STRUCTURES OF SU-RANK OMEGA WITH A DENSE INDEPENDENT SUBSET OF GENERICS

ALEXANDER BERENSTEIN, JUAN FELIPE CARMONA, AND EVGUENI VASSILIEV

Abstract. Extending the work done in [5, 9] in the o-minimal and geometric settings, we study expansions of models of a supersimple theory of SU-rank ω with a "dense codense" independent collection H of elements of SU-rank ω, where density of H means it intersects any definable set of SU-rank omega. We show that under some technical conditions, the class of such structures is first order. We prove that the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, we describe a natural "geometry of generics modulo H" associated with such expansions and show it is modular.

1. Introduction

There are several papers that deal with expansions of simple theories with a new unary predicate. For example, there is the expansion with a random subset [8] that gives a case where the new theory is again simple and forking remains the same, in contrast to the case of lovely pairs [2, 15], where the pair is usually much richer and the complexity of forking is related to the geometric properties of the underlying theory [15].

In [5] the first and the third authors studied, in the setting of geometric structures, adding a predicate for an algebraically independent set H which is dense and codense in a model M (meaning every non-algebraic formula in a single variable has a realization in H and a realization generic over H and its parameters). The paper generalized ideas developed in the framework of o-minimal theories in [9]. The key tool used in [5] was that the closure operator acl has the exchange property and thus gives a matroid that interacts well with the definable subsets. A special case under consideration was SU-rank one theories, where forking independence agrees with algebraic independence. In this stronger setting the authors characterized forking and gave a description of canonical bases in the expansion. As in the lovely pair case, the complexity of forking is related to the underlying geometry of the base theory T.

In this paper we start with a theory T that has SU-rank omega and we use the closure operator associated to the weight of generic types, namely for M ⊧ T, a ∈ M, A ⊂ M, we have a ∈ cl(A) if SU(a/A) < ω. This closure operator has the exchange property and many of the results obtained in [5] can be proved in the new framework: we expand M by a new predicate consisting in a cl-dense cl-codense family of independent generics (see Definition 2.3). In particular, the extension is supersimple and we get a clear description of canonical bases in the expansion, up to interalgebraicity (see Proposition 5.6).

1991 Mathematics Subject Classification. 03C45.

Key words and phrases. supersimple theories, SU-rank omega, unary predicate expansions, one-basedness, ampleness, CM-triviality .

In the special case where the theory of M is superstable with a unique type of U-rank ω, the predicate H is a Morley sequence of generics; this case is related to the work done in [1]. Our work is also related to work of Fornasiero on lovely pairs of closure operators [10].

Of special interest is the effect of our expansion on the geometric complexity, namely the ampleness hierarchy. Following the ideas of [7], we show that the expansion preserves CM-triviality, but one-basedness is preserved only in the trivial case.

We then use this expansion to study the underlying geometry of the closure operator localized in H. We show that if T is a one-based supersimple theory of SU-rank ω, (N, H) a sufficiently (e.g. |T |^{+}-) saturated H-structure, then the localized closure operator cl(− ∪ H) is modular and its associated geometry is a disjoint union of projective geometries over division rings and trivial geometries.

This paper is organized as follows. In section 2 we define H-structures associated to models M of a theory T. We show that two H-structures associated to the same theory are elementary equivalent and call T^{ind} this common theory. Finally we prove that that under some technical conditions (elimination of the quantifier ∃^{large} and the type definablity of the predicates Q_{φ,ψ}) the saturated models of T^{ind} are again H-structures.

In section 3 we study four different examples of theories of SU-rank ω: differentially closed fields, vector spaces with a generic automorphism, H-pairs and lovely pairs of geometric theories. In each case we show the corresponding theory of H-structures is first order.

In section 4 we analyze the definable sets in the expansion, we prove that every definable set is a boolean combination of old formulas bounded by existential quantifiers over the new predicate. In section 5 we characterize forking in the expansion and characterize canonical bases. In section 6 we study the question of preservation of one-basedness and CM-triviality under our expansion. Finally in section 7 we study the geometry of cl(− ∪ H).

2. H-structures: definition and first properties

Let T be a simple theory of SU-rank ω. Let H be a new unary predicate and let cl_{H} = cl ∪{H}. Let T′ be the L_{H}-theory of all structures (M, H), where M ⊧ T and H(M) is an independent subset of generic elements of M, that is, all elements have SU-rank ω. Note that saying that H(M) is an independent collection of generics is a first order property, it is simply the conjunctions of formulas of the form ¬φ(x_{1}, …, x_{n}), where SU(φ(x_{1}, …, x_{n})) < ωn.

For M ⊧ T, A ⊂ M and b ∈ M, we write b ∈ cl(A) and say that b is small over A if SU(b/A) < ω. By the additivity properties of SU rank we have that L gives a pregeometry on M. We write dim_{cl}(φ(x_{1}, …, x_{n})) = n and say that φ(x_{1}, …, x_{n}) is large if SU(φ(x_{1}, …, x_{n})) = ωn

We will assume that for every formula φ(x, y⃗) there is a formula ψ(y⃗) such that for any a⃗ ∈ M φ(x, a⃗) is large if and only if ψ(a⃗). We write ∃^{large}φ(x, y⃗) if ψ(y⃗) holds.

There is a strong analogy to what happens in geometric theories (see [3]), we change the pregeometry acl for the pregeometry L and the quantifier ∃^{∞} for the quantifier ∃^{large}.

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

Notation 2.2. Throughout this paper independence means independence in the sense of T and we use the familiar symbol | ⌣. We write tp(a⃗) for the L-type of a and dcl, acl for the definable closure and the algebraic closure in the language L. Similarly we write dcl_{H}, acl_{H}, tp_{H} for the definable closure, the algebraic closure and the type in the language L_{H}.

Definition 2.3. We say that (M, H(M)) is an H-structure if (1) (M, H(M)) ⊧ T′

(2) (Density/coheir property) If A ⊂ M is finite and q ∈ S_{1}(A) is the type of a generic element (of SU-rank ω), there is a ∈ H(M) such that a ⊧ q.

(3) (Co-density/extension property) If A ⊂ M is finite and q ∈ S_{1}(A), there is a ∈ M, a ⊧ q and a | ⌣_{A} H(M).

Lemma 2.4. Let (M, H(M)) ⊧ T′. Then (M, H(M)) is an H-structure if and only if:

(2’) (Generalized density/coheir property) If A ⊂ M is finite and q ∈ S_{n}(A) has

SU-rank ωn, then there is a⃗ ∈ H(M)^{n} such that a⃗ ⊧ q. (3’) (Generalized co-density/extension property) If A ⊂ M is finite dimensional

and q ∈ S_{n}(A), then there is a⃗ ∈ M^{n} realizing q such that tp(a⃗/A ∪ H(M)) does not fork over A.

Proof. We prove (2’) and leave (3’) to the reader. Let b⃗ ⊧ q, we may write b⃗ = (b_{1}, …, b_{n}). Since (M, H(M)) is an H-structure, applying the density property we can find a_{1} ∈ H(M) such that tp(a_{1}/A) = tp(b_{1}/A). Let q(x, b_{1}, A) = tp(b_{2}, b_{1}, A) and let A_{1} = A ∪ {a_{1}}. Finally consider the type q(x, a_{1}, A) over A_{1}, which is the type of a generic element. Applying the density property we can find a_{2} ∈ H(M) such that tp(a_{2}, a_{1}/A) = tp(b_{2}, b_{1}/A). We continue inductively to find the desired tuple (a_{1}, a_{2}, …, a_{n}). □

Note that if (M, H(M)) is an H-structure, the extension property implies that M is ℵ_{0}-saturated.

Definition 2.5. Let A be a subset of an H-structure (M, H(M)). We say that A is H-independent if A is independent from H(M) over H(A).

Lemma 2.6. Any model M of T with a distinguished independent subset H(M) can be embedded in an H-structure in an H-independent way.

Proof. Given any model M with a distinguished independent subset H(M) of generics, we can always find an elementary extension N of M and a set H(N) extending H(M) such that for every generic 1-type p(x, acl(m⃗)) (i.e. SU(p(x)) = ω), where m⃗ ∈ M, there is d ∈ N such that d ⊧ p(x, acl(m⃗)) and d | ⌣_{H(M)} m⃗. Add a similar statement for the extension property. Now apply a chain argument. □

In particular, for a SU-rank ω theory T, H-structures exist.

Lemma 2.7. Let (M, H) and (N, H) be sufficiently saturated H-structures, a⃗ ∈ M and a′ ∈ N H-independent tuples such that tp(a⃗, H(a⃗)) = tp(a′, H(a′)). Then tp_{H}(a⃗) = tp_{H}(a′).

Proof. Write a⃗ = a⃗_{0}a⃗_{1}h⃗, where a⃗_{0} is independent over H(M), h⃗ = H(a⃗) ∈ H(M) and a⃗_{1} ∈ L(a⃗_{0}h⃗). Similarly write a′ = a′_{0}a′_{1}h⃗′.

It suffices to show that for any b ∈ M there are h⃗_{1} ∈ H(M), h⃗′_{1} ∈ H(N) and b′ ∈ N such that a⃗h⃗_{1}b and a′h⃗′_{1}b′ are each H-independent, tp(a⃗_{0}a⃗_{1}h⃗h⃗_{1}b) = tp(a′_{0}a′_{1}h⃗′h⃗′_{1}b′), and b ∈ H(M) iff b′ ∈ H(N).

Case 1: b ∈ cl(a⃗) ∩ H(M). By H-independence of a⃗, we must have b ∈ cl(h⃗) and since H forms an independent set we must have b ∈ h⃗. Let b′ ∈ h⃗′ be such that tp(b′a′) = tp(ba⃗) and the result follows. Here we can take h⃗_{1} and h⃗′_{1} to be empty

Case 2: b ∈ H(M) and is non small over a⃗. Then tp(b/a⃗) is generic. By the density property, we can find b′ ∈ H(N) such that tp(b′a′) = tp(ba⃗). Here again we can take h⃗_{1} and h⃗′_{1} to be empty.

Case 3: b ∈ cl(a⃗). We claim that b | ⌣a⃗ H(M). Indeed let h⃗_{1} (say of length k) in H(M) \ h⃗. Since a⃗ is H-independent, the elements in H(M) \ h⃗ are independent over a⃗ and thus SU(h⃗_{1}/a⃗) = SU(h⃗_{1}/h⃗) = ωk. On the other hand SU(b/a⃗) < ω, so the types tp(b/a⃗), tp(h⃗_{1}/a⃗) are orthogonal and the claim follows.

Thus the tuple a⃗b is H-independent. Let p(x, a⃗) = tp(b/a⃗). Now use the extension property to find b′ ∈ N′ such that b′ ⊧ p(x, a′), b′ | ⌣_{a′} H(N), so by transitivity a′b′ is H-independent.

Case 4: b ∈ cl(H(M)a⃗). Add a tuple h⃗_{1} ∈ H(M) such that a⃗bh⃗_{1} is Hindependent, and use Case 2 and Case 3.

Case 5: b ∉ cl(H(M)a⃗). By the extension property, there is b′ ∈ N such that b′ ∉ cl(H(N)a′) and tp(b′a′) = tp(ba⃗). The tuples stay H-independent, so again we can take h⃗_{1} and h⃗′_{1} to be empty.

□

The previous result has the following consequence:

Corollary 2.8. All H-structures are elementarily equivalent.

We write T^{ind} for the common complete theory of all H-structures of models of T.

Definition 2.9. We say that T^{ind} is first order if the |T |^{+}-saturated models of T^{ind} are again H-structures.

