LOVELY PAIRS AND DENSE PAIRS OF O-MINIMAL STRUCTURES

ALEXANDER BERENSTEIN

Abstract. We study the theory of Lovely pairs of þ-rank one theories, in particular O-minimal theories. We show that the class of ℵ_{0}-saturated dense pairs of O-minimal structures studied by van den Dries [6] agrees with the corresponding class of lovely pairs. We also prove that the theory of lovely pairs of O-minimal structures is super-rosy of rank ≤ ω.

1. Introduction

This paper brings together results on dense pairs by van den Dries [6] and lovely pairs of rank one simple theories developed by Vassiliev [14] using the framework of rosy theories. In [14] Vassiliev studies lovely pairs of a SU-rank one simple theory T and, provided T eliminates the quantifier ∃^{∞}, shows that the theory of the lovely pairs of T exists and it is simple. The first goal of this paper is the generalize Vassiliev’s work and show that the theory of lovely pairs of þ-rank one rosy theories exists provided the original theory eliminates the quantifier ∃^{∞}.

In [6] van den Dries studies dense pairs of O-minimal theories that expand the theory of ordered abelain groups, generalizing the classical work of Robinson on the completeness of the theory of real closed fields with a predicate for a real dense closed subfield [13]. The author shows that the theory of dense pairs is complete and gives a description of definable sets. It is well known that dense O-minimal theories eliminate the quantifier ∃^{∞} and are of þ-rank one (see section 5 of [10]), so we can consider the corresponding theory of lovely pairs. In this paper we show that the theory of lovely pairs of O-minimal theories expanding the theory of ordered abelian groups agrees with the corresponding theory of dense pairs. Part of the goals of this paper is to extend the ideas presented in [6] to the more general framework of lovely pairs.

Berenstein, Ealy and Gunaydin showed in [4] that the theory of dense pairs of O-minimal theories that expand the theory of ordered abelain groups is super-rosy of rank ≤ ω. The tools used in the proof depended mainly on the description of definable sets given by van den Dries in [6].

A key idea throughout this paper is the notion of smal set and smal closure. For (M, P (M)) a lovely pair, and X ⊂ M definable, we say that X is small if it is a subset of an image of a cartesian power of P (M) under a definable function. For a ∈ M and B ⊂ M, we write a ∈ scl(B) if there are g_{1}, …, g_{n} ∈ P (M) such that a ∈ dcl(B, g_{1}, …, g_{n}); that is, if a belongs to small subset defined over B. In this paper we follow the ideas of van den Dries (used for dense pairs) to describe definable subsets of lovely pairs of O-minimal structures in terms of small sets. Using this description of definable sets we prove that the theory of dense pairs of O-minimal structures is super-rosy; the proof follows the steps of the corresponding result in [4].

Date: August 2007.

Key words and phrases. rosy theories, O-minimal theories, dense pairs, lovely pairs.

I would like to thank Evgueni Vassiliev for some useful remarks.

Finally, following ideas of Buechler and Vassiliev [2, 14], we prove that the rank of a generic type is determined by the local geometry of the underlying O-minimal structure:

Main Theorem Let M be an O-minimal structure, let P (M) ⪯ M and assume that (M, P (M)) is a lovely pair. Given a ∈ M the following holds:

(1) If M is trivial in a neighborhood of a, then U^{þ}(tp(a)) ≤ 1 and equality holds if a is not algebraic.

(2) If M induces the structure of an ordered vector space over an ordered division ring in a neighborhood of a, then U^{þ}(tp(a)) ≤ 2 and equality holds if a ∉ scl(∅).

(3) If M induces the structure of an O-minimal expansion of a real closed field in a neighborhood of a, then U^{þ}(tp(a)) ≤ ω and equality holds if a ∉ scl(∅).

This paper is divided as follows. In the second section we study lovely pairs of þ-rank one structures that eliminate the quantifier ∃^{∞}. In the third section we study the definable sets of a lovely pair. In section four we give a more detailed analysis of definable sets of a lovely pair of O-minimal structures. In section five we prove that the theory of lovely pairs of O-minimal structures is super-rosy of rank ≤ ω. In section six we prove the Main Theorem.

We assume throughout this paper that the reader is familiar with the basic ideas of rosy theories presented in [10], [1]. We follow the notation from [4], we write capital leters such as C, D, X, Y for definable sets and sometimes we write Cb⃗ to emphazise that C is definable over b⃗. We may write b⃗ ∈ Cy⃗ to mean that b⃗ is a tuple of the same arity as y⃗ whose components belong to C.

2. Lovely pairs of þ-rank one structures

We begin by translating to the setting of þ-rank one theories, the definitions used by Vassiliev in [14]. Let T be a þ-rank one theory (see [10]) with quantifier elimination in a language L. By symmetry of thorn forking, in any model of T acl has the exchange property and defines a pregeometry. Examples of such theories includes strongly minimal theories, SU-rank one simple theories with quantifier elimination and O-minimal theories. Let P be a new unary predicate and let L_{P} = L ∪ {P }. Let T′ be the L_{P} -theory of all structures (M, P), where M ⊧ T and P (M) is an L-algebraically closed subset of M. Let T_{pairs} be the theory of elementary T -pairs, that is, the theory of structures of the form (M, P (M)) where P (M) ⪯ M and M ⊧ T.

Notation 2.1. Let (M, P (M)) ⊧ T′ and let A ⊂ M. We write P (A) for P (M)∩A.

Notation 2.2. Throughout this section independence means acl_{L}-independent, where acl_{L} means algebraic closure in the sense of L. We write tp(a⃗) for the L-type of a.

