat87, 3.10].

Idea for proof. Assume for contradiction that G is not Abelian. Let Z := Z(G) and \overline{G} := G/Z. Since Z < G and Z ≠ G, we have dim(Z) = 0, and therefore \overline{G} is also connected and of dimension 1.

For every a ∈ G, let U_{a} be the set of conjugates of a.

cl:conjugate Claim 20. If a ∉ Z, then dim(U_{a}) = 1.

By general group theory, U_{a} ≡ G/C(a), where C(a) is the centraliser of a. Since a ∉ Z, C(a) is not all of G; moreover, C(a) < G, therefore dim(C(a)) = 0, and thus dim(U_{a}) = 1, and similarly for U_{b}.

cl:conjugate-2 Claim 21. For every a, b ∈ G \ Z, a is a conjugate of b.

In fact, by connectedness and the above claim, U_{a} ∩ U_{b} ≠ ∅, and thus U_{a} = U_{b}. Claim 22. For every x, y ∈ \overline{G} \ \{1\}, x is a conjugate of y.

In fact, x = ā and y = b̄ for some a, b ∈ G \ Z. By Claim 21, a and b are conjugate in G, and thus x and y are conjugate in \overline{G}.

Thus, \overline{G} is a definable (in the imaginary sorts) connected group of dimension 1, such that any two elements different from the identity are in the same conjugacy class (and therefore \overline{G} is torsion-free and has trivial centre).

One should now prove that such a group \overline{G} cannot exists.

Notice that the above conjecture is false if m > 1.

Lemma 11.10. Assume that m = 1 and G is Abelian. Let p be a prime number. Then, either pG = 0, or G is divisible by p.

Proof. Let H := pG and K := {x ∈ G : px = 0}. If dim(H) = 1, then G = H and therefore G is p-divisible. If dim(H) = 0, then dim(K) = 1, thus G = K and pG = 0.

Notice that the above lemma needs the hypothesis that m = 1. For instance, let 𝕄 be the algebraic closures of 𝔽_{p}, and let G := 𝕄 × 𝕄^{∗} (where 𝕄^{∗} is the multiplicative group of 𝕄).

Theorem 11.11. Assume that G expands an integral domain (and ⟨G, +⟩ is connected). Then, G is an algebraically closed field.

Cf. Macintyre’s Theorem [Poizat87, 3.1, 6.11].

Proof. Let ⟨G^{∗}, ·⟩ be the multiplicative semigroup of G. By Lemma 11.4, ⟨G^{∗}, ·⟩ is a group, and therefore G is a field. For every n ∈ ℕ, consider the map f_{n} : G^{∗} → G^{∗} x ↦ x^{n}. Since f_{n} has finite kernel, Lemma 11.6 implies that f_{n} is surjective, and therefore every element of G has an nth root in G. In particular, G is perfect.

Let p := char(G). If p > 0, consider the map h : G → G, x ↦ x^{p} + x. Notice that h is an additive homomorphism with finite kernel; hence, h is surjective.

Since G^{l} is also connected for every 0 < l ∈ ℕ, the above is true not only for G, but also for every finite-degree extension G_{1} of G.

The rest of the proof is the same as in [Poizat87, 3.1]: G contains all roots of 1 (because G^{∗} is divisible), and, if G were not algebraically closed, there would exists a finite extension G_{1} and normal finite extension L of G_{1}, such that the Galois group of L/G_{1} is cyclic and of prime order q. If q ≠ p, then L/G_{1} is a Kummer extension, absurd. If q = p, then L/G_{1} is an Artin-Schreier extension, also absurd.

In the above theorem it is essential that G is connected. For instance, if 𝕄 is a formally p-adic field, then 𝕄 itself is a non-algebraically closed field (of dimension 1). Notice also that the first step in the proofs of [Poizat87, 3.1, 6.11] is showing that G is connected.

Question 11.12. Can we weaken the hypothesis in the above theorem from “G expands an integral domain” to “G expands a ring without zero divisors”?