To axiomatize T^{ind} and to show that T^{ind} is first oder, we follow the ideas of [15, Prop 2.15], [3] and [2]. Here we use for the first time that T eliminates ∃^{large}. Recall that whenever T eliminates ∃^{large} the expression the formula φ(x, b⃗) is large is first order.

We also need the following definition from [2, Definition 2.4]:

Definition 2.10. Let ψ(y⃗, z⃗) and φ(x⃗, y⃗) be L-formulas. Q_{φ,ψ} is the predicate which is defined to hold of a tuple c⃗ (in M) if for all b⃗ satisfying ψ(y⃗, c⃗), the formula φ(x⃗, b⃗) does not divide over c⃗.

The following result follows word by word from the proof of [2, Proposition 4.5], changing the elementary substructure for the predicate H:

Proposition 2.11. The following are equivalent:

(1) Q_{φ,ψ} is type-definable (in M ) for all L-formulas φ(x⃗, y⃗),ψ(y⃗, z⃗).

(2) The extension property is first order.

(3) Any |T |^{+}-saturated model of T^{ind} satisfies the extension property.

Corollary 2.12. Let T be a simple theory of SU-rank omega that satisfies wnfcp. Then the extension property is first order.

Proposition 2.13. Assume T eliminates ∃^{large} and that the predicates Q_{φ,ψ} are L-type-definable for all L-formulas φ(x⃗, y⃗),ψ(y⃗, z⃗). Then T^{ind} is first order.

Proof. The theory T^{ind} is described by T′, the density property and the extension property.

T′ is a first order property.

The density property can be described in first order by the scheme:

For all L-formulas φ(x, y⃗) ∀y⃗(φ(x, y⃗) large ⟹ ∃x(φ(x, y⃗) ∧ x ∈ H)).

Thus all saturated models of the scheme satisfy the density property. Finally by Proposition 2.11 any |T |^{+}-saturated model of T^{ind} satisfy the extension property.

□

Notation 2.14. Let (M, H(M)) be an H-structure and let A ⊂ M. We write cl_{H}(A) for cl(AH(M)) and we call it the small closure of A over H.

3. examples

In this section we give a list of examples of simple theories of SU-rank ω that eliminate ∃^{large} and where the extension property is first order. We also list some examples that eliminate the quantifier ∃^{large} but where it remains as an open question if the extension property is first order.

3.1. Differentially closed fields. Let T = DCF_{0}, the theory of differentially closed fields. This theory is stable of U rank ω and also RM(DCF_{0}) = ω.

Let p(x) be the unique generic type of the theory. This type is complete, stationary and definable over ∅. Let φ(x, y⃗) be a formula and let ψ(y⃗) be its p-definition. Then for (K, d) ⊧ DCF_{0}, a⃗ ∈ K, the formula φ(x, a⃗) is large iff ψ(a⃗). Thus this theory eliminates the quantifier ∃^{large}.

Now let us study the extension property. Recall that DCF_{0} has quantifier elimination [12, Theorem 2.4] and eliminates imaginaries [12, Theorem 3.7]. It is proved in [12, Theorem 2.13] that DCF_{0} has uniform bounding (i.e. it eliminates ∃^{∞}) and thus it has nfcp. This is also explicitly explained in [12, page 52]. It follows by Corollary 2.12 that the extension property is first order.

3.2. Free pseudoplane-infinite branching tree. Let T be the theory of the free pseudoplane, that is, a graph without cycles such that every vertex has infinitely many edges. The theory of the free pseudoplane is stable of U-rank ω and MR(T) = ω. For every A, acl(A) = dcl(A) = A ∪ {x| there are points a, b ∈ A and a path connecting them passing trough x}. For A algebraically closed and a a single element, U(a/A) = d(a, A) where d(a, A) is the minimum length of a path from a to an element of A or ω if there is no path; in this last case we say that a is at infinite distance to A or that a is not connected to A. Note that there is a unique generic type over A, namely the type of an element which is not connected to A.

The generic type is definable over ∅ and thus by definability of types T eliminates the quantifier ∃^{large}.

An H-structure (M, H) associated to T is an infinite collection of trees with an infinite collection of selected points H(M) at infinite distance one from the other and with infinite many trees not connected to them. If (N, H) ⊧ T h(M, H), then N has infinitely many selected points H(N) at infinite distance one from the other.

If (N, H) is ℵ_{0}-saturated, then by saturation it also has infinitely many trees which are not connected to the points H(N). We will prove that in this case (N, H) is an H-structure. The density property is clear. Now let A ⊂ N be finite and assume that A = dcl(A) and let c ∈ N. If U(c/A) = ω choose a point b in a tree not connected to A ∪ H, then tp(c/A) = tp(b/A) and b | ⌣_{A} H. If U(c/A) = 0 there is nothing to prove. If U(c/A) = n > 0, let a be the nearest point from A to c. Since there is at most one point of H connected to a and the trees are infinitely branching, we can choose a point b with d(b, a) = n and such that d(b, A ∪ H) = n; then tp(c/A) = tp(b/A) and b | ⌣_{A} H. This proves that (N, H) is an H-structure and that that T^{ind} is first order.

3.3. Vector space with a generic automorphism. Let T be the theory of (infinite-dimensional) vector spaces over a division ring F, and let T_{σ} by its (unique) generic automorphism expansion.

This theory has a unique generic, which is definable over ∅. By definability of types, T_{σ} eliminates the quantifier ∃^{large}.

Now we prove that the extension property is first order.

Let (M, H) be an H-structure associated to T_{σ}, let (N, H) ⊧ T h(M, H) be |T |^{+}-saturated and let a, b⃗ ∈ N.

Note that the type of the element a over a tuple b⃗ in T_{σ} is determined by

qftp^{−}(σ^{ℤ}(a)/σ^{ℤ}(b⃗)),

where the superscript − refers to the language of T, and

σ^{ℤ}(c⃗) = …, σ^{−1}(c⃗), c⃗, σ(c⃗), σ^{2}(c⃗), ….

There are three possible situations for tp(a/b⃗):

(1) a ∈ span(σ^{ℤ}(b⃗))

(2) a, σ(a), …, σ^{n−1}(a) are independent over σ^{ℤ}(b⃗), but

σ^{n}(a) ∈ span(a, σ(a), …, σ^{n−1}(a)σ^{ℤ}(b⃗))

(3) σ^{ℤ}(a) is independent over σ^{ℤ}(b⃗)

For the first case, we have that a ∈ dcl(b⃗) and thus a | ⌣b⃗ H.

For the second case, assume now that σ^{n}(a) ∈ span(a, σ(a), …, σ^{n−1}(a)σ^{ℤ}(b⃗)). Since M is an ℵ_{0}-saturated, we can find a′, b⃗′ ∈ M such that tp(a, b⃗) = tp(a′, b⃗′) and since (M, H) is an H-structure we may assume that a′ | ⌣b⃗′ H. In particular, the elements a′, σ(a′), …, σ^{n−1}(a′) do not satisfy any nontrivial linear combination with elements in dcl(b⃗′H(M)). Since (N, H) ⊧ T h((M, H)) is |T |^{+}-saturated, we can find (a′′, b⃗) ⊧ tp(a′, b⃗′) such that a′′, σ(a′′), …, σ^{n−1}(a′′) do not satisfy any nontrivial linear combination with elements in dcl(b⃗H(N)). This shows that a′′ | ⌣b⃗ H as we wanted.

For the third case, since (N, H) ⊧ (M, H), we have that H(N) is an infinite collection of independent generics. Let a_{0}, …, a_{n^{2}−1} ∈ H(N) be distinct and consider c_{0} = a_{0} + ⋯ + a_{n−1}, …, c_{n−1} = a_{n^{2}−n} + ⋯ + a_{n^{2}−1}. Then the elements c_{0}, …, c_{n−1} are independent generics and neither one can be written as a linear combination of less that n elements in H. Since (N, H) is |T |^{+}-saturated, we can find infinitely many independent generics that are independent over H(N). If σ^{ℤ}(a) is independent over σ^{ℤ}(b⃗) we can choose a′ generic independent from b⃗H(N) and thus a′ | ⌣b⃗ H.

3.4. Theories of Morley rank omega with definable Morley rank. Let T be a ω-stable theory of rank ω and let M ⊧ T be |T |^{+}-saturated. Assume also that the Morley rank is definable, that is, for every formula φ(x, y⃗) without parameters and every α ∈ {0, 1, …, ω} there is a formula ψ_{α}(y⃗) without parameters such that for a⃗ ∈ M, MR(φ(x, a⃗)) ≥ α if and only if ψ_{α}(a⃗). To simplify the notation, we will write MR(φ(x, a⃗)) ≥ α instead of ψ_{α}(a⃗). We will prove that T^{ind} is first order.

Elimination of ∃^{large}. Consider first φ(x, y⃗) and let b⃗ ∈ M. Then φ(x, b⃗) is large if and only if MR(φ(x, b⃗)) ≥ ω, so T eliminates the quantifier ∃^{large}.

Extension property. Now assume that (M, H) is an H-structure and let (N, H) ⊧ T h(M, H) be |T |^{+}-saturated. Let a ∈ N and let b⃗ ∈ N. If MR(tp(a/b⃗)) = 0 there is nothing to prove. Assume then that MR(tp(a/b⃗)) = n > 0.

Let φ(x, y⃗) ∈ tp(a, b⃗) with MR(φ(x, b⃗)) = n and Md(φ(x, b⃗)) = Md(tp(a/b⃗)). Let (a′, b⃗′) ⊧ tp(a, b⃗) belong to M. Since (M, H) is an H-structure, we may assume that a′ | ⌣b⃗′ H and thus for every formula θ(x, y⃗, z⃗) and every tuple h⃗ ∈ H, if MR(θ(x, b⃗′, h⃗)) < MR(φ(x, b⃗′)) = n then ¬θ(x, b⃗′, h⃗) ∈ tp(a′/b⃗′H). So (M, H) ⊧ ∀d′MR(φ(x, d⃗′)) ≥ n ⟹ ∃cφ(c, d⃗′) ∧ ∀h⃗ ∈ H(MR(θ(x, d⃗′, h⃗)) < n ⟹ ¬θ(c, d⃗′, h⃗)).

Since (N, H) ⊧ T H(M, H) is |T |^{+}-saturated, we can find a′ such that MR(φ(a′, b⃗)) ≥ n and whenever h⃗ ∈ H(N) and θ(x, b⃗, h⃗) is a formula with Morley rank smaller than n we have ¬θ(a′, b⃗, h⃗). This shows that MR(a′/b⃗H) = MR(a′/b⃗) = MR(a/b⃗), Md(a′/b⃗) = Md(a/b⃗), both a and a′ are generics of the formula φ(x, b⃗) and thus tp(a/b⃗) = tp(a′/b⃗). Finally by construction a′ | ⌣b⃗ H. It follows that T^{ind} is first order.

3.5. H-triples. Recall from [3] that if T_{0} is supersimple SU-rank one theory whose pregeometry is not trivial, then T = T_{0}^{ind} has SU-rank omega. The models of T are structures of the form (M, H_{1}), where M ⊧ T_{0} and H_{1} is a acl_{0}-dense and acl_{0}-codense subset of M. We write L_{0} for the language associated to T_{0} and L for the language associated to T. Similarly, we write acl_{0} for the algebraic closure in the language L_{0} and for A ⊂ M ⊧ T_{0}, we write S_{n}^{0}(A) for the space of L_{0}-n-types over A.

We will assume that T_{0} has a strong form of non-triviality, namely for all L_{0}- definable infinite sets φ(x), there is an algebraic triangle inside φ(x). So there is a set B and there are a ⊧ φ(x) and there are b, c with each of a, b, c acl_{0}- independent from B and such that a ∈ acl_{0}(bcB) \ acl_{0}(bB). With this assumption, if (M, H) ⊧ T, A ⊂ M and a ∉ acl_{0}(AH_{1}), then SU(tp(a/A)) = ω and the generics in the sense of (M, H_{1}) have SU as required for the present paper.