Definition 2.3. We say that a structure (M, P (M)) is a lovely pair of models of T if

(1) (M, P (M)) ⊧ T′

(2) (Density property) If A ⊂ M is algebraically closed and finite dimensional and q ∈ S_{1}(A) is non-algebraic, there is a ∈ P (M) such that a ⊧ q.

(3) (Extension property) If A ⊂ M is algebraically closed and finite dimensional and q ∈ S_{1}(A) is non-algebraic, there is a ∈ M, a ⊧ q and a ∉ acl(A ∪ P (M)).

Equivalently, we could follow the approach from [3] and define, for κ ≥ |T |^{+}, the class of κ-lovely pairs, replacing the condition A ⊂ M is algebraically closed and finite dimensional in the clauses (2) and (3) above for A ⊂ M is algebraically closed of dimension ≤ κ.

Note that if (M, P (M)) is a lovely pair, the extension property implies that M is ℵ_{0}-saturated. If (M, P (M)) is a κ-lovely pair, the extension property implies that M is κ-saturated and that M \ P (M) is non-empty. Assume now that T is an O-minimal theory and that a, b ∈ M are such that a < b; then the partial type a < x < b is non-algebraic and by the density property it is realized in P (M). Thus, the density property implies that P (M) is dense in M.

Lemma 2.4. Any lovely T -pair is an elementary T -pair.

Proof. We apply the Tarski-Vaught test. Let (M, P (M)) be a lovely T pair, let φ(x, y⃗) be an L-formula and let b⃗ ∈ P (M)y⃗. Assume that there is a ∈ M such that M ⊧ φ(a, b⃗). If a is algebraic over b⃗, since P (M) is algebraically closed we get a ∈ P (M). If a is not algebraic over b⃗, the type tp(a/b⃗) is not algebraic and by the density property there is a′ ∈ P (M) such that a′ ⊧ tp(a/b⃗); in particular, M ⊧ φ(a′, b⃗). □

We follow now section 3 of [3]. The existence of κ-lovely pairs follows from [3, Lemma 3.5]. The proof presented there does not use the Independence Theorem, in fact it only uses transitivity and the existence of non-forking extensions. Exchanging the word independence for þ-independence gives a proof in our setting.

Definition 2.5. Let A be a subset of a lovely pair (M, P (M)) of models of T. We say that A is P -independent if A is independent from P (M) over P (A).

Lemma 2.6. Let (M, P (M)) and (N, P (N)) be lovely pairs of modells of T. Let a⃗, b⃗ be finite tuples of the same length from M, N respectively, which are both P - independent. Assume that a⃗, b⃗ have the same quantifier free L_{P} -type. Then a⃗, b⃗ have the same L_{P} -type.

Proof. It is a similar argument to the one presented in [3, Lemma 3.8]. □

The previous result has the following consequence:

Corollary 2.7. All lovely pairs of T are elementary equivalent.

We write T_{P} for the common complete theory of all lovely pairs of T.

To axiomatize T_{P} we follow the ideas of [14, Prop 2.15]. Here we need an additional hypothesis, namely, we assume that T eliminates ∃^{∞}. It follows from a result of Hrushovski (Lemma 4.2 [9]) that a supersimple theory of SU-rank 1 satisfies this property. It is also well known, by uniform finiteness, that a dense O-minimal theory eliminates this quantifier. Recall that whenever T eliminates ∃^{∞} the expression the formula φ(x, b̄) is nonalgebraic is first order.

Theorem 2.8. Assume T eliminates ∃^{∞}. Then the theory T_{P} is axiomatized by:

(1) T′

(2) For all L-formulas φ(x, y⃗)

∀y⃗(φ(x, y⃗) nonalgebraic ⟹ ∃x(φ(x, y⃗) ∧ x ∈ P)).

(3) For all L-formulas φ(x, y⃗), m ∈ ω, and all L-formulas ψ(x, z_{1}, …, z_{m}, y⃗) such that for some n ∈ ω ∀z⃗∀y⃗∃^{≤n}xψ(x, z⃗, y⃗) (so ψ(x, y⃗, z⃗) is always algebraic in x) ∀y⃗(φ(x, y⃗) nonalgebraic ⟹ ∃x(φ(x, y⃗) ∧ x ∉ P) ∧ ∀w_{1} … ∀w_{m} ∈ P ¬ψ(x, w_{1}, …, w_{m}, y⃗))

The second scheme of axioms corresponds to the density property and the third scheme to the extension property.

Proof. Let T_{0} be the theory axiomatized by the scheme of axioms described above. Claim Any lovely T -pair is a model of T_{0}. Let (M, P (M)) be a lovely T -pair. Clearly it is a model of T′. Now let φ(x, y⃗) be

a formula, let b⃗ ∈ My⃗ and assume that φ(x, b⃗) is non-algebraic. Let B = acl(b⃗) and let p(x) be a non algebraic L-type over B extending φ(x, b⃗). Since (M, P (M)) is a lovely pair, by the density property p(x) is realized in P (M) and thus the second axiom holds. Now assume that ψ(x, z⃗, y⃗) is a formula such that there is n with the property that for all z⃗, y⃗ there are at most n realizations of ψ(x, z⃗, y⃗). Let φ(x, y⃗) be a formula, let b⃗ ∈ My⃗ be such that φ(x, b⃗) is non-algebraic. Let B = acl(b⃗) and let p(x) be a non algebraic L-type over B extending φ(x, b⃗). By the extension property there is c ∈ M realizing p and independent from P (M) over B. Let d⃗ ∈ P (M)z⃗, then c is not algebraic over d⃗b⃗, so M ⊧ ¬ψ(c, d⃗, b⃗) and the third axiom holds.

Claim Any ℵ_{0}-saturated model of T_{0} is a lovely pair.