12. Ultraproducts

12 Ultraproducts

Let I be an infinite set, and μ be an ultrafilter on I. For every i ∈ I, let ⟨𝕂_{i}, cl_{i}⟩ be a pair given by first-order L-structure 𝕂_{i} and an existential matroid cl_{i} on 𝕂_{i}. Let K be the family (⟨𝕂_{i}, cl_{i}⟩)_{i∈I}, and 𝕂 := Π_{i}𝕂_{i}/μ be the corresponding ultraproduct.

We will give some sufficient condition on the family K, such that there is an existential matroid on 𝕂 induced by the family of cl_{i}. Denote by d_{i} the dimension induced by cl_{i}.

Definition 12.1. We say that the dimension is uniformly definable (for the family K) if, for every formula ϕ(x̄, ȳ) without parameters, x̄ = ⟨x_{1}, …, x_{n}⟩, ȳ = ⟨y_{1}, …, y_{m}⟩, and for every l ≤ n, there is a formula ψ(ȳ), also without parameters, such that, for every i ∈ I,

{ȳ ∈ 𝕂_{i}^{m} : d_{i}(ϕ(𝕂_{i}, ȳ)) = l} = ψ(𝕂_{i}).

We denote by d_{ϕ}^{l} the formula ψ.

Remark 12.2. The dimension is uniformly definable if, for every formula ϕ(x, ȳ) without parameters, ȳ = ⟨y_{1}, …, y_{m}⟩, there is a formula ψ(ȳ), also without parameters, such that, for every i ∈ I,

{ȳ ∈ 𝕂_{i}^{m} : d_{i}(ϕ(𝕂_{i}, ȳ)) = 1} = ψ(𝕂_{i}).

For instance, if every 𝕂_{i} expands a ring without zero divisors, then the dimension is uniformly definable: given ψ(x, ȳ), define ψ(ȳ) by

∀z ∃x_{1}, … x_{4} (z = F^{4}(x_{1}, …, x_{4}) & ⋀_{i=1}^{4} ϕ(x_{i}, ȳ)).

For the remainder of this section, we assume that the dimension is uniformly definable for K.

Definition 12.3. Let d be the function from definable sets in 𝕂 to {−∞}∪ℕ defined in the following way: given a 𝕂-definable set X = Π_{i∈I}X_{i}/μ and l ∈ ℕ, d(X) = l if, for μ-almost every i ∈ I, d(X_{i}) = l.

Theorem 12.4. d is a dimension function on 𝕂. Let cl be the existential matroid induced by d. Then, a ∈ cl(b̄) implies that, for μ-almost every i ∈ I, a_{i} ∈ cl_{i}(b̄_{i}), but the converse is not true.

Remark 12.5. Let X ⊆ 𝕂^{n} be definable with parameters c̄; let ϕ(x̄, c̄) be the formula defining X. Given l ∈ N, d(X) = l iff, for μ-almost every i ∈ I, 𝕂_{i} ⊧ d_{ϕ}^{l}(c̄_{i}).

Lemma 12.6. If each 𝕂_{i} is cl-minimal, then 𝕂 is also cl-minimal.

Proof. By Remark 10.5.

Examples 12.7. The ultraproduct 𝕂 of strongly minimal structures is not strongly minimal in general (it will not even be a pregeometric structure), but if each structure expands a ring without zero divisors, then 𝕂 will have a (unique) existential matroid, and will be cl-minimal.

It is easy to find a family K = (𝕂_{i})_{i∈ℕ} of strongly minimal structures expanding a field, such that any non-principal ultraproduct of 𝕂 K is not pregeometric, does satisfy the Independence Property, and has an infinite definable subset with a definable linear ordering. Moreover, one can also impose that the trivial chain condition for uniformly definable subgroups of ⟨𝕂, +⟩ fails in 𝕂 [Poizat87, 1.3]. However, 𝕂 will satisfy the following conditions:

1. Every definable associative monoid with left cancellation is a group [Poizat87, 1.1];

2. Given G a definable group acting in a definable way on a definable set E, if E is a definable subset of A and g ∈ G such that g · A ⊆ A, then g · A = A [Poizat87, 1.2].