In this subsection we change our notation and we let H_{2} be a new predicate symbol that will be interpreted by a dense and codense T -generic subset of (M, H_{1}).

The structures (M, H_{1}, H_{2}) were already studied in [3]. We recall the definitions and the main result. The main tool for studying T^{ind} is to take into account the base theory T_{0} and use triples.

Definition 3.1. We say that (M, H_{1}(M), H_{2}(M)) is an H-triple associated to T_{0} if:

(1) M ⊧ T_{0}, H_{1}(M) is an acl_{0}-independent subset of M, H_{2}(M) is an acl_{0}- independent subset of M over H_{1}.

(2) (Density property for H_{1}) If A ⊂ M is finite dimensional and q ∈ S_{1}^{0}(A) is non-algebraic, there is a ∈ H_{1}(M) such that a ⊧ q.

(3) (Density property for H_{2}/H_{1}) If A ⊂ M is finite dimensional and q ∈ S_{1}^{0}(A) is non-algebraic, there is a ∈ H_{2}(M) such that a ⊧ q and a ∉ acl_{0}(A ∪ H_{1}(M)).

(4) (Extension property) If A ⊂ M is finite dimensional and q ∈ S_{1}^{0}(A) is non-algebraic, there is a ∈ M, a ⊧ q and a ∉ acl_{0}(A ∪ H_{1}(M) ∪ H_{2}(M)).

It is observed in [3] that if (M, H_{1}(M), H_{2}(M)), (N, H_{1}(N), H_{2}(N)) are Htriples, then T h(M, H_{1}(M), H_{2}(M)) = T h(N, H_{1}(N), H_{2}(N)) and we denote the common theory by T_{0}^{tri}.

The folowing result is proved in [3] for geometric theories.

Proposition 3.2. Let T be an SU rank one strongly non-trivial supersimple theory, let M ⊧ T and let H_{1}(M) ⊂ M, H_{2}(M) ⊂ M be distinguished subsets. Then (M, H_{1}(M), H_{2}(M)) is a H_{2}-structure associated to T if and only if (M, H_{1}(M), H_{2}(M)) is an H-triple.

Thus, to show that the class of H_{2}-structures associated to T is first order, it suffices to prove that this is the case for H-triples associated to T_{0}. As pointed out in [3] we have:

Proposition 3.3. The theory T^{tri} is axiomatized by:

(1) T.

(2) M ⊧ T_{0}, H_{1}(M) is an acl_{0}-independent subset of M, H_{2}(M) is an acl_{0}- independent subset of M over H_{1}.

(3) For all L-formulas φ(x, y⃗)

∀y⃗(φ(x, y⃗) nonalgebraic ⟹ ∃x(φ(x, y⃗) ∧ x ∈ H_{1})).

(4) 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 ∈ H_{2}) ∧ ∀w_{1} … ∀w_{m} ∈ H_{1}¬ψ(x, w_{1}, …, w_{m}, y⃗))

(5) 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 ⟹ ∃xφ(x, y⃗) ∧ ∀w_{1} … ∀w_{m} ∈ H_{1} ∪ H_{2}¬ψ(x, w_{1}, …, w_{m}, y⃗))

Furthermore, if (M, H, H_{2}) ⊧ T^{tri} is |T |^{+}-saturated, then (M, H, H_{2}) is an H-triple.

Thus when T_{0} is a strongly non-trivial supersimple SU-rank one theory, T^{ind} = T^{tri} is first order.

3.6. H structures of lovely pairs of SU-rank one theories. Let T be a geometric theory, T_{P} its lovely pairs expansion, and let

cl(−) = acl(− ∪ P (M))

be the small closure operator in a lovely pair (M, P). Our goal is to expand T_{P} to a theory T_{P}^{ind} in the language L_{P H} = L_{P} ∪ {H}, by adding a cl-independent dense set to a model of T_{P}.

The following definition is analogous to Definition 3.1.

Definition 3.4. We say that an L_{P H}-structure (M, P, H) is a P H-structure of T if

(1) P (M) is an elementary substructure of M;

(2) H(M) is acl-independent over P (M);

(3) for any non-algebraic type q ∈ S_{1}^{T} (A) over a finite-dimensional set A ⊂ M,

q is realized in (density of P over H) P (M)\ acl(H(M)A); (density of H over P) H(M)\ acl(P (M)A); (extension) M\ acl(P (M)H(M)A).

Remark 3.5. (a) It suffices to require P (M) to be dense in the usual sense, i.e. q having a realization in P (M).

(b) We can get a P H-structure from an H-triple (M, H_{1}, H_{2}) (see previous example), by letting P (M) = acl(H_{1}).

(c) A usual elementary chain argument shows that any L_{P H} structure (M, P, H) satisfying (1,2) embeds in a P H-structure (N, P, H) so that H(N) | ⌣_{H(M)} MP (N) and P (N) | ⌣_{P (M)} MH(N). In particular, P H-structures exist.

(d) Reducts (M, P) and (M, H) of (M, P, H) are lovely pairs and H-structures, respectively.

While in linear examples the SU-rank of T_{P} is two instead of ω, the machinery for this paper still goes through we our current assumptions for cl.

Definition 3.6. We say that (M, P, H) is an cl-structure if (1) (M, P) is a lovely pair and H is an cl-independent set

(2) (Density/coheir property for cl) If A ⊂ M is finite dimensional and q ∈ S_{1}^{P} (A) is large, there is a ∈ H(M) such that a ⊧ q.

(3) (Extension property) If A ⊂ M is finite dimensional and q ∈ S_{1}^{P} (A) is large, there is a ∈ M, a ⊧ q and a ∉ cl(A ∪ H(M)).

Proposition 3.7. (M, P, H) is an cl-structure if and only if (M, P, H) is a P Hstructure.

Proof. Assume first that (M, P, H) is a cl-structure. Then the pair (M, P) is lovely and thus (M, P, H) satisfies the density axiom for P. Now let A ⊂ M be finite dimensional and let q ∈ S_{1}(A) be non-algebraic. Let q̂ ∈ S_{1}^{P} (A) be an extension of q that contains no small formula with parameters in A. Then by the Density/coheir property for cl it follows that there is a ∈ H(M) such that a ⊧ q̂. In particular, a ⊧ q and a ∉ cl(A) and thus we get the density property for H over P. Finally, since the same q̂ is not small, there is c ∈ M, c ⊧ q̂ and c ∉ cl(A ∪ H(M)) = acl(A ∪ P (M) ∪ H(M)). Thus the extension property holds as well.

Now assume that (M, P, H) is an P H-structure. Then H is an cl-independent set, and by the density property for P and the extension property it follows that (M, P) is a lovely pair. Now let A ⊂ M be finite dimensional and let q̂ ∈ S_{1}^{P} (A) be non-small. We may enlarge A and assume that A is P -independent. Let q be the restriction of q̂ to the language L. Note that q̂ is the unique extension of q to a non-small type. By the density for H over P, there is a ∈ H(M) such that a ⊧ q, a ∉ cl(A) and thus a ⊧ q̂. Finally the extension property follows from the extension property for P H-structures. □

We will now show that the class of P H-structures is "first order", that is, that there is a set of axioms whose |T |^{+}-saturated models are the P H-structures. The axiomatization works as in H-triples.

Proposition 3.8. Assume T eliminates ∃^{∞}. Then the theory T_{P H} is axiomatized by:

(1) T

(2) axioms saying that P distinguishes an elementary substructure.

(3) For all L-formulas φ(x, y⃗)

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

(4) 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 ∈ H) ∧ ∀w_{1} … ∀w_{m} ∈ P ¬ψ(x, w_{1}, …, w_{m}, y⃗))

(5) 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 ∧ x ∉ H) ∧ ∀w_{1} … ∀w_{m} ∈ P ∪ H¬ψ(x, w_{1}, …, w_{m}, y⃗))

Furthermore, if (M, P, H) ⊧ T_{P H} is |T |^{+}-saturated, then (M, P, H) is a P H-structure.

Now we list a family of structures of SU-rank ω where we do know if the corresponding theory of H-structures is axiomatizable. In both cases it is open whether or not the extension property is first order.

3.7. ACFA. Let T = ACF A, (a completion) of the theory of algebraically closed fields with a generic automorphism. This theory is simple of SU rank ω and it is unstable.

Let p(x) be the generic type of the theory, namely the type of a transformally independent element. This type is complete, stationary and definable over ∅. Let φ(x, y⃗) be a formula and let ψ(y⃗) be its p-definition. Then for (K, σ) ⊧ ACF A, a⃗ ∈ K, the formula φ(x, a⃗) is large iff ψ(a⃗). Thus this theory eliminates the quantifier ∃^{large}.

Question Does the extension property hold for ACFA? Does T_{0} satisfy wnfcp?

3.8. Hrushovski amalgamation without collapsing. In this subsection we follow the presentation of Hrushovski amalgamations from [16], all the results we mention can be found in [16]. Let L = {R} where R stands for a ternary relation. We let C be the class of L-structures where R is symmetric and not reflexive. For A ∈ C a finite structure we let δ(A) = |A| − |R(A)| and we let C_{fin}^{0} be the subclass of C consisting of all finite L-structures M where for A ⊂ M we have δ(A) ≥ 0. Finally M^{0} stands for the Fraïssé limit of the class C_{fin}^{0}. Let T_{0} be the theory of M^{0}, then MR(T_{0}) = ω and Md(T_{0}) = 1.

Now let M ⊧ T_{0} and for A ⊂ M finite we define d(A) = inf{δ(B) : A ⊂ B}. Then d is the dimension function of a pregeometry and that for an element a and a set B, d(a/B) = 1 if and only if MR(a/B) = ω if and only if U(a/B) = ω. Thus the pregeometry studied in [16] corresponds to the pregeometry associated to cl. Since the theory T_{0} has a unique generic type, by definability of types the theory T_{0} eliminates the quantifier ∃^{large}.

Question Does the extension property hold for T_{0}? Does T_{0} satisfy nfcp?

4. Definable sets in H-structures

Fix T a SUrank ω theory and let (M, H(M)) ⊧ T^{ind}. Our next goal is to obtain a description of definable subsets of M and H(M) in the language L_{H}.

Notation 4.1. Let (M, H(M)) be an H-structure. Let a⃗ be a tuple in M. We denote by etp_{H}(a⃗) the collection of formulas of the form ∃x_{1} ∈ H … ∃x_{m} ∈ Hφ(x⃗, y⃗), where φ(x⃗, y⃗) is an L-formula such that there exists h⃗ ∈ H with M ⊧ φ(h⃗, a⃗).

Lemma 4.2. Let (M, H(M)), (N, H(N)) be H-structures. Let a⃗, b⃗ be tuples of the same arity from M, N respectively. Then the following are equivalent:

(1) etp_{H}(a⃗) = etp_{H}(b⃗).
(2) a⃗, b⃗ have the same L_{H}-type.

Proof. Clearly (2) implies (1). Assume (1), then tp(a⃗) = tp(b⃗).

Claim dim_{cl}(b⃗/H) = dim_{cl}(a⃗/H).

Let h⃗ = (h_{1}, …, h_{l}) ∈ H(M) be such k := dim_{cl}(a⃗/h⃗) = dim_{cl}(a⃗/H(M)). We may assume that a^{1} = (a_{1}, …, a_{k}) are independent over H and a^{2} = (a_{k+1}, …, a_{n}) ∈ cl(a_{1}, …, a_{k}, h_{1}, …, h_{l}). Choose ψ(x⃗, y⃗, z⃗) such that for any b⃗ ∈ M, c⃗ ∈ M ψ(b⃗, c⃗, z⃗) is always small in z⃗ and M ⊧ ψ(h⃗, a^{1}, a^{2}). Since etp_{H}(a⃗) = etp_{H}(b⃗) we get that dim_{cl}(b⃗/H) ≤ k. A similar argument shows that dim_{cl}(a⃗/H(M)) ≤ dim_{cl}(b⃗/H(N)).