Let (M, P (M)) ⊧ T_{0} be ℵ_{0}-saturated and let A ⊂ M be algebraically closed and finite dimensional. Let p(x) be a non-algebraic L-type over A. First consider the L_{P} partial type p(x) ∧ P (x). By the second axiom this partial type is finitely realizable and by ℵ_{0}-saturation it is realized in (M, P (M)). Thus (M, P (M)) satisfies the density property. Now consider the partial type p(x) ∪ {∀w⃗ ∈ P ¬ψ(x, w⃗, a⃗) : ψ is as in (3), a⃗ ∈ Ay⃗}. By the third axiom this type is finitely realizable in (M, P (M)) and by ℵ_{0}-saturation it is realized in (M, P (M)). Thus (M, P (M)) satisfies the extension property. □

We now compare lovely pairs with the dense pairs studied by van den Dries in [6]. We start by recalling some definitions from that paper:

Let L = {<, 0, 1, +, −, … } be a language and let T be an O-minimal L-theory that extends the theory of ordered abelian groups with a positive element 1.

Definition 2.9. A dense pair is an elementary pair (so P (M) ⪯ M) such that P (M) ≠ M and P (M) is dense in M.

Clearly any lovely T pair (M, P (M)) is a dense pair. It is proved in [6, Theorem 2.5] that the common theory of dense pairs is complete and thus it coincides with T_{P}. The study of T_{P} can be seen as a generalization of van den Dries’ work on dense pairs of O-minimal structures.

3. Definable sets

Fix T a þ-rank one theory that eliminates quantifiers and eliminates ∃^{∞} and let (M, P (M) ⊧ T_{P}. Our next goal is to obtain a description of definable subsets of M and P (M) in the language L_{P}.

We start by considering the L_{P} -definable subsets of M, we follow the ideas from [3, Corollary 3.11]. We will extend the language adding new relation symbols. Let L′_{P} be L_{P} together with new relation symbols R_{φ}(y⃗) for each L-formula φ(x⃗, y⃗). Let T′_{p} be the theory T_{P} together with the sentences ∀y⃗(R_{φ}(y⃗) ↔ ∃x⃗(P (x⃗) ∧ φ(x⃗, y⃗))). Since T_{p} is a complete theory so is T′_{p}. We will show that T′_{p} has quantifier elimination. We should point out that this result is also proved in [6, Theorem 2.5] for dense pairs of O-minimal structures that extends the theory of ordered abelian groups.

Lemma 3.1. Let (M, P (M)), (N, P (N)) be lovely pairs. Let a⃗, b⃗ be tuples of the same arity from M, N respectively. Then the following are equivalent:

(1) a⃗, b⃗ have the same quantifier-free L′_{P} -type.

(2) a⃗, b⃗ have the same L_{P} -type.

Proof. Clearly (ii) implies (i). Assume (i). Since L has quantifier elimination, tp(a⃗) = tp(b⃗). Since T is super-rosy, there is A ⊂ P (M) finite such that tp(a⃗/P (M)) does not þ-fork over A. Let q(z⃗, a⃗) be the L-type of A over a⃗. Since the quantifier free L′_{P} type of a⃗ agrees with the quantifier free L′_{P} type of b⃗, q(z⃗, b⃗) does not þ-fork over P (N). Since (N, P (N)) is a lovely pair, q(z⃗, b⃗) is realized in P (N), say by B.

Claim b⃗ is þ-free from P (N) over B.

Say b⃗ = (b_{1}, …, b_{n}) and assume that for some k ≤ n, (b_{1}, …, b_{k}) are B independent and b⃗ ∈ acl(B, b_{1}, …, b_{k}). If the claim does not hold, dim(b⃗/B ∪ P (N)) < k say b_{k} ∈ acl(b_{1}, …, b_{k−1}, B, P (N)). Let d_{1}, …, d_{m} ∈ P (N) such that b_{k} ∈ acl(b_{1}, …, b_{k−1}, B, d_{1}, …, d_{m}). Since the quantifier free L′_{P} type of a⃗, A agrees with the quantifier free L′_{P} type of b⃗, B, there are c_{1}, …, c_{m} ∈ P (M) such that a_{k} ∈ acl(a_{1}, …, a_{k−1}, A, d_{1}, …, d_{m}), a contradiction.

Also note that a⃗A, b⃗B have the same quantifier free L_{P} -type, so the result follows from Lemma 2.6. □

Now we are interested in the L_{P} -definable subsets of P (M). For this material we follow the presentation from [6, Theorem 2].

Proposition 3.2. Let (M, P (M)) be a lovely pair and let Y ⊂ P (M)^{n} be L_{P} - definable. Then there is X ⊂ M^{n} L-definable such that Y = X ∩ P (M)^{n}.

Proof. Let (M_{1}, P (M_{1})) ⪰ (M, P (M)) be κ-saturated where κ > |M| + |L| and let a⃗, b⃗ ∈ P (M_{1})^{n} such that tp(a⃗/M) = tp(b⃗/M). We will prove that tp_{P} (a⃗/M) = tp_{P} (b⃗/M) and the result will follow by compactness. Since a⃗, b⃗ ∈ P (M_{1})^{n}, we get that Ma⃗, Mb⃗ are P -independent sets and thus by Lemma 2.6 we get tp_{P} (a⃗/M) = tp_{P} (b⃗/M). □

4. More on definable sets: the O-minimal case

Fix T an O-minimal theory that eliminates the quantifier ∃^{∞}.

Definition 4.1. Let (M, P (M)) be a lovely pair of models of T. A definable set D ⊂ M^{k} is small if and only if there is some m, and an L-definable function f : M^{m} → M^{k} such that D ⊂ f(P (M)^{m}). Let F be a cell and let S ⊂ F be definable. We say S is large in F if F \ S is small. A definable subset D ⊂ M^{k} is basic small if it is small and of the form ∃y_{1} ∈ P … ∃y_{n} ∈ P φ(x⃗, y⃗), where φ(x⃗, y⃗) is an L-formula.