We do not know if conditions (1) and (2) in the above example are true for an arbitrary cl-minimal structure expanding a field.

Remark 12.8. Assume that each 𝕂_{i} is a first-order topological structure, and that the definable basis of the topology of each 𝕂_{i} is given by the same function Φ(x, ȳ). Then, 𝕂 is also a first-order topological structure, and Φ(x, ȳ) defines a basis for the topology of 𝕂. If each 𝕂_{i} is d-minimal, then 𝕂 has an existential matroid, but it needs not be d-minimal.

Assume that each 𝕂 is d-minimal and satisfies the additional condition

(*) Every definable subset of 𝕂_{i} of dimension 0 is discrete.

Then, 𝕂 is also d-minimal and satisfies condition (*).

Example 12.9. The ultraproduct of o-minimal structures is not necessarily o-minimal, but it is d-minimal, and satisfies condition (*).

13. Dense tuples of structures

13 Dense tuples of structures

In this section we assume that T expands the theory of integral domains. We will extend the results of §8 to dense tuples of models of T.

Definition 13.1. Fix n ≥ 1. Let L^{n} be the expansion of L by (n − 1) new unary predicates P_{1}, …, P_{n−1}. Let T^{n} be the L^{n}-expansion of T, whose models are sequences 𝕂_{1} ≺ ⋯ ≺ 𝕂_{n−1} ≺ 𝕂_{n} ⊧ T, where each 𝕂_{i} is a proper cl-closed elementary substructure of 𝕂_{i+1}. Let T^{nd} be the expansion of T^{n+1} saying that 𝕂_{1} is dense in 𝕂_{n}. We also define T^{0d} := T.

For instance, T^{1} = T, T^{2} is the theory we already defined in §8, and T^{1d} = T^{d}.

Lemma 13.2. If T is cl-minimal, then T^{n} is complete for every n ≥ 1 (and therefore coincides with T^{(n−1)d}). Moreover, T^{n} has a (unique) existential matroid cl^{n}: given ⟨𝕂_{n}, …, 𝕂_{1}⟩ ⊧ T^{n}, we have b ∈ cl^{n}(A) iff b ∈ cl(A𝕂_{n−1}). Finally, T^{n} is cl^{n}-minimal.

Proof. By induction on n: iterate n times Lemma 10.7.

Corollary 13.3. Assume that T is strongly minimal. Then, T^{n} is complete, and coincides with the theory of tuples 𝕂_{1} ≺ ⋯ ≺ 𝕂_{n} ⊧ T.

Proof. One can use either the above Lemma, or reason as in [Keisler64], using Lemma 8.10.

rem:pair-rank Remark 13.4. Let ⟨𝔹, 𝔸⟩ be a κ-saturated model of T^{d}. Let U ⊆ 𝔹 be 𝔹-definable and of dimension 1. Then rk(U ∩ 𝔸) ≥ κ.

Theorem 13.5. T^{nd} is complete. There is a (unique) existential matroid on T^{nd}.

Proof. By induction on n, we will prove T^{nd} coincides with (… (T^{d})^{d} …)^{d} iterated n times. This implies both that T^{nd} is complete, and that it has an existential matroid.

It suffices to treat the case n = 2. Notice that ⟨𝕂_{2}, 𝕂_{1}⟩ ≺ ⟨𝕂_{3}, 𝕂_{1}⟩ ⊧ T^{d}. It suffices to show that 𝕂_{2} is Scl-dense in ⟨𝕂_{3}, 𝕂_{1}⟩. W.l.o.g., we can assume that ⟨𝕂_{3}, 𝕂_{2}, 𝕂_{1}⟩ is κ-saturated.

Let X ⊆ 𝕂_{3} be definable in ⟨𝕂_{3}, 𝕂_{1}⟩ (with parameters from 𝕂_{3}), such that Sdim(X) = 1. We need to show that X intersects 𝕂_{2}. By Corollary 8.35, there exists U and S subsets of 𝕂_{3}, such that X is definable in 𝕂_{3}, S is definable in ⟨𝕂_{3}, 𝕂_{1}⟩ and small, and X = U Δ S. Therefore, dim(U) = 1. If, by contradiction, X ∩ 𝕂_{2} = ∅, then 𝕂_{2} ∩ U ⊆ S; therefore, Srk(𝕂_{2} ∩ U) < ω (where Srk is the rank induced by Scl), contradicting Remark 13.4.