Claim tp_{H}(b⃗) = tp_{H}(a⃗).

As before, let h⃗ = (h_{1}, …, h_{l}) ∈ H(M) be such that k := dim_{cl}(a⃗/h⃗) = dim_{cl}(a⃗/H(M)). Then a⃗h⃗ is H-independent. Since N is saturated as an L-structure there are h⃗′ = (h′_{1}, …, h′_{l}) ∈ H such that tp(a⃗, h⃗) = tp(b⃗, h⃗′). By the claim above b⃗h⃗′ is H-independent, so the result follows from Lemma 2.7. □

Now we are interested in the L_{H}-definable subsets of H(M). This material is very similar to the results presented in [5].

Lemma 4.3. Let (M_{0}, H(M_{0})) ⪯ (M_{1}, H(M_{1})) and assume that (M_{1}, H(M_{1})) is |M_{0}|-saturated. The M_{0} (seen as a subset of M_{1}) is a H-independent set.

Proof. Assume not. Then there are a_{1}, …, a_{n} ∈ M_{0} \ H(M_{0}) such that a_{n} ∈ cl(a_{1}, …, a_{n−1}, H(M_{1})) and a_{n} ∉ cl(a_{1}, …, a_{n−1}, H(M_{0})). Let φ(x, y⃗, z⃗) be a formula which is always small on x and b⃗ ∈ H(M_{1})z⃗ be a tuple such that φ(a_{n}, a_{1}, …, a_{n−1}, b⃗) holds. Since (M_{0}, H(M_{0})) ⪯ (M_{1}, H(M_{1})) there is b⃗′ ∈ H(M_{0})z⃗ such that

φ(a_{n}, a_{1}, …, a_{n−1}, b⃗′) ∧ ¬∃^{large}xφ(x, a_{1}, …, a_{n−1}, b⃗′)

holds, so a_{n} ∈ acl(a_{1}, …, a_{n−1}, H(M_{0})), a contradiction. □

Proposition 4.4. Let (M, H(M)) be an H-structure and let Y ⊂ H(M)^{n} be L_{H}- definable. Then there is X ⊂ M^{n} L-definable such that Y = X ∩ H(M)^{n}.

Proof. Let (M_{1}, H(M_{1})) ⪰ (M, H(M)) be κ-saturated where κ > |M| + |L| and let a⃗, b⃗ ∈ H(M_{1})^{n} be such that tp(a⃗/M) = tp(b⃗/M). We will prove that tp_{H}(a⃗/M) = tp_{H}(b⃗/M) and the result will follow by compactness. Since a⃗, b⃗ ∈ H(M_{1})^{n}, we get by Lemma 4.3 that Ma⃗, Mb⃗ are H-independent sets and thus by Lemma 2.7 we get tp_{H}(a⃗/M) = tp_{H}(b⃗/M). □

Definition 4.5. Let (M, H) ⊧ T^{ind} be saturated. We say that an L_{H}-formula ψ(x, c⃗) defines a H-large subset of M if there is b ⊧ ψ(x, c⃗) such that b ∉ cl(Hc⃗). This is equivalent as requiring that there are infinitely many realizations of ψ(x, c⃗) that are not small over H(M)c⃗.

Definition 4.6. Let (M, H) ⊧ T^{ind} be κ-saturated and let A ⊂ M be smaller than κ. Let b⃗ ∈ M be a tuple. We say that b⃗ is in the H-small closure of A if b⃗ ∈ cl(AH(M)). Let X ⊂ M^{n} be A-definable. We say that X is H-small if X ⊂ cl(A ∪ H(M)).

Proposition 4.7. Let (M, H(M)) be an H-structure. Let a⃗ = (a_{1}, …, a_{n}) ∈ M. Then there is a unique smallest tuple h⃗ ∈ H(M) such that a⃗ | ⌣h⃗ H.

Proof. Since T is supersimple, there is a finite tuple h⃗ ∈ H such that a⃗ | ⌣h⃗ H. Choose such a tuple so that |h⃗| (the length of the tuple) is minimal. We will now show such a tuple h⃗ is unique (up to permutation).

We can write a⃗ = (a⃗_{1}, a⃗_{2}) so that a⃗_{1} is independent tuple of generics which is independent from H(M) and a⃗_{2} ∈ cl(a⃗_{1}). If a⃗_{2} = ∅, then h⃗ = ∅ and the result follows. So we may assume that a⃗_{2} ≠ ∅.

Then a⃗_{2} ∈ cl(a⃗_{1}, h⃗). Let h⃗′ be another such tuple. Let h⃗_{1} be the list of common elements in both h⃗ and h⃗′, so we can write h⃗ = (h⃗_{1}, h⃗_{2}) and h⃗′ = (h⃗_{1}, h⃗′_{2}).

Claim h⃗_{2} = h⃗′_{2} = ∅.

Assume otherwise. Then there is c ∈ a⃗_{2} such that c ∈ cl(a⃗_{1}, h⃗_{1}, h⃗_{2}) \ cl(a⃗_{1}, h⃗_{1}). Since a⃗ | ⌣h⃗′ H, we must have that c ∈ cl(a⃗_{1}, h⃗_{1}, h⃗′_{2}) \ cl(a⃗_{1}, h⃗_{1}). By the exchange property dim_{cl}(h⃗′_{2}/a⃗_{1}h⃗_{1}h⃗_{2}) < dim_{cl}(h⃗′_{2}/a⃗_{1}h⃗_{1}). Since a⃗_{1} is a tuple of generic elements that are independent over H we get that dim_{cl}(h⃗′_{2}/h⃗_{1}h⃗_{2}) < dim_{cl}(h⃗′_{2}/h⃗_{1}) and since H is independent, h⃗_{2} has a common element with h⃗′_{2}, a contradiction. □

Remark 4.8. Let (M, H(M)) be an H-structure. Let a⃗ = (a_{1}, …, a_{n}) ∈ M and let C ⊂ M be such that C is H-independent. As before, there is a unique smallest tuple h⃗ ∈ H(M) such that a⃗ | ⌣_{h⃗C} H.

Notation 4.9. Let (M, H(M)) be an H-structure. Let a⃗ = (a_{1}, …, a_{n}) ∈ M. Let h⃗ ∈ H(M) be the smallest tuple such that a⃗ | ⌣h⃗ H. We call h⃗ the H-basis of a⃗ and we denote it as HB(a⃗). Given C ⊂ M such that C is H-independent, let h⃗ ∈ H(M) the smallest tuple such that a⃗ | ⌣_{Ch⃗} H. We call h⃗ the H-basis of a⃗ over C and we denote it as HB(a⃗/C). Note that H-basis is unique up to permutation, therefore we will view the H-basis h⃗ = (h_{1}, …, h_{k}) either as a finite set {h_{1}, …, h_{k}} or as the imaginary representing this finite set. If we view it as a tuple, we will explicitly say so.

Proposition 4.10. Let (M, H(M)) be an H-structure. Let a_{1}, …, a_{n}, a_{n+1} ∈ M and let C ⊂ M be such that C is H-independent. Then HB(a_{1}, …, a_{n}, a_{n+1}/C) = HB(a_{1}, …, a_{n}/C) ∪ HB(a_{n+1}/Ca_{1}, …, a_{n}HB(a_{1}, …, a_{n}/C)), where all H-basis are seen as sets.

Proof. Let h⃗_{1} = HB(a_{1}, …, a_{n}/C). First note that since a_{1}, …, a_{n} | ⌣_{Ch⃗_{1}} H, then

the set a_{1}, …, a_{n}Ch⃗_{1} is H-independent and we can define h⃗_{2} = HB(a_{n+1}/Ca_{1}, …, a_{n}h⃗_{1}). Finally, let h⃗ = HB(a_{1}, …, a_{n}, a_{n+1}/C).

Claim h⃗ ⊂ h⃗_{1}h⃗_{2}.

We have a_{1}, …, a_{n} | ⌣_{Ch⃗_{1}} H and a_{n+1} | ⌣_{Ch⃗_{1}h⃗_{2}a_{1},...,a_{n}} H, so by transitivity,

a_{1}, …, a_{n}a_{n+1} | ⌣_{Ch⃗_{1}h⃗_{2}} H and by the minimality of an H-basis, we have h⃗ ⊂

h⃗_{1}h⃗_{2}. Claim h⃗ ⊃ h⃗_{1}h⃗_{2}. By definition, a_{1}, …, a_{n}a_{n+1} | ⌣_{Ch⃗} H, so a_{1}, …, a_{n} | ⌣_{Ch⃗} H and by minimality

we have h⃗_{1} ⊂ h⃗. We also get by transitivity that a_{n+1} | ⌣_{Ca_{1},...,a_{n}h⃗_{1}h⃗} H and by the

minimality of H-basis we get h⃗_{2} ⊂ h⃗ as desired. □

Proposition 4.11. Let (M, H(M)) be an H-structure. Let a_{1}, …, a_{n} ∈ M and let C ⊂ D ⊂ M be such that C, D are H-independent. Assume that there is h ∈ HB(a_{1}, …, a_{n}/C) \ HB(a_{1}, …, a_{n}/D). Then h ∈ D.

Proof. Write h_{D} = HB(a_{1}, …, a_{n}/D) and see it as a set. Then a_{1}, …, a_{n}D | ⌣_{h_{D}H(D)} H and a_{1}, …, a_{n} | ⌣_{h_{D}CH(D)} H. By minimality of HB(a_{1}, …, a_{n}/C) we get that HB(a_{1}, …, a_{n}/C) ⊂ h_{D}H(D) and thus if h ∈ HB(a_{1}, …, a_{n}/C) \ h_{D}, we must have h ∈ H(D). □

We will now apply the H-basis to characterize definable sets in terms of Ldefinable sets.

Proposition 4.12. Let (M, H(M)) be an H-structure and let Y ⊂ M be L_{H}- definable. Then there is X ⊂ M L-definable such that Y △X is H-small, where △ stands for a boolean connective for the symmetric difference.

Proof. If Y is H-small or H-cosmall, the result is clear, so we may assume that both Y and M \Y are H-large. Assume that Y is definable over a⃗ and that a⃗ = a⃗HB(a⃗). Let b ∈ Y be such that b ∉ cl(a⃗H) and let c ∈ M \ Y be such that c ∉ cl(a⃗H). Then ba⃗, ca⃗ are H-independent and thus there is X_{bc} an L-definable set such that b ∈ X_{bc} and c ∉ X_{bc}. By compactness, we may get a single L-definable set X such that for b′ ∈ Y and c′ ∈ M \ X not in the H-small closure of a⃗, we have b′ ∈ X and c′ ∈ M \ Z. This shows that Y △X is H-small. □

Our next goal is to characterize the algebraic closure in H-structures. The key tool is the following result:

Lemma 4.13. Let (M, H(M)) be an H-structure, and let A ⊂ M be acl-closed and H-independent. Then A is acl_{H}-closed.

Proof. Suppose a ∈ M, a ∉ A. If a ∉ cl(AH), then A ∪ {a} is H-independent, and using the extension property, we can find a_{i}, i ∈ ω, acl-independent over A∪H(M), realizing tp(a/A). By Lemma 2.7, each a_{i} realizes tp_{H}(a/A), and thus a ∉ acl_{H}(A).