The definition above is what is called P (M)-bound in [4] and it turns out to be equivalent to the notion of small set from [4] (see Corollary 2.16). Note that if D_{1}, D_{2} ⊂ M^{k} are small their union is also small.

We need to refine the description of L_{P} -definable subsets of M that we obtained in the previous section. In particular, we want to generalize Theorem 4 of [6] to general lovely pairs of O-minimal structures. We will follow the strategy from [6] and we start by reproving Lemma 4.3 of [6]. The proof we present is the one given in [6], we include it for completeness.

Lemma 4.2. Let X ⊂ M be small. Then X is a finite union of sets of the form f(P (M)^{m} ∩ E) where E is an L-definable open cell in M^{m} and f : E → M is L-definable and continuous.

Proof. Since X is small, X ⊂ f(P (M)^{m}) for some L-definable function f from M^{m} into M. Thus we may write X = f(X′) for some L_{P} -definable set X′ ⊂ P (M)^{m}. By Proposition 3.2 we have X′ = P (M)^{m} ∩ Y for some L-definable Y ⊂ M^{m}. The rest of the proof is by induction on m. The case m = 0 is trivial, as X is either empty or a single point. So assume the result holds for values lower than m and we will prove it for m. We can subdivide Y into smaller cells E so that f ↾_{E} is continuous. If E is an open cell in M^{m} we get the conclusion of the lemma. If E is not open and dim(E) = d < m, there are indices 1 ≤ i_{1} < i_{2} < ⋯ < i_{d} ≤ m such that the projection map π : M^{m} → M^{d}, π(x_{1}, …, x_{n}) = (x_{i_{1}}, …, x_{i_{d}}) is homeomorphism from E into the open cell E′ = π(E) of M^{d}. Let μ be the inverse of this map. Then f(P (M)^{m} ∩ E) = (f ∘ μ)(P (M)^{d} ∩ E′ ∩ μ^{−1}(P (M)^{m})) and by Proposition 3.2 there is an L-definable set F′ ⊂ E′ such that P (M)^{d} ∩ E′ ∩ μ^{−1}(P (M)^{m}) = P (M)^{d} ∩ F′. By the induction hypothesis, f(P (M)^{m} ∩E) = (f ∘μ)(P (M)^{d} ∩F′) is of the desired form. □

Lemma 4.3. Let C ⊂ M^{k} be a cell. Then there is a partition C_{1}, …, C_{n} of C into cells such that C_{i} ∩ P (M)^{k} is either empty or a dense subset of C_{i}.

Proof. The proof is by induction on k. The result is clear for k = 0. Assume now that the result holds for values smaller than or equal to k and we will prove it for k + 1. First assume that C is the set of realizations of the formula f(y_{1}, …, y_{k}) < x < g(y_{1}, …, y_{k}) for y⃗ in a cell D and f, g continuous functions. By induction hypothesis we need to consider two cases. If D ∩ P (M)^{k} is dense in D, then C ∩ P (M)^{k+1} is dense in C. If D ∩ P (M)^{k} is empty, then so is C ∩ P (M)^{k+1}.

Now assume that C is of the form x = f(y_{1}, …, y_{k}) for y⃗ in a cell D and f a continuous function. Then there is d ≤ k and there are indices 1 ≤ i_{1} < i_{2} < ⋯ < i_{d} ≤ k + 1 such that the projection map π : M^{k+1} → M^{d}, π(x_{1}, …, x_{k+1}) = (x_{i_{1}}, …, x_{i_{d}}) is homeomorphism from C into the open cell C′ = π(C) of M^{d}. Let μ be the inverse of this map. Note that μ is a definable function. Then P (M)^{k+1} ∩ C = μ(P (M)^{d} ∩ C′ ∩ μ^{−1}(P (M)^{k+1})) and by Proposition 3.2 there is an L-definable set F ⊂ C′ such that P (M)^{d} ∩ C′ ∩ μ^{−1}(P (M)) = P (M)^{d} ∩ F. By the induction hypothesis we can find a finite partition F into cells {F_{j} : j ≤ n_{1}} such that either F_{j} ∩ P (M)^{d} = ∅ or F_{j} ∩ P (M)^{d} is dense in F_{j}. Furthermore, we can extend the partition {F_{j} : j ∈ J} to a partition {C′_{i} : i ≤ n_{2}} of C′ with the same properties. Since μ is a homeomorphism, {μ(C′_{j}) : j ∈ J} forms a partition of C into cells. Let C_{j} = μ(C′_{j}). Note that if C_{k} ∩ P (M)^{k+1} ≠ ∅, then π(C_{k}) ∩ P (M)^{d} ∩ μ^{−1}(P (M)^{k+1}) ≠ ∅, so π(C_{k}) = F_{j} for some j such that F_{j} ∩ P (M)^{d} is dense in F_{j}. Then μ(F_{j} ∩ P (M)^{d}) is a dense subset of C_{j}. Since P (M)^{d} ∩ F_{j} ⊂ P (M)^{d} ∩ C′ ∩ μ^{−1}(P (M)^{k+1}), μ(F_{j} ∩ P (M)^{d}) ⊂ P (M)^{k+1}, so C_{j} ∩ P (M)^{k+1} is a dense subset of C_{j}. □

Now we generalize Lemma 2.15 from [4]:

Proposition 4.4. Let D ⊂ M be definable in (M, P (M)) over d⃗. Then there is a partition −∞ = a_{0} < ⋯ < a_{n} = ∞ and basic small dense sets S_{1}, …, S_{n} such that D ∩ [a_{i−1}, a_{i}] is either contained in the set S_{i} or contains the set S_{i}^{c} ∩ [a_{i−1}, a_{i}], and each S_{i} is defined from d⃗.

