GEOMETRIC STRUCTURES WITH A DENSE INDEPENDENT SUBSET

ALEXANDER BERENSTEIN AND EVGUENI VASSILIEV

Abstract. We generalize the work of [13] on expansions of o-minimal structures with dense independent subsets, to the setting of geometric structures. We introduce the notion of an H-structure of a geometric theory T, show that H-structures exist and are elementarily equivalent, and establish some basic properties of the resulting complete theory T^{ind}, including quantifier elimination down to “H-bounded” formulas, and a description of definable sets and algebraic closure. We show that if T is strongly minimal, supersimple of SUrank 1, or superrosy of thorn rank 1, then T^{ind} is ω-stable, supersimple, and superrosy, respectively, and its U-/SU-/thorn rank is either 1 (if T is trivial) or ω (if T is non-trivial). In the supersimple SU-rank 1 case, we obtain a description of forking and canonical bases in T^{ind}. We also show that if T is (strongly) dependent, then so is T^{ind}, and if T is non-trivial of finite dp-rank, then T^{ind} has dp-rank greater than n for every n < ω, but bounded by ω. In the stable case, we also partially solve the question of whether any group definable in T^{ind} comes from a group definable in T.

1. Introduction

We say a theory T is geometric if for any model M ⊧ T the algebraic closure satisfies the exchange property and T eliminates the quantifier ∃^{∞} (see [20, Def. 2.1], [17]). There are many examples of geometric theories, among them dense ominimal theories, strongly minimal theories, SU-rank 1 theories, the p-adics in a single sort, etc.

Expansions of geometric theories with a unary predicate have been studied extensively. There are expansions where the underlying model M is an algebraically closed or a real closed field and the predicate is interpreted as a multiplicative subgroup, for example to study groups with the Mann property [15]. This expansion created a nice framework for studying groups satisfying the Mordell-Lang property inside a fixed field. In the same way, the work on rational points of eliptic curves from [18] gives connections with number theory.

Another such expansion corresponds to lovely pairs [6, 3]. Let L be the language of T, let M ⊧ T and let H be a new unary predicate that does not belong to L. For M ⊧ T, we say that (M, H(M)) is a lovely pair if H(M) is an elementary substructure of M, the predicate satisfies the density property (for any infinite Lformula φ(x, b⃗) with parameters in M, φ(H(M)) ≠ ∅) and M satisfies the extension property over H(M) (for any infinite L-formula φ(x) with parameters in M, (φ(M)\ acl(H(M)b⃗)) ≠ ∅). Lovely pairs are a tool for understanding the properties of the underlying geometry such as linearity. The structure imposed on the predicate, i.e. being an elementary substructure, a subgroup, allows one to use the expansion to get an insight into different properties of T or structures living inside a model of T.

2000 Mathematics Subject Classification. 03C10, 03C45, 03C64.

Key words and phrases. geometric structures, strongly minimal theories, U-rank one theories, rosy theories, strongly dependent theories, definable groups.

Generic (random) predicate expansions have also been studied extensively, e.g. by Chatzidakis and Pillay in [11]. In the strongly minimal case they provide “natural” examples of unstable supersimple structures of SU-rank 1. Baldwin and Benedikt [1] have also considered expansions by indiscernible sequences.

In this paper we will explore an expansion, introduced in the o-minimal case in [13], which in some sense is dual to the lovely pairs expansion. We will assume that H(M) is a collection of algebraically independent elements satisfying the density and extension properties. The construction is a dual in the sense that instead of assuming the predicate to be an elementary substructure, we assume it is a collection of “geometrically unrelated” (algebraically independent) elements. We call such an expansion (M, H) an H-structure, and we write T^{ind} for its theory.

Examples of T^{ind} include the theory of a vector space with a distinguished basis and the theory of a real closed field with a distinguished dense transcendence basis.

Some properties of this expansion are very similar to those of lovely pairs. For example, we show in Section 2 that saturated models of T^{ind} are again H-structures. We show in Section 3 that the definable subsets of H(M) are just intersections of Ldefinable formulas with H. We also show that the definable subsets of (M, H(M)) come as boolean combinations of L-formulas enlarged by existential quantifiers over H. While the question of elimination of ∃^{∞} in T^{ind} remains open, we show that it holds for formulas where parameters are assumed to be in H(M). As in the pairs setting, one of the central notions in the study of H-structures is that of the large and small set. What is different from the case of pairs is that in H-structures we also have the notion of “H-basis” of a tuple (over ∅ or another set). On one hand, this notion allows one to “coordinatize” the structure by elements of H and elements orthogonal to H, while on the other hand it generates a variety of new definable functions from definable sets in (M, H) to H(M).