If a ∈ cl(AH), take a minimal tuple h⃗ ∈ H(M) such that a ∈ acl(Ah⃗). Using conjugates of h⃗ over A it is easy to see that tp(a/A) has ∞-many realizations. □

Corollary 4.14. Let (M, H(M)) be an H-structure, and let A ⊂ M. Then acl_{H}(A) = acl(A, HB(A)).

Proof. By Proposition 4.7, it is clear that HB(A) ∈ acl(A), so acl_{H}(A) ⊃ acl(A, HB(A)). On the other hand, A ∪ HB(A) is H-closed, so by the previous Proposition, acl(A ∪ HB(A)) = acl_{H}(A ∪ HB(A)) and thus acl_{H}(A) ⊂ acl(A, HB(A)) □

5. Supersimplicity

In this section we prove that T^{ind} is supersimple and characterize forking in T^{ind}.

Theorem 5.1. The theory T^{ind} is supersimple.

Proof. We will prove that non-dividing has local character.

Let (M, H(M)) ⊧ T^{ind} be saturated. Let C ⊂ D ⊂ M and assume that C = acl_{H}(C) and D = acl_{H}(D). Note that both C and D are H-independent. Let a⃗ ∈ M. We will find a collection of conditions for the type of a⃗ over C that guarantee that tp_{H}(a⃗/D) does not divide over C.

We may write a⃗ = (a⃗_{1}, a⃗_{2}) ∈ M so that a⃗_{1} is an independent tuple of generics over DH, a⃗_{2} is a tuple such that a⃗_{2} ∈ cl(a⃗_{1}DH).

Assume that the following conditions hold for C:

(1) HB(a⃗/D) = HB(a⃗/C).

(2) SU(a⃗_{2}/Ca⃗_{1}H) = SU(a⃗_{2}/Da⃗_{1}H)

Claim tp_{H}(a⃗/D) does not divide over C.

Let p(x⃗, D) = tp(a⃗_{1}, D). Let {D_{i} : i ∈ ω} be an L_{H}-indiscernible sequence over C. Since a⃗_{1} is an independent tuple of generics over D, tp(a⃗_{1}/D) does not divide over C and ∪_{i∈ω}p(x⃗, D_{i}) is consistent. We can find a′_{1} ⊧ ∪_{i∈ω}p(x⃗, D_{i}) such that {a′_{1}D_{i} : i ∈ ω} is indiscernible and a′_{1} is an independent tuple of generics over ∪_{i∈ω}D_{i}. By the generalized extension property, we may assume that a′_{1} is independent over ∪_{i∈ω}D_{i}H. Note that a⃗_{1}D is H-independent, a′_{1}D_{i} is also Hindependent for any i ∈ ω. So by Lemma 2.7 tp_{H}(a⃗_{1}D) = tp_{H}(a′_{1}D_{i}) for any i ∈ ω.

Now let h⃗ = HB(a⃗/C) (viewed as a tuple) and let q(y⃗, a⃗_{1}, D) = tp(h⃗, a⃗_{1}, D). Note that h⃗ is an independent tuple of generics over a⃗_{1}D (as well as an independent tuple over a⃗_{1}C). Since {D_{i}a′_{1} : i ∈ ω} is an L-indiscernible sequence, there is h⃗′ ⊧ ∪_{i∈ω}q(y⃗, a′_{1}, D_{i}). We may assume that h⃗′ is independent from ∪_{i∈ω}D_{i}a′_{1} and thus it is a tuple of generics over ∪_{i∈ω}D_{i}a′_{1}. Furthermore we may assume that the sequence {D_{i}a′_{1} : i ∈ ω} is indiscernible over h⃗′.

By the generalized coheir/density property, we may assume that h⃗′ ∈ H. Note that since each a′_{1}D_{i} is H-independent, then h⃗′a′_{1}D_{i} is also H-independent. On the other hand, tp(h⃗, a⃗_{1}, D) = tp(h⃗′, a′_{1}, D_{i}) for each i, so by Lemma 2.7 we have tp_{H}(h⃗, a⃗_{1}, D) = tp_{H}(h⃗′, a′_{1}, D_{i}). This shows that tp(a⃗_{1}, h⃗/D) does not divide over C.

Now consider t(z⃗, a⃗_{1}, h⃗, D) = tp(a⃗_{2}, a⃗_{1}, h⃗, D). By the assumption SU(a⃗_{2}/Ca⃗_{1}h⃗) = SU(a⃗_{2}/Da⃗_{1}h⃗), so tp(a⃗_{2}/a⃗_{1}h⃗D) does not divide over a⃗_{1}h⃗C. Since tp(a⃗_{1}h⃗D) = tp(a′_{1}h⃗′D_{0}), then t(z⃗, a′_{1}, h⃗′, D_{0}) does not divide over a′_{1}h⃗′C.

Since {D_{i} : i ∈ ω} is an L-indiscernible sequence over Ca′_{1}h⃗′, there is a′_{2} ⊧ ∪_{i∈ω}t(z⃗, a′_{1}, h⃗′, D_{i}). We may assume as before that a′_{2} | ⌣_{a′_{1}h⃗′C} ∪_{i}D_{i}.

By the extension property, we may assume that a′_{2} | ⌣_{a′_{1}h⃗′∪_{i}D_{i}} H. Using transitivity we also have a′_{2} | ⌣_{a′_{1}h⃗′C} H ∪_{i} D_{i} and it follows that a′_{2} | ⌣_{a′_{1}h⃗′D_{i}} H, so we have a′_{1}a′_{2} | ⌣_{h⃗′D_{i}} H, and thus, also a′_{1}a′_{2}D_{i} | ⌣_{h⃗′H(D_{i})} H for each index i.

Since both a⃗_{1}a⃗_{2}h⃗D, a′_{1}a′_{2}h⃗′D_{i} are H-independent and tp(a′_{1}a′_{2}h⃗′D_{i}) = tp(a⃗_{1}a⃗_{2}h⃗D) by Lemma 2.7 tp_{H}(a′_{1}a′_{2}h⃗′D_{i}) = tp_{H}(a⃗_{1}a⃗_{2}h⃗D) .

This shows that tp(a⃗/D) does not divide over C.

Since T is supersimple, for any D and a⃗ we can always choose a finite subset C_{0} of D such that C = acl_{H}(C_{0}) satisfies the conditions (1) and (2) above. This shows that T^{ind} is supersimple. □

Proposition 5.2. Let (M, H) ⊧ T^{ind} be saturated, let C ⊂ D ⊂ M be such that C = acl_{H}(C), D = acl_{H}(D) and let a ∈ M. Then tp(a/D) forks over C iff a ∈ D\C or a ∈ cl(HD) \ cl(HC) or HB(a/C) ⊋ HB(a/D) or, HB(a/C) = HB(a/D) and SU(a/CH) ≠ SU(a/DH) .

Proof. In the proof of Theorem 5.1 we showed that if a ∈ C or if, HB(a/C) = HB(a/D) and SU(a/CH) = SU(a/DH) then tp(a/D) does not fork over C. So it remains to show the other direction, which we do case by case.

Case 1: Assume that a ∈ D \ C, then a becomes algebraic over D and tp(a/D) forks over C.

Case 2: Assume that a ∈ cl(DH) \ cl(CH). Then SU(tp(a/DH)) < ω and SU(tp(a/CH)) = ω. We will prove that tp_{H}(a/D) divides over C.

Let d⃗ ∈ D and let c⃗ ∈ C be such that a ∈ cl(c⃗d⃗H), so SU(tp(a/d⃗c⃗H)) < ω. By additivity of Lascar rank, we can choose d⃗ to be independent generics over HC. Let h⃗ ∈ H be such that a ∈ cl(c⃗d⃗h⃗). Let p(x, y⃗) = tp_{H}(a, d⃗/C).

Let {d⃗_{i} : i ∈ ω} be an L -indiscernible sequence in tp(d⃗/C) over C such that {d⃗_{i} : i ∈ ω} is independent over C. By the generalized extension property, we may assume that {d⃗_{i} : i ∈ ω} is independent over HC. Note that by Lemma 2.7 {d⃗_{i} : i ∈ ω} is an L_{H}-indiscernible sequence of generics over C. Assume, in order to get a contradiction, that there is a′ ⊧ ∪_{i∈ω}p(x, d⃗_{i}). Then there are {h⃗_{i} : i ∈ ω} such that a′ ∈ cl(d⃗_{i}, c⃗, h⃗_{i}) for every i, that is, SU(a′/d⃗_{i}, c⃗, h⃗_{i}) < ω. But a′ ∉ cl(CH), so d⃗_{0} |̸ ⌣_{CH} d⃗_{1}, a contradiction.

Case 3: Assume that HB(a/D) ≠ HB(a/C) and a ∈ cl(CH). Then HB(a/D) is a proper subset of HB(a/C). Write h⃗_{C} = HB(a/C), h⃗_{D} = HB(a/D) and let h⃗_{E} ∈ H be such that h⃗_{C} = h⃗_{D}h⃗_{E}. Note that h⃗_{E} ≠ ∅ and that h⃗_{E} is an independent tuple over C.

Let p(x, y⃗) = tp_{H}(a, h⃗_{E}/C). Let {h⃗_{E}^{i} : i ∈ ω} be an L-indiscernible sequence in tp(h⃗_{E}/C) such that {h⃗_{E}^{i} : i ∈ ω} is independent over C. Then by the generalized density property, we may assume that the sequence {h⃗_{E}^{i} : i ∈ ω} belongs to H. Note that by Lemma 2.7, the sequence {h⃗_{E}^{i} : i ∈ ω} is L_{H}-indiscernible over C. We will show that ∪_{i∈ω}p(x, h⃗_{E}^{i}) is inconsistent.

Assume, not, so there is a′ ⊧ ∪_{i∈ω}p(x, h⃗_{E}^{i}). Then we can find h⃗_{D_{i}} in H such that HB(a′/C) = h⃗_{D_{i}}h⃗_{E}^{i}. Since the h⃗_{E}^{i} are independent, we get that the HB basis of a′ over C is not unique, a contradiction.

Case 4: Assume that a ∈ cl(HC), that HB(a/D) = HB(a/C) and SU(a/CH) < SU(a/DH).

Let h⃗ = HB(a/D) = HB(a/C), so SU(a/Ch⃗) < SU(a/Dh⃗) and a |̸ ⌣_{Ch⃗} D. Write h⃗_{D} = H(D) \ H(C). Note that h⃗ is independent over D. Let p(x, h⃗, D) = tp(a, h⃗, D) and let {D_{i} : i ∈ ω} be an L-Morley sequence in tp(D/Ch⃗) such that {p(x, h⃗, D_{i}) : i ∈ ω} is k-inconsistent. Let h⃗_{D_{i}} be such that tp(D, h⃗_{D}) = tp(D_{i}, h⃗_{D_{i}}), then {h⃗_{D_{i}} : i ∈ ω} is a L-Morley sequence in tp(h⃗_{D}/Ch⃗). By the density property, we may choose the elements in H. By the extension property, we can realize tp((D_{i} : i ∈ ω)/C ∪_{i∈ω} h⃗_{D_{i}}) independent from H over C ∪_{i∈ω} h⃗_{D_{i}}. Then tp(D, h⃗_{D}/Ch⃗) = tp(D_{i}, h⃗_{D_{i}}/Ch⃗) so by Lemma 2.7, tp_{H}(D, h⃗_{D}/Ch⃗) = tp_{H}(D_{i}, h⃗_{D_{i}}/Ch⃗) so we get that {D_{i} : i ∈ ω} is a sequence in tp_{H}(D/Ch⃗) such that {p(x, h⃗, D_{i}) : i ∈ ω} is k-inconsistent. Using Erdö-Rado, we can change {D_{i} : i ∈ ω} for an indiscernible sequence in tp_{H}(D/Ch⃗) with the property that {p(x, h⃗, D_{i}) : i ∈ ω} is k-inconsistent. This proves that tp_{H}(a/Dh⃗) divides over Ch⃗. But since h⃗ ∈ acl_{H}(Ca) we also get that tp_{H}(a/D) divides over C.