Proof. We first show the result for sets D defined by formulas of the form ∃y_{1} … ∃y_{n}P (y_{1})∧ ⋯ ∧ P (y_{n}) ∧ φ(y_{1}, …, y_{n}, x), where φ(y_{1}, …, y_{n}, x) is a cell.

Assume the cell φ(y_{1}, …, y_{n}, x) is of the form f(y_{1}, …, y_{n}) < x < g(y_{1}, …, y_{n}) for y⃗ in a cell C and f, g continuous functions. Then, after subdividing C if necessary, we obtain two cases. If P (M)^{n} ∩ C is empty, then D is empty. If P (M)^{n} ∩ C is dense in C, then D is an open interval.

Now assume that the cell φ(y_{1}, …, y_{n}, x) is of the form x = f(y_{1}, …, y_{n}) for y⃗ in a cell C and f is a continuous function, which is either constant, strictly increasing or strictly decreasing. As above, after subdividing C if necessary, we obtain the following cases. If P (M)^{n} ∩ C is empty, then D is empty. If P (M)^{n} ∩ C is dense in C and f is constant, then D is a point. If f is strictly monotone, then D is a dense small subset of f(C).

Clearly if the conclusion of the Proposition holds for a set D, then it also holds for the complement of D. It remains to see what happens with intersections. Assume that D_{1}, D_{2} are definable over d⃗ and that there is a partition −∞ = a_{0} < ⋯ < a_{n} = ∞ and basic small dense sets S_{11}, …, S_{1n}, S_{21}, …, S_{2n} as prescribed by the Proposition for D_{1}, D_{2} respectively. If D_{1} ∩ [a_{i}, a_{i+1}] ⊂ S_{i1}, then (D_{1} ∩ D_{2}) ∩ [a_{i}, a_{i+1}] ⊂ S_{i1}. On the other hand, if D_{1} ∩ [a_{i}, a_{i+1}] ⊃ S_{i1}^{c} ∩ [a_{i}, a_{i+1}], D_{2} ∩ [a_{i}, a_{i+1}] ⊃ S_{i2}^{c} ∩ [a_{i}, a_{i+1}], then D_{1} ∩ D_{2} ∩ [a_{i}, a_{i+1}] ⊃ (S_{i1} ∪ S_{i2})^{c} ∩ [a_{i}, a_{i+1}]. □

Proposition 4.5. If X ⊂ M is L_{P} -definable and small, then there is a partition −∞ = b_{0} < b_{1} < ⋯ < b_{k+1} = ∞ of M such that for each i = 0, …, k, either X ∩(b_{i}, b_{i+1}) = ∅, or X ∩(b_{i}, b_{i+1}) as well as (b_{i}, b_{i+1})\X are dense in (b_{i}, b_{i+1}). If X ⊂ M is L_{P} -definable then there is a partition −∞ = b_{0} < b_{1} < ⋯ < b_{k+1} = ∞ of M such that for each i = 0, …, k, either X ∩ (b_{i}, b_{i+1}) = ∅, or X ∩ (b_{i}, b_{i+1}) = (b_{i}, b_{i+1}) or X ∩ (b_{i}, b_{i+1}) as well as (b_{i}, b_{i+1}) \ X are dense in (b_{i}, b_{i+1}).

Proof. Let X ⊂ P (M) be small. By Lemma 4.2 we can write X as a finite union of sets f(P (M)^{m} ∩ E) where E ⊂ M^{m} is an open cell and f is L-definable continuous function. If X is a single point there is nothing to prove, so we may assume that f(E) is an interval possibly with endpoints. The set f(P (M)^{m} ∩ E) is dense in f(E) and by the extension property f(E) \ f(P (M)^{m} ∩ E) is also dense in f(E). The second part of the Proposition follows from the first part and from Proposition 4.4. □

5. þ-rank

In [4, Theorem 3’] it is shown that:

Theorem 5.1. Suppose that (R, +, …) is an o-minimal expansion of a group in the language L. Consider the expansion ℜ = (R, P, +, …) in the language L_{P} = L ∪ {P } where P is a unary predicate such that:

(1) P (R) is small, and contained in some interval, (a, ∞) ⊆ R, in which it is dense.

(2) Each L_{P} -formula ψ(x) is equivalent to a boolean combination of formulas of the form ∃y⃗(P (y_{1}) ∧ ⋯ ∧ P (y_{n}) ∧ φ(x, y⃗)) where φ is an L-formula.

(3) For each definable D ⊆ P (R)^{k} there is an L-definable set E such that

D = E ∩ P (R)^{k} Then ℜ is rosy of þ-rank less than or equal to ω and þ-rank of P (R) is 1. Moreover, if ℜ includes a field structure, þ-rank of ℜ equals ω.

Our goal in this section is to prove that for (M, P (M)) ⊧ T_{p}, þ-rank (x = x) = ω and þ-rank(P (x)) = 1. Our approach follows the ideas of [4], but we need to modify slightly the proofs as we do not have necessarily the structure of an ordered abelian group.

Notation 5.2. For (M, P (M)) ⊧ T_{P}, a ∈ M, B ⊂ M we write tp(a/B) for the L-type of a over B and tp_{P} (a/B) for the L_{P} -type of a over B.

We start by recalling some technical results from [7] on þ-rank:

Fact 5.3. Let T be a complete theory and let N ⊧ T. If D ⊂ N^{k} is definable and has þ-rank α, then D^{n} has þ-rank at least αn and equality holds if α = 1.

Fact 5.4. Let T be a complete theory and let N ⊧ T. Let D ⊂ N^{k}, E ⊂ N^{l} be definable and assume there is a definable function f : D ↠ E. Then if D has þ-rank α, then E has þ-rank ≤ α. Furthermore, if the fibers are finite , we have equality.