Corollary 13.6. Assume that T is d-minimal. Then, T^{nd} coincides with the theory of (n + 1)-tuples 𝕂_{1} ≺ ⋯ ≺ 𝕂_{n} ≺ 𝕂_{n+1} ⊧ T, such that 𝕂_{1} is (topologically) dense in 𝕂_{n+1}.

Proof. Notice that if ⟨𝕂_{n}, …, 𝕂_{1}⟩ satisfy the assumption, then, by Corollary 9.14, each 𝕂_{i} is cl-closed in 𝕂_{n}.

13.1 Dense tuples of topological structures

Assume that T expands the theory of integral domains. Assume that 𝕄 has both an existential matroid cl and a definable topology (in the sense of [Pillay87]. Let ϕ(x, ȳ) be a formula such that the family of sets

Bb̄ := ϕ(𝕄, b̄),

as b̄ varies in 𝕄^{k}, is a basis of the topology of 𝕄. If b̄ = ⟨b̄_{1}, …, b̄_{m}⟩, we

denote by Bn ; b̄ := Bb̄_{1} × ⋯ × Bb̄_{m} ⊆ 𝕄^{m}.

Assume the following conditions:

Hypothesis. I. Every definable non-empty open subset of 𝕄 has dimension 1.

II. For every m ∈ ℕ, every U open subset of 𝕄^{m}, and every ā ∈ U, the set {b̄: ā ∈ Bb̄} has non-empty interior.

Remark 13.7. Assumption I implies that a definable subset of 𝕄^{m} with non-empty interior has dimension m (but the converse is not true: there can be definable subsets of dimension m but with empty interior). Moreover, it implies that a cl-dense subset of 𝕄^{m} is also topologically dense (but, again, the converse is not true).

Example 13.8. 1. If 𝕄 is either a valued field (with the valuation topology) or a linearly ordered field (with the order topology), then it satisfies Assumption II.

2. If 𝕄 is a d-minimal structure, then it satisfies Assumption I.

3. Let 𝕄 be either a formally p-adic field, or an algebraically closed valued field, or a d-minimal expansion of a linearly ordered definably complete field (cf. Example 9.5). Then, 𝕄 satisfies both assumptions.

We have two notions of closure and of density on 𝕄: the ones given by the topology and the ones given by the matroid; to distinguish them, we will 14. The (pre)geometric case

speak about topological closure and cl-closure respectively (and similarly for density).

The following theorem follows easily from [BH10] (we are assuming that the Hypothesis holds).

Theorem 13.9 ([BH10, Corollary 3.4]). Let ℂ := ⟨𝔹, 𝔸_{n−1}, …, 𝔸_{1}⟩ ⊧ T^{nd}. Let c̄ ⊂ 𝔹 be cl-independent over c̄∩𝔸_{n−1}. Let U ⊆ 𝔹^{m} be open and definable in ℂ, with parameters c̄. Then, U is definable in 𝔹, with parameters c̄.

In the terminology of [DMS08], the above theorem proves that 𝔹 is the open core of ℂ.

Proof. By induction on n, it suffices to do the case when n = 2, i.e. when 𝔻 = ⟨𝔹, 𝔸⟩. W.l.o.g., 𝔻 is λ-saturated and λ-homogeneous, for some |T | < λ < κ We want to verify that the hypothesis of [BH10, Corollary 3.1] is satisfied. Let D_{m} := {b̄ ∈ 𝔹^{m} : Srk(b̄/c̄) = m}.

1. If V ⊆ 𝔹^{n} is 𝔹-definable and of dimension m, then V ∩D_{m} is non-empty: therefore, D_{m} is topologically dense in 𝔹^{m}.