□

Corollary 5.3. Let (M, H) ⊧ T^{ind} be saturated, let C ⊂ D ⊂ M be such that C and D are H-independent and let a_{1}, …, a_{n} ∈ M. We may reorder the tuple and assume that there is k ≤ n such that a_{1}, …, a_{k} are independent generics over CH and a_{k+1}, …, a_{n} ∈ cl(a_{1}, …, a_{k}, C, H). Then tp_{H}(a_{1}, …, a_{n}/D) forks over C iff

(1) dim_{cl}(a_{1}, …, a_{n}/ cl(HD)) < dim_{cl}(a_{1}, …, a_{n}/ cl(HD)) or

(2) dim_{cl}(a_{1}, …, a_{n}/ cl(HD)) = dim_{cl}(a_{1}, …, a_{n}/ cl(HD)) and HB(a_{1}, …, a_{n}/C) ⊋

HB(a_{1}, …, a_{n}/D) or,

(3) dim_{cl}(a_{1}, …, a_{n}/ cl(HD)) = dim_{cl}(a_{1}, …, a_{n}/ cl(HD)), HB(a_{1}, …, a_{n}/C) =

HB(a_{1}, …, a_{n}/D) and
SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}CH) > SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}DH).

Proof. The proof is by induction on n. For n = 1 the result follows from Proposition 5.2 noticing that the arguments in the proposition work with the weaker assumption that the base sets C, D are H-independent sets.

Assume the result holds for n and consider a_{1}, …, a_{n+1} ∈ M.

Assume that tp(a_{1}, …, a_{n}/D) forks over C or that tp(a_{n+1}/a_{1}, …, a_{n}D) forks over a_{1}, …, a_{n}C. We apply the induction hypothesis and Proposition 5.2.

If dim_{cl}(a_{1}, …, a_{n}/HD) < dim_{cl}(a_{1}, …, a_{n}/HC) or if dim_{cl}(a_{n+1}/a_{1}, …, a_{n}HD) < dim_{cl}(a_{n+1}/a_{1}, …, a_{n}HC), then dim_{cl}(a_{1}, …, a_{n+1}/HD) < dim_{cl}(a_{1}, …, a_{n+1}/HC) as we wanted.

Assume now that dim_{cl}(a_{1}, …, a_{n}/HC) = dim_{cl}(a_{1}, …, a_{n}/HD) and that

dim_{cl}(a_{1}, …, a_{n+1}/HC) = dim_{cl}(a_{1}, …, a_{n+1}/HD).

If HB(a_{1}, …, a_{n}/C) ⊋ HB(a_{1}, …, a_{n}/D), then there is h ∈ HB(a_{1}, …, a_{n}/C) with h ∈ D. If a_{n+1} | ⌣ HDa_{1}, …, a_{n}, then HB(a_{1}, …, a_{n+1}/D) = HB(a_{1}, …, a_{n}/D) and HB(a_{1}, …, a_{n+1}/C) = HB(a_{1}, …, a_{n}/C).

So HB(a_{1}, …, a_{n+1}/C) ⊋ HB(a_{1}, …, a_{n+1}/D) as needed.

If a_{n+1} ∈ cl(HDa_{1}, …, a_{n}) then by the condition on dim_{cl} we must also have that a_{n+1} ∈ cl(HCa_{1}, …, a_{n}) and thus

HB(a_{n+1}/Da_{1}, …, a_{n}) ⊂ HB(a_{n+1}/Ca_{1}, …, a_{n}).

Thus we get again HB(a_{1}, …, a_{n+1}/D) ⊊ HB(a_{1}, …, a_{n+1}/D).

If HB(a_{1}, …, a_{n}/C) = HB(a_{1}, …, a_{n}/D) and

HB(a_{n+1}/Da_{1}, …, a_{n}) ⊊ HB(a_{n+1}/Ca_{1}, …, a_{n})

then there is h ∈ HB(a_{n+1}/Ca_{1}, …, a_{n}) with h ∈ D. Then HB(a_{1}, …, a_{n+1}/C) ⊊ HB(a_{1}, …, a_{n+1}/D).

Assume now that dim_{cl}(a_{1}, …, a_{n+1}/ cl(HD)) = dim_{cl}(a_{1}, …, a_{n+1}/ cl(HD)), HB(a_{1}, …, a_{n+1}/C) = HB(a_{1}, …, a_{n+1}/D).

Case 1. SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}CH) > SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}DH). Then if a_{n+1} ∈ cl(a_{1}, …, a_{k}, C, H) we also get by additivity of SU-rank that SU(a_{k+1}, …, a_{n+1}/a_{1}, …, a_{k}CH) > SU(a_{k+1}, …, a_{n+1}/a_{1}, …, a_{k}DH) as desired.

If a_{n+1} ∉ cl(a_{1}, …, a_{k}, C, H), then by the assumptions on dim_{cl} we also have a_{n+1} ∉ cl(a_{1}, …, a_{k}, D, H) and tp(a_{n+1}/DHa_{1}, …, a_{k}) is orthogonal to

tp(a_{k+1}, …, a_{n}/DHa_{1}, …, a_{k})

and

SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}a_{n+1}CH) > SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}a_{n+1}DH),

as desired.

Case 2. SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}CH) = SU(a_{k+1}, …, a_{n}/a_{1}, …, a_{k}DH) and SU(a_{n+1}/a_{1} …, a_{n}CH) > SU(a_{n+1}/a_{1} …, a_{n}DH). Then by additivity of SUrank we have SU(a_{k+1}, …, a_{n+1}/a_{1}, …, a_{k}CH) > SU(a_{k+1}, …, a_{n+1}/a_{1}, …, a_{k}DH) as desired.

The other direction is proved in a similar way. □

We use the above result to give a different perspective on H-basis.

Lemma 5.4. Let (M, H(M)) be an H-structure. Let a⃗ = (a_{1}, …, a_{n}) ∈ M and let C ⊂ M be such that C is H-independent. Let h⃗ be a minimal tuple such that dim_{cl}(a⃗/Ch⃗) = dim_{cl}(a⃗/CH), then h⃗ = HB(a⃗/C).

Proof. Write a⃗ = a⃗_{1}a⃗_{2}, where a⃗_{1} are independent generics over CH and a⃗_{2} ∈ cl(a⃗_{1}CH). Choose h⃗ minimal so that a⃗_{2} ∈ cl(a⃗_{1}Ch⃗). Then a⃗_{1} are independent generics over CH and SU(a⃗_{2}/a⃗_{1}Ch⃗) < ω. Then tp(a⃗_{1}/C) is independent from H and tp(a⃗_{2}/a⃗_{1}Ch⃗) is orthogonal to H. We get a⃗ | ⌣_{Ch⃗} H and HB(a⃗/C) ⊂ h⃗. For the other direction, a⃗ | ⌣_{CHB(a⃗/C)} H implies that dim_{cl}(a⃗/CHB(a⃗/C)) = dim_{cl}(a⃗/CH)

and by minimality of h⃗ we get HB(a⃗/C) ⊂ h⃗. □

We are interested in characterizing canonical bases. We start with the following result which holds also in the geometric setting:

Lemma 5.5. Let (M, H) be a sufficiently saturated H-structure of T, B ⊂ M an H-independent set, and a⃗ ∈ M, h = HB(a⃗/B) (viewed as an imaginary representing a finite set). Suppose e ∈ acl^{eq}(B) (in the original theory) is such that a⃗h | ⌣_{e} B. Then a⃗ | ⌣_{e}^{ind} B.

Proof. We may assume that a⃗ = a⃗_{1}a⃗_{2}, where a⃗_{1} are generics over B ∪ H(M), a⃗_{2} ∈ cl(H(M)Ba⃗_{1}). Note that a⃗_{2} ∈ cl(a⃗_{1}Bh), so a⃗h | ⌣_{e} B implies that a⃗_{2} ∈ cl(a⃗_{1}eh), SU(tp(a⃗_{2}/Ba⃗_{1}h)) = SU(tp(a⃗_{2}/ea⃗_{1}h)) and also HB(a⃗/B) = HB(a⃗/e). Since HB(a⃗/B) = HB(a⃗/e) and a⃗h | ⌣_{e} B by our characterization of forking in T^{ind} we get a⃗ | ⌣_{e}^{ind} B. □

Finally, the following result on canonical bases can be proved doing very small modifications to the argument presented in [5]:

Proposition 5.6. Let (M, H) be a sufficiently saturated H-structure of T, B ⊂ M an H-independent set, and a⃗ ∈ M. Then Cb_{H}(a⃗/B) and Cb(a⃗HB(a⃗/B)/B)) are interalgebraic.

Proof. Let e = Cb(a⃗HB(a⃗/B)/B)). We saw in the previous lemma that a⃗ | ⌣_{e}^{ind} B and thus Cb_{H}(a⃗/B) ∈ acl^{eq}(e).

We will now prove that e is in the algebraic closure of any Morley sequence in stp_{H}(a⃗/B).

Let {a⃗_{i} : i < ω} be an L_{H}-Morley sequence in tp_{H}(a⃗/ acl_{H}^{eq}(B)). Let h_{j} = HB(a⃗_{j}/B) (viewed as an imaginary representing a finite set), so we have h_{j} ∈ dcl_{H}(a⃗_{j}B). Thus {a⃗_{i}h_{i} : i < ω} is also an L_{H}-Morley sequence over B. This implies h_{j} = HB(a⃗_{j}/Ba⃗_{<j}h_{<j}). We can write a⃗_{j} = a⃗_{j1}a⃗_{j2} and hence by our characterization of forking in T^{eq} we have that a⃗_{j1}h_{j} is an independent tuple of L generics over Ba⃗_{<j}h_{<j}) and SU(a⃗_{j2}/Ba⃗_{<j}h_{<j}a⃗_{j1}h_{j}) = SU(a⃗_{j2}/Ba⃗_{j1}h_{j}). Then it follows that tp(a⃗_{j}h_{j}/Ba⃗_{<j}h_{<j}) does not fork (in the sense of L) over B. Thus, {a⃗_{i}h_{i} : i < ω} is also an L-Morley sequence over B in tp(a⃗h/B). Since tp(a⃗_{0}h_{0}/{a⃗_{i}h_{i} : 0 < i < ω}B) is a free extension of tp(a⃗_{0}h_{0}/{a⃗_{i}h_{i} : 0 < i < ω}) we also get that e = Cb(a⃗_{0}h_{0}/{a⃗_{i}h_{i} : 0 < i < ω}). It follows that e ∈ acl^{eq}({a⃗_{i}h_{i} : i < ω}).

Since T^{ind} is supersimple there is N ∈ ω such that for all n ≥ N, a⃗_{n} | ⌣ind ; a⃗_{<N} B. By Proposition 4.7 acl_{H}(a⃗_{<N}) is H-independent. By our characterization of nonforking, HB(a⃗_{n}/B) = HB(a⃗_{n}/Ba⃗_{<N}) = HB(a⃗_{n}/ acl_{H}(a⃗_{<N})) and in particular h_{n} ∈ acl_{H}(a⃗_{i} : i < ω) for every n ≥ N. We then get e ∈ acl_{H}^{eq}({a⃗_{i} : N ≤ i < ω}). Now, since {a⃗_{i} : i < ω} is a Morley sequence in tp_{H}(a⃗/ acl_{H}^{eq}(B)), we have

{a⃗_{i} : N ≤ i < ω} | ⌣_{Cb_{H}(a⃗/B)}^{ind} B,

and thus also

{a⃗_{i} : N ≤ i < ω} | ⌣_{Cb_{H}(a⃗/B)}^{ind} e.