The following Lemma is a generalization of Lemma 44 in [4].

Lemma 5.5. If φ(x, b⃗) φ(x, b⃗) is an infinite set definable in L, then φ(x, b⃗) does not þ-divide over the empty set.

Proof. Assume, for a contradiction, that φ(x, b⃗) does þ-divide over the empty set. So tp_{P} (b⃗) is non-algebraic and there is some θ(y⃗, c⃗) ∈ tp(b⃗) and some k ∈ ℕ such that whenever b⃗_{1}, …, b⃗_{k} are distinct elements of θ(My⃗^{eq}, c⃗), we have that φ(x, b⃗_{1}) ∧ ⋯ ∧ φ(x, b⃗_{k}) is inconsistent. Since φ defines an infinite L-definable set, by the O-minimality of M|_{L}, it defines a finite collection of points and open intervals.

We may assume that each φ(x, b⃗) defines a single interval, modifying φ and θ if necessary. Since (M, <) is a linear order, after modifying φ and θ we may assume that φ(x, b⃗) 2-þ-divides. Then ∃y⃗(θ(y⃗)∧φ(x, y⃗)) defines an infinite union of disjoint open intervals, a contradiction with Proposition 4.5. □

Theorem 5.6. 𝔐 = (M, P (M)) is rosy of þ-rank less than or equal to ω and þ-rank of P (M) is 1.

Proof. First we wish to show that the þ-rank of P (M) is 1. For a contradiction, suppose that some formula φ(x, b⃗) which defines a infinite subset of P (M) þ-divides over the empty set, where b⃗ may come from any sort in (M, P (M))^{eq}. Say that k, θ(y⃗, c⃗) are such that ⋀_{i<k} φ(x, b⃗_{i}) is inconsistent for any k distinct elements b⃗_{1}, …, b⃗_{k} satisfying θ(x, c⃗).

Then, by Proposition 3.2 φ(x, b⃗) is of the form ψ(x, b⃗) ∩ P (M), where ψ(x, b⃗) is an L formula. Without loss of generality, we may assume that ψ(x, b⃗) defines a single open interval. By the previous lemma, ψ(x, b⃗) does not þ-divide, so we may find an infinite set {b⃗_{i} : i ∈ K} such that b⃗_{i} ∈ θ(My⃗^{eq}, c⃗) and ⋀_{i∈K} ψ(M, b⃗_{i}) is nonempty and, hence, contains an open set (d_{1}, d_{2}).

Then ⋀_{i∈K} φ(M, b⃗_{i}) = ⋀_{i∈K}(ψ(M, b⃗_{i}) ∩ P (M)) ⊃ P (M) ∩ (d_{1}, d_{2}) ≠ ∅, so ⋀_{i∈K} φ(M, b⃗_{i}) is non-empty, a contradiction.

Second, we wish to show that the þ-rank of x = x is no larger than ω. Suppose that φ(x, b⃗) k-þ-divides over the empty set, where, again, b⃗ may come from any sort in (M, P (M))^{eq}. We observe that it suffices to show that Db⃗ := φ(M, b⃗) must be a small set and by Fact 5.4 and Fact 5.3 we can conclude that it has finite þ-rank. Then we will have shown that any formula, φ(x, b⃗), which þ-divides has finite þ-rank, and, thus, þ-rank(x = x) ≤ ω.

Now assume for a contradiction that φ(x, b⃗) is not a small set. By Proposition 4.4 there is some open interval Ib⃗ such that Db⃗ is large in Ib⃗. Suppose that θ(y⃗, c⃗) is such that for any b⃗_{1}, …, b⃗_{k} different realizations of θ(y⃗, c⃗), one has

Db⃗_{1} ∩ ⋯ ∩ Db⃗_{k} = ∅.

Claim J := Ib⃗_{1} ∩ ⋯ ∩ Ib⃗_{k} = ∅.

Otherwise 𝕁 is an open interval (d_{1}, d_{2}). Let Sb⃗ be a small set such that Db⃗ = I_{b} \ Sb⃗. Then (Db⃗_{1} ∩ ⋯ ∩ Db⃗_{k}) ∩ (d_{1}, d_{2}) = J \ (Sb⃗_{1} ∪ ⋯ ∪ Sb⃗_{k}) ≠ ∅ by the extension property.

Thus, if ψ(x, b⃗) defines Ib⃗, we see that ψ(x, b⃗) also þ-divides. But since intervals are L-definable, this contradicts the previous lemma. Thus we conclude that þ-rank(x = x) is no greater than ω.

□

Note that we also proved the following result, that will prove useful later

Corollary 5.7. If φ(x, b⃗) þ-forks over A, then φ(x, b⃗) defines a small set.

6. þ-rank and the trichotomy Theorem

There is strong relationship between the pregeometry associated to an SU-rank one theory and the rank of the associated lovely pair. It was shown by Buechler in [2] that for a strongly minimal theory T, T_{P} is totally transcendental and that MR(T_{P}) = 1 if T is trivial, MR(T_{P}) = 2 is T is locally modular non-trivial and that MR(T_{P}) = ω in all other cases. This result was generalized by Vassiliev in [14], where he showed that for T a simple theory of SU-rank one, SU(T_{P}) = 1 if T is trivial, SU(T_{P}) = 2 is T is locally modular non-trivial and that SU(T_{P}) = ω in all other cases.

In [4] Berenstein, Ealy and Günaydin showed that for T = T h(ℝ, +, 0, 1, <), þrank(T_{P}) = 2 and for T = T h(ℝ, +, ×, 0, 1, <), þ-rank(T_{P}) = ω. When these results were proved, the authors were interested in showing that there was an analogy between the rank of the dense pairs of O-minimal theories and the ranks of lovely pairs of simple theory of SU-rank one.