2. Let d̄ ∈ D_{m}, and U ⊆ 𝕄^{m} be open, such that p := tp^{1}(d̄/c̄) is realized in U. We have to show that p is realized in U ∩ D_{m}. Let d̄′ ∈ U be such a realization, and let b̄ ⊂ 𝔹 such that d̄′ ∈ Bb̄. Since d̄′ ≡1 ; c̄ d̄, we have that d̄′ is cl-independent over c̄. By changing b̄ if necessary, we can also assume that d̄′ | ⌣ b̄c̄ (cf. the proof of Lemma 9.16), and thus d̄′ is cl-independent over b̄c̄. Finally, since 𝔸 is cl-dense in 𝔹, there exists d̄′′ ≡_{b̄c̄}^{1} d̄ such that d̄′′ is cl-independent over b̄c̄𝔸.

3. By Proposition 8.16, for every dv ∈ D_{m}, tp^{2}(d̄/c̄) is determined by tp^{1}(d̄/c̄) in conjunction with “d̄∈ D_{m}”.

Hence, we can apply [BH10, Corollary 3.1] and we are done.

14 The (pre)geometric case

Remember that 𝕄 is a pregeometric structure if acl satisfies EP. If moreover 𝕄 eliminates the quantifier ∃^{∞}, then 𝕄 is geometric.

In this section we gather various results about (pre)geometric structures, mainly in order to clarify and motivate the general case of structures with an existential matroid.

Remember that 𝕄 has geometric elimination of imaginaries if every for imaginary tuple ā there exists a real tuple b̄ such that ā and b̄ are interalgebraic.

Remark 14.1. T is pregeometric iff T is a real-rosy theory of real þ-rank 1. Moreover, if T is pregeometric and has geometric elimination of imaginaries, then | ⌣^{þ} = | ⌣^{acl}, and dim^{acl} is equal to the þ-rank: see [EO07] for definitions and proofs.

Remark 14.2. The model-theoretic algebraic closure acl is a definable closure operator.

Fact. Let 𝕄 be a definably complete and d-minimal expansion of a field. Then, 𝕄 has elimination of imaginaries an definable Skolem functions; moreover, 𝕄 is rosy iff it is o-minimal. In particular, an ultraproduct of o-minimal structures expanding a field is rosy iff it is o-minimal.

The proof of the above fact will be given elsewhere.

For the remainder of this section, 𝕄 is pregeometric (and T is its theory).

Remark 14.3. acl is an existential matroid on 𝕄. The induced independence relation | ⌣^{acl} coincides with real þ-independence | ⌣^{þ} and with the Mdividing notion | ⌣^{M} of [Adler05]. A formula is x-narrow (for acl) iff it is algebraic in x.

Remark 14.4. Let X ⊆ 𝕄^{n} be definable. dim^{acl}(X) = 0 iff X is finite.

Remark 14.5. 𝕄 is geometric iff dim^{acl} is definable.

Remark 14.6. 𝕄 is acl-minimal iff it is strongly minimal.

In §6 we defined ãcl, the extension of acl to the imaginary sorts.

Remark 14.7. If a is real and B is imaginary, then a ∈ ãcl(B) iff a ∈ acl^{eq}(B).

Remark 14.8. T.f.a.e.:

1. acl^{eq} coincides with the extension of acl defined in §6;

2. T is superrosy of þ-rank 1 [EO07];

3. T is surgical [Gagelman05].

rem:geometric-density Remark 14.9. X is dense in 𝕄 iff for every U infinite definable subset of 𝕄, U ∩ X ≠ ∅. If 𝔽 ⪯ 𝕂, then 𝔽 is acl-closed in 𝕂.

Remark 14.10. Assume that T is geometric. Then, T^{2} is the theory of pairs ⟨𝕂, 𝔽⟩, with 𝔽 ≺ 𝕂 ⊧ T, and T^{d} is the theory of pairs ⟨𝕂, 𝔽⟩ ⊧ T^{2}, such that 𝔽 is dense in 𝕂. For every X ⊆ 𝕂, Scl(X) = acl(𝔽X) (cf. Question 8.38).

For more on the theory T^{d} in the case when T is geometric, and in particular when T is o-minimal, see [Berenstein07].