It follows that e ∈ acl_{H}^{eq}(Cb_{H}(a⃗/B)), as needed. □

6. Ampleness

In this section we examine the relation between the ampleness of T and T^{ind}. In [4] it is shown an example of an one-based geometric theory T such that T^{ind} is not one-based. We follow the ideas on [7] to understand exactly when one-based is preserved and to show that non 2-ampleness is also preserved. In this section we will assume that T eliminates imaginaries.

Remark 6.1. If T eliminates imaginaries then canonical bases are interalgebraic with real tuples. By Proposition 5.6 canonical bases in T^{ind} are also interalgebraic with real tuples. Hence T^{ind} has geometric elimination of imaginaries.

Example 6.2. Let G be an one-based stable group of U-rank ω and T = T h(G). Notice that T^{ind} is again a stable theory so (M, H) is a stable group but clearly H is not a boolean combination of cosets of subgroups, so T^{ind} is not one-based.

Definition 6.3. A pregeometry (X, cl) is trivial if for every A ⊂ X, cl(A) = ⋃_{a∈A} cl(a) .

Notice that if G is a group of U-rank ω then cl is not trivial (take a | ⌣ b both of rank ω and c = a + b, then c ∈ cl(a, b) \ cl(a) ∪ cl(b)).

Remark 6.4. In the theory of the free pseudoplane (see example 3.2 ) the pregeometry generated by cl is trivial: for A algebraically closed and a a single element, U(a/A) = d(a, A) where d(a, A) is the minimum length of a path from a to an element of A (or ω if there is no path). If b ∈ cl(A) it means that there is a path to some element a ∈ A so cl(A) = ⋃_{a∈A} cl(a).

We will now prove that one-basedness is only preserved in T^{ind} when the pregeometry cl is trivial. It is worth to notice that, unlike the U-rank 1 case, the triviality of cl does not imply that T is one-based. In fact, the theory of the free pseudoplane is the canonical example of a CM-trivial theory which is not one-based. This is the reason why the statement of the following proposition is a little bit different from [7].

Lemma 6.5. If cl is trivial in T then for every a⃗ and for every B = acl_{H}(B),

HB(a⃗/B) ⊂ HB(a⃗).

Proof. Let h = HB(a⃗/B) = {h_{i}|i ∈ I}. By minimality of H-bases for every i ∈ I a⃗ |̸ ⌣_{Bh\h_{i}} h_{i}, then h_{i} ∈ cl(a⃗Bh \ h_{i}). As B is H-independent and h_{i} ∉ B then h_{i} | ⌣ Bh \ h_{i}, hence h_{i} ∉ cl(Bh \ h_{i}). By triviality it means that h_{i} ∈ cl(a_{i}) for some a_{i} ∈ a⃗. By exchange property a_{i} ∈ cl(h_{i}), this implies a_{i} |̸ ⌣ h_{i} and a_{i} | ⌣_{h_{i}} H because tp(a_{i}/h_{i}) is orthogonal to H. We conclude that h_{i} = HB(a_{i}) and HB(a⃗/B) = {h_{i}|i ∈ I} = ⋃_{a_{i}∈A} HB(a_{i}) ⊂ HB(a⃗). □

Proposition 6.6. Assume T is one-based, then T^{ind} is one-based if and only if cl is trivial in T.

Proof of Proposition 6.6. (⇐) Assume cl is trivial, let a⃗ be a tuple, B an algebraic closed set in (M, H) and h⃗ = HB(a⃗/B). By the characterization of canonical bases, acl_{H}(cb_{H}(a⃗/B)) = acl_{H}(cb(a⃗h⃗/B)), as T is one-based, cb(a⃗h⃗/B) ⊂ acl(a⃗h⃗). By the previous lemma, h⃗ ⊂ HB(a⃗) then cb_{H}(a⃗/B) ⊂ acl_{H}(a⃗HB(a⃗)) = acl_{H}(a⃗), i.e. T^{ind} is one-based.

(⇒) Assume T^{ind} is one-based and cl is not trivial, then there are a tuple a⃗ and elements b and h such that b ∈ cl(a⃗h) and b ∉ cl(a⃗) ∪ cl(h). We can take a⃗ cl independent tuple minimal with this property and, by the generalized extension property, we may assume that a⃗ | ⌣ H . Moreover, as h ∉ cl(a⃗), we may assume also that h belongs to H by the generalized density property.

As b ∈ cl(a⃗h) and a⃗h is H-independent, tp(b/a⃗h) is orthogonal to H, i.e. b | ⌣_{ha⃗} H. Recall that b |̸ ⌣a⃗ h and h is a single element, then h = HB(b/a⃗). By hypothesis T^{ind} is one-based, then acl_{H}(cb_{H}(b/a⃗)) = acl_{H}(b) ∩ acl_{H}(a⃗). Now, acl_{H}(a⃗) = acl(a⃗) as a⃗ | ⌣ H. On the other hand, as a⃗ | ⌣ H, and b | ⌣_{ha⃗} H we have b | ⌣_{h}H. By hypothesis b ∉ cl(h), hence b | ⌣ h (recall that b is a single element)

and by transitivity b | ⌣ H. So HB(b) = ∅ and acl_{H}(b) = acl(b). This means acl_{H}(cb_{H}(b/a⃗)) = acl(b) ∩ acl(a⃗).

Recall that acl_{H}(cb_{H}(b/a⃗)) = acl_{H}(cb(bh/a⃗)). So a maximal cl-independent subset d⃗ of cb(bh/a⃗) satisfies that b ∈ cl(d⃗h) and b ∉ cl(d⃗) ∪ cl(h). The minimality of the length of a⃗ yields cl(cb(bh/a⃗)) = cl(a⃗), hence cl(a⃗) = cl(acl(a) ∩ acl(b)) ⊂ cl(a⃗) ∩ cl(b), then a⃗ ∈ cl(b) and h ∈ cl(a⃗b) ⊂ cl(b). This is a contradiction. □

The notion of ampleness, defined by Pillay, captures forking complexity. He proved in [13] that a theory T is one-based if and only if is not 1-ample, a theory T is CM-trivial if and only if is not 2-ample. Moreover if T interprets a field then it is n-ample for every n.

Definition 6.7. A supersimple theory T is CM-trivial if for every tuple c and for every A ⊂ B, if acl^{eq}(cA) ∩ acl^{eq}(B) = acl^{eq}(A) then cb(c/A) ⊂ acl^{eq}(cb(c/B))

Definition 6.8. A supersimple theory T is n-ample if (possibly after naming some parameters) there exist tuples a_{0}, ..., a_{n} in M^{eq} satisfying the following conditions:

For all 1 ≤ i ≤ n − 1.

(1) a_{i+1} | ⌣_{a_{i}}a_{i−1}...a_{0},

(2) acl^{eq}(a_{0}...a_{i−1}a_{i+1}) ∩ acl^{eq}(a_{0}...a_{i−1}a_{i}) = acl^{eq}(a_{0}...a_{i−1}).

(3) a_{n} |̸ ⌣_{acl^{eq}(a_{1})∩acl^{eq}(a_{0})}a_{0}.

Following [7] we prove that CM-triviality is preserved in T^{ind}. First we need the following lemma.

Lemma 6.9. Let A ⊂ B, A = acl_{H}(A) y B = acl_{H}(B). If acl_{H}(cA) ∩ B = A then HB(c/A) ⊂ HB(c/B).

Proof. It is clear that

HB(cA) ⊂ HB(cB).

By transitivity

HB(cA) = HB(c/A) ∪ HB(A),

and the same with HB(cB), hence

HB(c/A) ∪ HB(A) ⊂ HB(c/B) ∪ HB(B),

in particular HB(c/A) ⊂ HB(c/B) ∪ HB(B).

Now, if HB(c/A) ∩ H(B) = ∅ we are done, but

HB(c/A) ∩ HB(B) ⊂ acl_{H}(cA) ∩ B = A

and HB(c/A) ∩ A = ∅. □

Proposition 6.10. Let T be a SU-rank ω theory eliminating imaginaries, then T is CM-trivial if and only if T^{ind} is CM-trivial.

Proof. Assume T is 2-ample. Let a_{0}, a_{1}, a_{2} be tuples such that:

(1) a_{2} | ⌣_{a_{1}}a_{0},

(2) acl(a_{0}a_{2}) ∩ acl(a_{0}a_{1}) = acl(a_{0}),

(3) a_{2} |̸ ⌣_{acl(a_{1})∩acl(a_{0})}a_{0}.

By the generalized extension property, there are a′_{0}, a′_{1}, a′_{2} such that tp(a′_{0}a′_{1}a′_{2}) = tp(a_{0}a_{1}a_{2}) and a′_{0}a′_{1}a′_{2} | ⌣ H.

As the H-bases of any subset of {a′_{0}a′_{1}a′_{2}} are empty, the algebraic closure in T^{ind} of any of these sets is the same as in T. So condition (2) holds in T^{ind}.

By the characterization of canonical bases, since H-bases are empty then condition (1) holds also in T^{ind}. But if

a′_{2} | ⌣_{acl_{H}(a′_{1})∩acl_{H}(a′_{0})}^{H} a′_{0}

then

a′_{2}  | ⌣_{acl(a′_{1})∩acl(a′_{0})}a′_{0}.

This is a contradiction.

Assume T is not 2-ample, so it is CM-trivial. Let us see that T^{ind} is CM-trivial. Let c be a tuple and A ⊂ B be algebraically closed sets (in T^{ind}) such that

acl_{H}(cA) ∩ B = A. Define h = HB(c/A), h′ = HB(c/B) and c′ = ch. By Proposition 5.6 we have acl_{H}(cb_{H}(c/A)) = acl_{H}(cb(ch/A)) and by Lemma 6.9 h ⊂ h′. Note that acl(c′A) ∩ acl(B) = acl(A) because acl(c′A) ⊂ acl_{H}(cA), A = acl(A) and B = acl(B). So, by CM-triviality of T, cb(c′/A) ⊂ acl(cb(c′/B)). Recall that c′ = ch. Hence

acl_{H}(cb_{H}(c/A)) = acl_{H}(cb(ch/A))
⊂ acl_{H}(cb(ch/B))
⊂ acl_{H}(cb(ch′/B))
= acl_{H}(cb_{H}(c/B)).
□

We can adapt the previous proof in order to prove that if T^{ind} is n-ample then T is n-ample for every n. In [7] the converse has been proved for SU-rank 1 theories with a predicate, but we could not adapt that proof to this context.

7. Geometry modulo H in the one-based case

In this section we consider the case when T is one-based, and follow the proofs of Theorem 5.13 [15] and the results of Section 6 of [15], and Section 4 of [4], to study the geometry induced by cl localized at H(M). Many of the proofs are nearly identical to the ones from [15] and [4], we include them for completeness.

Let (M, H) be a sufficiently saturated model of T^{ind}. Let cl_{H} be the localization of the operator cl at H(M), i.e. cl_{H}(A) = cl(A ∪ H(M)). Thus, a ∈ cl_{H}(B) means SU(a/B ∪ H(M)) < ω.

Proposition 7.1. Suppose T is one-based. Then the pregeometry (M, cl_{H}) is modular.