Our goal in this subsection is to study, for (M, P (M)) a lovely pair and a ∈ M, the relation between the rank of tp_{P} (a) and the local L-structure that M induces on a neighborhood of a. According to the Trichotomomy Theorem of Peterzil and Stacherko [11, 12], for any a in M, either a is trivial (in which case we prove that U^{þ}(tp_{P} (a)) = 1 for a non-algebraic), or there is a convex neighborhood of a where the structure is an ordered vector space over an ordered division ring (in which case we show that U^{þ}(tp_{P} (a)) = 2 for a sufficiently general), or there is a neighborhood where the structure is that of an expansion of a real closed field (and then U^{þ}(tp_{P} (a)) = ω for generic a).

We start with some preliminary observations.

Proposition 6.1. Assume that the structure around a is that of an ordered vector space over an ordered division ring R. Then for B ⊂ M a ∈ scl(B) if and only if there is b ∈ dcl(B) and n ∈ R^{>0} such that a ∈ b + P (M)/n.

Proof. Right to left is clear. Now assume that a ∈ scl(B). By Proposition 4.4, a is contained in S, a basic small set defined over B. Let ∃y⃗ ∈ P (M)^{k}(φ(x, y⃗)) be a formula defining S. For each g⃗, φ(M, g⃗) is a finite union of points and intervals. However, if for any g⃗ in P (M)^{k}, φ(M, g⃗) contains a non-empty open interval, then S is not small. Thus, we may reduce to the case where φ(x, y⃗) is x = f(y⃗), where

f(y⃗) = b + ∑_{i=1}^{k} \frac{m_{i}}{n_{i}} y_{i}

for some b ∈ dcl(B), m_{i} ∈ R and n_{i} ∈ R^{>0}. Let n be the least common multiple of the n_{i}. Thus f(G^{k}) is contained in b + P (M)/n, and a ∈ b + P (M)/n. □

We are ready to show our main Theorem for this section:

Main Theorem Let M be an O-minimal structure, let P (M) ⪯ M and assume that (M, P (M)) is a lovely pair. Given a ∈ M the following holds:

(1) If M is trivial in a neighborhood of a, then U^{þ}(tp(a)) ≤ 1 and equality holds if a is not algebraic.

(2) If M induces the structure of an ordered vector space over an ordered division ring in a neighborhood of a, then U^{þ}(tp(a)) ≤ 2 and equality holds if a ∉ scl(∅).

(3) If M induces the structure of an O-minimal expansion of a real closed field in a neighborhood of a, then U^{þ}(tp(a)) ≤ ω and equality holds if a ∉ scl(∅).

Proof. (1) Trivial case. If a ∈ scl(∅) then by triviality there is b ∈ P (M) such that a ∈ dcl(b). Since þ-rk(P (M)) = 1, we get U^{þ}(tp(a)) ≤ 1. So assume that a ∉ scl(∅) and that B ⊂ M is such that tp(a/B) þ-forks over ∅. Then a ∈ scl(B), so a ∈ dcl(B ∪ P (M)). Since M is trivial in a neighborhood of a and a ∉ P (M) we get that a ∈ dcl(B) so U^{þ}(tp(a/B)) = 0 and U^{þ}(tp(a)) ≤ 1.

(2) Locally modular case. By Proposition 6.1, every small subset of M has þrank at most one, and by Corollary 5.7, a þ-forking extension of tp(a) must include a formula defining a small set. Thus U^{þ}(tp(a)) ≤ 2. We want to show that for a ∉ scl(∅) we have U^{þ}(tp(a)) = 2.

Let g ∈ P (M) be such that tp(g) is non-algebraic and g | ⌣^{þ} a. Then we get U^{þ}(tp(a)) = U^{þ}(tp(a/g)) = U^{þ}(tp(a + g/g)).

Claim a + g | ⌣^{þ} g.

Otherwise, by Corollary 5.7 we would have a + g ∈ scl(g) = scl(∅), and thus a ∈ scl(∅), a contradiction.

Thus U^{þ}(tp_{P} (a)) = U^{þ}(tp_{P} (a + g/g)) = U^{þ}(tp_{P} (a + g)), and it suffices to show that U^{þ}(tp_{P} (a + g)) = 2.

Consider the chain tp_{P} (a + g/∅) ⊂ tp_{P} (a + g/a) ⊂ tp_{P} (a + g/a, g). First note that tp_{P} (a + g/a) contains the formula saying x ∈ P (M) + a. This formula is true of a + g and þ-divides over the empty set. Thus, tp_{P} (a + g/a) is a þ-forking extension of tp_{P} (a + g). Second, note that tp_{P} (a + g/a, g) is algebraic, and thus it is a þ-forking extension of tp_{P} (a + g/a). We just proved that the chain described above þ-forks at every step, so U^{þ}(tp(a)) = 2.

(3) Field case. Assume that in an open neighborhood V of a, M induces a field structure. By Corollary 5.7 for any set B, if tp(a/B) þ-forks over ∅ we get that a ∈ scl(B). In particular, there are g_{1}, …, g_{n} ∈ P (M) such that a ∈ dcl(g_{1}, …, g_{n}, B). By Lascar’s inequality this implies U^{þ}(tp_{P} (a/B)) ≤ n and we get that U^{þ}(tp_{P} (a)) ≤ ω.

To show the other direction, let us assume that a ∉ scl(∅) and we show that for every n ≥ 0, there exists B such that U^{þ}(tp_{P} (a/B)) = n. Let c_{1}, …, c_{n} ∈ V be such that c_{1} ∉ scl(a), c_{2} ∉ scl(a, c_{1}), …, c_{n} ∉ scl(a, c_{1}, …, c_{n_{1}}) (these elements exist by the extension property). Now let g_{1}, …, g_{n} ∈ P (M) be independent from each other and independent from c_{1}, …, c_{n}, a.