References  References

Acknowledgments

I thank H. Adler, A. Berenstein, G. Boxall, J. Ramakrishnan, K. Tent, and M. Ziegler for helping me to understand the subject of this article.

References

adler [Adler05] H. Adler. Explanation of Independence. Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau, June 2005.

BPV [BPV03] I. Ben-Yaacov, A. Pillay, E. Vassiliev. Lovely pairs of models. Pure Appl. Logic 122 (2003), no. 1–3, 235–261.

berenstein [Berenstein07] A. Berenstein. Lovely pairs and dense pairs of o-minimal structures. Preprint, 2007.

boxall [Boxall09] G. Boxall. Lovely pairs and dense pairs of real closed fields. Dissertation, University of Leeds, July 2009.

BH [BH10] G. Boxall and P. Hieronymi. Expansions which introduce no new open sets. Preprint, 2010.

CF [CF04] E. Casanovas, Enrique and R. Farré. Weak forms of elimination of imaginaries. MLQ Math. Log.0 Q. 50 (2004), no. 2, 126–140.

cdm [CDM92] Z. Chatzidakis, L. van den Dries, and A. Macintyre. Definable sets over finite fields. J. Reine Angew. Math. 427 (1992), 107–135.

DMS [DMS08] A. Dolich, C. Miller, and C. Steinhorn. Structures having o-minimal open core. Preprint.

dries89 [Dries89] L. van den Dries. Dimension of definable sets, algebraic boundedness and Henselian fields. Stability in model theory, II (Trento, 1987). Ann. Pure Appl. Logic 45 (1989), no. 2, 189–209.

vdd-dense [Dries98] L. van den Dries. Dense pairs of o-minimal structures. Fundamenta Mathematicae 157 (1998), 61–78.

EO [EO07] C. Ealy, and A. Onshuus. Characterizing rosy theories. J. Symbolic Logic 72 (2007), no. 3, 919–940.

gagelman [Gagelman05] J. Gagelman. Stability in geometric theories. Ann. Pure Appl. Logic 132 (2005), no. 2-3, 313–326. hp [HP94] E. Hrushovski, A. Pillay. Groups definable in local fields and pseudo-

finite fields. Israel J. Math. 85 (1994), no. 1-3, 203–262.

keisler [Keisler64] H.J. Keisler. Complete theories of algebraically closed fields with distinguished subfields. Michigan Math. J. 11 1964 71–81.

macintyre [Macintyre75] A. Macintyre. Dense embeddings. I. A theorem of Robinson in a general setting. Model theory and algebra (A memorial tribute to Abraham Robinson), pp. 200–219. Lecture Notes in Math., Vol. 498, Springer, Berlin, 1975.

miller [Miller05] C. Miller. Tameness in expansions of the real field. Logic Colloquium ’01, 281–316, Lect. Notes Log., 20, Assoc. Symbol. Logic, Urbana, IL, 2005.

pillay87 [Pillay87] A. Pillay. First order topological structures and theories. J. Symbolic Logic 52 (1987), no. 3, 763–778.

poizat85 [Poizat85] B. Poizat. Cours de théorie des modèles. Une introduction à la logique mathématique contemporaine. Nur al-Mantiq wal-Ma’rifah. Bruno Poizat, Lyon, 1985. vi+584 pp.

poizat87 [Poizat87] B. Poizat. Stable groups. Mathematical Surveys and Monographs, 87. American Mathematical Society, Providence, RI, 2001. xiv+129 pp.

robinson74 [Robinson74] A. Robinson. A note on topological model theory. Fund. Math. 81 (1973/74), no. 2, 159–171.

ZT [TZ08] K. Tent and M. Ziegler. Model theory. Preprint, 2009-05-27. V. 0.2.414.

wagner [Wagner00] F.O. Wagner. Simple theories. Mathematics and its Applications, 503. Kluwer Academic Publishers, Dordrecht, 2000. xii+260 pp.

wood [Wood76] C. Wood. The model theory of differential fields revisited. Israel J. Math. 25 (1976), no. 3-4, 331–352.