Proof. It suffices to show that for any a, b ∈ M and a small set C ⊂ M, if a ∈ cl_{H}(bC) then there exists d ∈ cl_{H}(C) such that a ∈ cl_{H}(bd). We may assume that a, b ∉ cl_{H}(C). Let h⃗ ∈ H(M) be finite such that a ∈ cl(bCh⃗). Let e = Cb(ab/Ch⃗). Thus, by one-basedness of T, e ∈ acl^{eq}(ab) ∩ acl^{eq}(Ch⃗). By the density property, there is b′ ⊧ tp(b/ acl^{eq}(Ch⃗)), b′ ∈ H(M). Take a′ ∈ M such that tp(a′b′/ acl^{eq}(Ch⃗)) = tp(ab/ acl^{eq}(Ch⃗)). Then e ∈ acl^{eq}(a′b′). Clearly, a′ ∈ cl(b′Ch⃗) ⊂ cl_{H}(C). Also, ab | ⌣_{e} Ch⃗ implies SU(a/be) = SU(a/bCh⃗) < ω. Since e ∈ acl^{eq}(a′b′), we have SU(a/ba′b′) ≤ SU(a/be) < ω. Since b′ ∈ H(M), this implies a ∈ cl_{H}(ba′). Hence, taking d = a′, we have d ∈ cl_{H}(C) and a ∈ cl_{H}(bd), as needed. □

Let (M^{∗}, cl^{∗}) be the geometry associated with (M, cl_{H}) (i.e. M^{∗} is the set M\ cl_{H}(∅) modulo the relation cl_{H}(x) = cl_{H}(y)) . For any a ∉ cl_{H}(∅), let a^{∗} be the class of a modulo the relation cl_{H}(x) = cl_{H}(y). Define the relation ∼ by

a^{∗} ∼ b^{∗} ⟺ | cl^{∗}(a^{∗}, b^{∗})| ≥ 3 or a^{∗} = b^{∗}.

Lemma 7.2. For any a, b ∈ M, a^{∗} ∼ b^{∗} if and only if there exist d_{1}, …, d_{n} ∈ M such that

a^{∗} ∈ cl^{∗}(b^{∗}d_{1}^{∗} … d_{n}^{∗})\ cl^{∗}(d_{1}^{∗} … d_{n}^{∗}).

Proof. The "only if" direction is clear. For the "if" direction, suppose a^{∗} ≠ b^{∗} and a^{∗} ∈ cl^{∗}(b^{∗}d_{1}^{∗} … d_{n}^{∗})\ cl^{∗}(d_{1}^{∗} … d_{n}^{∗}). We may assume that n ≥ 1 is minimal such. Then a ∈ cl^{∗}(bd_{1} … d_{n}h_{1} … h_{k}) for some h_{1}, …, h_{k} ∈ H(M). We may assume that k is minimal such. Then the tuple abd_{2} … d_{n}h_{1} … h_{k} is cl-independent. By the density property, we can find d′_{2}, …, d′_{n} ∈ H(M) such that tp(d′_{2}, …, d′_{n}/abh⃗) = tp(d_{2}, …, d_{n}/abh⃗). Let d′_{1} ∈ M be such that

tp(d′_{1}, d′_{2}, …, d′_{n}/abh⃗) = tp(d_{1}, d_{2}, …, d_{n}/abh⃗).

Then d′_{1} ∉ cl_{H}(∅) and (d′_{1})^{∗} ∈ cl^{∗}(a^{∗}, b^{∗}), while (d′_{1})^{∗} ≠ a^{∗}, b^{∗}. Thus, | cl^{∗}(a^{∗}, b^{∗})| ≥ 3, as needed.

□

Lemma 7.3. The relation ∼ is an equivalence on M^{∗}.

Proof. Reflexivity and symmetry are clear. For transitivity, assume a^{∗} ∼ b^{∗} ∼ c^{∗}, with all three distinct. Then there exist d_{1}^{∗} ∈ cl^{∗}(a^{∗}b^{∗})\{a^{∗}, b^{∗}} and d_{2}^{∗} ∈ cl^{∗}(b^{∗}, c^{∗})\{b^{∗}, c^{∗}}. If d_{1}^{∗} = d_{2}^{∗}, then c^{∗} ∈ cl^{∗}(b^{∗}, d_{2}^{∗}) = cl^{∗}(b^{∗}, d_{1}^{∗}) = cl^{∗}(a^{∗}, d_{1}^{∗}), and therefore d_{1}^{∗} = d_{2}^{∗} ∈ cl^{∗}(a^{∗}, c^{∗})\{a^{∗}, c^{∗}}, hence a^{∗} ∼ c^{∗}.

Now, assume that d_{1}^{∗} ≠ d_{2}^{∗} and a^{∗} ∈ cl^{∗}(d_{1}^{∗}, d_{2}^{∗}). If a^{∗} = d_{2}^{∗}, then b^{∗} witnesses a^{∗} ∼ c^{∗}. If a^{∗} ≠ d_{2}^{∗}, then d_{2}^{∗} ∈ cl^{∗}(a^{∗}, d_{1}^{∗}). We also have b^{∗} ∈ cl^{∗}(a^{∗}, d_{1}^{∗}), c^{∗} ∈ cl^{∗}(b^{∗}, d_{2}^{∗}). Thus, c^{∗} ∈ cl^{∗}(a^{∗}, d_{1}^{∗}). If c^{∗} = d_{1}^{∗}, b^{∗} witnesses a^{∗} ∼ c^{∗}. If c^{∗} ≠ d_{1}^{∗}, then d_{1}^{∗} witnesses a^{∗} ∼ c^{∗}. Finally, assume that d_{1}^{∗} ≠ d_{2}^{∗} and neither a^{∗} ∉ cl^{∗}(d_{1}^{∗}, d_{2}^{∗}). Then

a^{∗} ∈ cl^{∗}(c^{∗}d_{1}d_{2}^{∗})\ cl^{∗}(d_{1}^{∗}d_{2}^{∗}).

Thus, by Lemma 7.2, a^{∗} ∼ c^{∗}.

□

For any a^{∗} ∈ M^{∗} let [a^{∗}] denote the ∼-class of a^{∗}.

Lemma 7.4. The ∼-classes are closed in the sense of cl^{∗}, i.e. for any a^{∗} ∈ M^{∗}, we have cl^{∗}([a^{∗}]) = [a^{∗}].

Proof. Assume c^{∗} ∈ cl^{∗}(b_{1}^{∗}, …, b_{n}^{∗}), b⃗^{∗} = (b_{1}^{∗}, …, b_{n}^{∗}) ∈ [a^{∗}] minimal such tuple, and n > 1 (if n = 1, we have c^{∗} = b_{1}^{∗}). Then b_{1}^{∗} … b_{n−1}^{∗} witnesses c^{∗} ∼ b_{n}^{∗}, by Lemma 7.2. □

For any geometry (X, Cl), a non-empty subset of X, with the closure operator induced by Cl, is referred to as a subgeometry of (X, Cl). Clearly, a subgeometry is itself a geometry. Next lemma shows that ∼ splits (M^{∗}, cl^{∗}) into disjoint subgeometries of the form ([a^{∗}], cl^{∗}), with no "interaction" between them.

Lemma 7.5. For any A ⊂ M^{∗}, cl^{∗}(A) = ⋃_{[a^{∗}]∈M^{∗}/∼} cl^{∗}(A ∩ [a^{∗}]).

Proof. Suppose c^{∗} ∈ cl^{∗}(A), and a_{1}^{∗}, …, a_{n}^{∗} ∈ A is a tuple such that c ∈ cl^{∗}(a_{1}^{∗}, …, a_{n}^{∗}), and n is minimal such. It suffices to show that a_{i}^{∗} all come from the same ∼-class. If n = 1, we are done. Suppose n > 1. Then c^{∗}a_{3}^{∗} … a_{n}^{∗} witnesses a_{1}^{∗} ∼ a_{2}^{∗} by Lemma 7.2. Similarly, a_{1}^{∗} ∼ a_{i}^{∗} for all 2 < i ≤ n. Thus, all a_{i}^{∗} belong to the same ∼-class. □

Next, we will show that the ∼-classes are either singletons or infinite dimensional (as geometries).

Lemma 7.6. If |[a^{∗}]| > 1, then dim([a^{∗}]) is infinite.

Proof. Suppose there exists b^{∗} ∼ a^{∗}, b^{∗} ≠ a^{∗}. Let c^{∗} ∈ cl^{∗}(a^{∗}, b^{∗})\{a^{∗}, b^{∗}}. Let a, b, c ∈ M be representatives of the classes a^{∗}, b^{∗} and c^{∗} modulo the relation cl_{H}(x) = cl_{H}(y), respectively.

Then SU(a/H(M)) = SU(b/aH(M)) = ω. By the extension property, we can find b_{i} ⊧ tp(b/a), i ∈ ω, independent over aH(M). Then, by Lemma 2.7, tp_{H}(b_{i}/a) = tp_{H}(b/a) for all i ∈ ω. Also, b_{i} are cl_{H}-independent over a. Let c_{i} be such that tp_{H}(b_{i}c_{i}/a) = tp_{H}(bc/a) for i ∈ ω. Passing to the geometry (M^{∗}, cl^{∗}), we get b_{i}^{∗} ∼ a^{∗} witnessed by c_{i}^{∗}, i ∈ ω, with b_{i} cl^{∗}-independent over a^{∗}. This shows that that ([a^{∗}], cl^{∗}) is infinite dimensional. □

Recall the following classical fact (see [11]) about projective geometries.

Fact 7.7. A non-trivial modular geometry of dimension ≥ 4 in which any closed set of dimension 2 has size ≥ 3 is a projective geometry over some division ring.

Lemma 7.8. If T is one-based and |[a^{∗}]| > 1 , the geometry ([a^{∗}], cl) is an infinite dimensional projective geometry over some division ring.

Proof. By Proposition 7.1, (M^{∗}, cl^{∗}) is modular. By Lemma 7.5, [a^{∗}] is closed in (M^{∗}, cl^{∗}), and hence ([a^{∗}], cl^{∗}) is also modular. Since |[a^{∗}]| > 1, ([a^{∗}], cl^{∗}) is nontrivial (there are two distinct point having a third one in its closure). Thus, the statement follows by Fact 7.7 and the definition of ∼. □

We are now ready to prove the characterization of the geometry of cl_{H}, as well as the original geometry of cl in the case when T is one-based.

Proposition 7.9. Suppose T is a one-based supersimple theory of SU-rank ω, (N, H) a sufficiently (e.g. |T |^{+}-) saturated models of T^{ind}, and M a small model of T (e.g. of size |T |). Then (1) The geometry (N^{∗}, cl^{∗}) of cl_{H} in (N, H) is a disjoint union of infinite dimensional projective geometries over division rings and/or a trivial geometry.

(2) The geometry of the closure operator cl in M is a disjoint union of subgeometries of projective geometries over division rings.

Proof. (1) Follows by Lemmas 7.5, 7.6 and 7.8.

(2) By Lemma 2.6, any structure of the form (M, H) where M ⊧ T, and H(M) is an independent set of generics, can be embedded, in an H-independent way, in a sufficiently saturated H-structure. Thus we may assume that (M, ∅) ⊂ (N, H) with M | ⌣_{∅} H(N). Then cl-independence over ∅ in M coincides with cl-independence in N over H(N), i.e. cl_{H}-independence. Thus, we have a natural embedding of the associated geometry of (M, cl) into (N^{∗}, cl^{∗}). The result now follows by (1). □

Remark 7.10. The previous proposition also holds with the weaker assumption that the pregeometry (N, cl_{H}) is modular instead of asking that is one-based. All the proofs depend on the properties of the closure operator, not the properties of forking in the full structure.

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

E-mail address: aberenst@uniandes.edu.co

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

Université de Lyon CNRS, Université Lyon 1, Institut Camille Jordan UMR5208, 43 boulevard du 11 novembre 1918, F–69622 Villeurbanne Cedex, France

E-mail address: jf.carmona658@uniandes.edu.co

Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL A2H 5G4, Canada

E-mail address: yvasilyev@grenfell.mun.ca