Claim g_{i} ∈ dcl(c_{1}g_{1} + … c_{n}g_{n}, c_{1}, …, c_{n}) for i ≤ n.

Consider the equation c_{1}x_{1} + ⋯ + c_{n}x_{n} = c_{1}g_{1} + … c_{n}g_{n}. If there is a solution (g′_{1}, …, g′_{n}) in P (M)^{n} different from (g_{1}, …, g_{n}) we get c_{1}(g_{1} − g′_{1}) + ⋯ + c_{n}(g_{n} − g′_{n}) = 0 and g_{j} − g′_{j} ≠ 0 for some j ≤ n. Then c_{j} ∈ scl(c_{1}, …, c_{j−1}, c_{j+1}, …, c_{n}) and this is a contradiction.

Let d = a + c_{1}g_{1} + … c_{n}g_{n} and B = {d, c_{1}, …, c_{n}}. Then a and c_{1}g_{1} + … c_{n}g_{n} are interdefinable over B and by the claim both this sets are interdefinable with {g_{1}, …, g_{n}} over B. Thus U^{þ}(tp(a/B)) = U^{þ}(tp(g_{1}, …, g_{n}/B)). On the other hand, a ∉ scl{c_{1}, …, c_{n}}, so d ∉ scl{c_{1}, …, c_{n}} and d | ⌣^{þ}{c_{1}, …, c_{n}, g_{1}, …, g_{n}}. This implies that U^{þ}(tp(g_{1}, …, g_{n}/B)) = U^{þ}(tp(g_{1}, …, g_{n}/{c_{1}, …, c_{n}})) = n and thus U^{þ}(tp(a/B)) = n as we wanted. □

We end this section with an example of a trivial dense pair.

Lemma 6.2. The structure (ℝ, <, ℚ) is a lovely pair.

Proof. We first show that the Density property holds. Let A ⊂ R be finite, say A = {a_{1}, …, a_{k}} with a_{1} < a_{2} < ⋯ < a_{k} and let q ∈ S_{1}(A) be non-algebraic. Then q is describing an open interval, either (−∞, a_{1}), (a_{i}, a_{i+1}) for some i, or (a_{k}, ∞). Since ℚ is dense in ℝ there is c ∈ P (ℝ) = ℚ such that c ⊧ q.

Now we show that the Extension property holds. Let A ⊂ R be finite, say A = {a_{1}, …, a_{k}} with a_{1} < a_{2} < ⋯ < a_{k} and let q ∈ S_{1}(A) be non-algebraic. Then q describes an open interval with endpoints in the set A. Since ℝ \ (A ∪ ℚ) is dense in ℝ, we can find a realization of q in ℝ \ (A ∪ ℚ). □

It is easy to check that the pair (ℝ, <, ℚ) is an expansion of (ℝ, <) with a generic predicate (in the sense of Chatzidakis, Pillay [5]). It is proved in [5, Corollary 2.6 part 3] that for such expansions, the algebraic closure in the extended language L_{p} coincides with the algebraic closure in the language L. In particular, algebraic independence inside the structure (ℝ, <, ℚ) satisfies the usual properties of an independence relation for real elements. On the other side, Sergio Fratarcangeli showed [8] that expansions of O-minimal structures with a generic predicate eliminate imaginaries. Thus algebraic independence inside the structure (ℝ, <, ℚ) defines an independence relation that extends to an independence relation for all elements in (ℝ, <, ℚ)^{eq} and thus T_{P} is rosy and acl_{L}-independence oincides with thorn-forking independence in the sense of T_{P}. Furthermore þ-rank(T h((ℝ, <, ℚ))) = 1, as we expected from the Main Theorem.

References

[1] H. Adler, A geometric introduction to forking and thorn-forking, preprint.

[2] S. Buechler, Pseudoprojective Strongly Minimal Sets are Locally Projective, Journal of Symbolic Logic, 56, (1991) , pp. 1184-1194

[3] I. Ben Yaacov, A. Pillay, E. Vassiliev, Lovely pairs of models, Annals of pure and applied logic 122 (2003), pp. 235–261.

[4] A. Berenstein, C. Ealy, A. Günaydin, Thorn independence in the field of real numbers with a small multiplicative group, to be published in Annals of Pure and Applied Logic.

[5] Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Annals of Pure and Applied Logic, (1998), pp. 71–92.

[6] L. van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), pp. 61–78.

[7] C. Ealy and A. Onshuus, Characterizing Rosy Theories, preprint.

[8] S. Fratarcangeli, Elimination of Imaginaries in expansions of o-minimal structures by generic sets, Journal of Symbolic Logic 70, (2005), pp. 1150–1160.

[9] E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

[10] A. Onshuus, Properties and consequences of thorn-independence, Journal of Symbolic Logic, 71 (2006), pp 1–21.

[11] Y. Peterzil and S. Starchenko, A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society, (2000), pp. 481–523.

[12] Y. Peterzil and S. Starchenko, Geometry, Calculus and Zil’ber’s conjecture, The bulletin of symbolic logic, vol 2, no 1, (1996), pp. 72–83.

[13] A. Robinson, Solution of a problem of Tarski, Fund. Math. 47 (1959), pp. 179–204.

[14] E. Vassiliev, Generic pairs of SU-rank 1 structures, Annals of Pure and Applied Logic, 120, (2003), pp. 103–149.

Universidad de los Andes, Cra 1 No 18A-10, Bogotá, Colombia

  E-mail address: aberenst@uniandes.edu.co
  URL: www.pentagono.uniandes.edu.co/~aberenst