In Section 4 we explore some additional topics related to H-structures motivated by the analogies with the pair expansions. In subsection 4.1 we compare H-structures with lovely pairs and show how to build a lovely pair out of an Hstructure. Subsection 4.2 iterates the construction of H-structures to tuples, following similar ideas of Poizat on beautiful pairs [24] and of Fornasiero for closure relations [16]. In subsection 4.3, we show elimination of ∃y ∈ H for the expansion of H(M) by externally definable sets.

In Section 5 we show that if T is strongly minimal (respectively T has SU-rank 1, thorn-rank 1), then MR(T^{ind}) ≤ ω (respectively SU-rank(T^{ind}) ≤ ω, thornrank(T^{ind}) ≤ ω). We obtain a description of forking and canonical bases in T^{ind} when T is supersimple of SU-rank 1. We also observe a (somewhat surprising) fact that that one-basedness is not preserved when passing to T^{ind}.

In the lovely pair case, the rank of the expansion captured the geometric complexity of the base theory (along the lines of the trivial/linear/non-linear trichotomy). Similar, but much less refined, connection takes place in the case of H-structures: non-triviality of the base theory guarantees that the expansion will have the maximal rank ω.

Finally in this section we show that if T is (strongly) dependent, T^{ind} is also (strongly) dependent. As before there is a connection between the triviality of T and the dp-rank. When T is trivial and T has dp-rank n, then so does T^{ind}, if T is not trivial and T has dp-rank n, then T^{ind} has dp-rank greater than or equal to k for all k.

In Section 6 we study groups definable in a H-structure (M, H). Since the geometry on H is trivial, we expected that adding H should not introduce new definable groups. We managed to show this claim only partially. When T is stable we show that every connected group definable in (M, H) is definably isomorphic to a group interpretable in M.

Acknowledgements. The authors thank Alf Dolich for many helpful discussions on the topic.

2. H-structures: definition and first properties

Let T be a complete geometric theory in a language L. Thus, in any model M ⊧ T, the algebraic closure satisfies the Exchange Property and T eliminates the quantifier ∃^{∞}. Let H be a new unary predicate and let L_{H} = L∪{H}. Let T′ be the L_{H}-theory of all structures (M, H), where M ⊧ T and H(M) is an L-algebraically independent subset of M. Note that saying that H(M) is independent is a first order property, it is simply the conjunctions of formulas of the form ¬φ(x_{1}, …, x_{n}), where dim(φ(x_{1}, …, x_{n})) < n.

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 (and the corresponding notation | ⌣) means acl-independence, where acl stands for the algebraic closure in the sense of L. 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}. For A ⊂ B sets and q ∈ S_{n}(B), we say that q is free over A or that q is a free extension of q ↾_{A} if for any (all) c⃗ ⊧ q, c⃗ is independent from B over A.

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 dimensional and q ∈ S_{1}(A) is non-algebraic, there is a ∈ H(M) such that a ⊧ q.

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

q ∈ S_{n}(A) has dimension n, then there is a⃗ ∈ H(M)^{n} such that a⃗ ⊧ q. (3’) (Generalized 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)) is free

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 n times the density property we can find a_{1}, …, a_{n} ∈ H(M) such that

tp(a_{1}, …, a_{n}/ acl(A)) = tp(b_{1}, …, b_{n}/ acl(A)).

□

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

Remark 2.5. Assume now that T is a geometric theory expanding DLO and that (M, H(M)) is an H-structure. Let a, b ∈ M be such that a < b; then the partial type a < x < b is non-algebraic and by the density property it is realized in H(M). Thus, the density property implies that H(M) is dense in M. The density property that we use in this paper can be traced back to Macintyre [21], it also appears under the name of coheir property in [2].

Definition 2.6. 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 2.7. Any model M of T with a distinguished independent subset H(M) can be embedded in a model of T′ in an H-independent way.

Proof. Given any model M with a distinguished independent subset H(M), we can always find an elementary extension N of M and a set H(N) extending H(M) such that for every non-algebraic 1-type p(x, acl(m⃗)), where m⃗ ∈ M, there are d ∈ N and b ∈ H(N) such that both b and d realize p(x, acl(m⃗)) and d ∉ acl(M, H(N)). Now apply a chain argument. □

In particular, for a geometric theory T, H-structures exist.

Lemma 2.8. Let (M, H) and (N, H) be sufficiently saturated models of T′, 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. Let a⃗ = a⃗_{0}a⃗_{1}h⃗, where a⃗_{0} is independent over H, h⃗ ∈ H and a⃗_{1} ∈ acl(a⃗_{0}h⃗). Similarly write a′ = a′_{0}a′_{1}h⃗′.

To prove the Lemma we show that the partial isomorphism that sends a⃗ to a′ can be extended, so 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 ∈ acl(a⃗). By H-independence, either b ∈ h⃗ or b ∉ H(M). Let b′ ∈ acl(a′) be such that tp(b′a′) = tp(ba⃗). Clearly, b ∈ H(M) iff b′ ∈ H(N). Here we can take h⃗_{1} and h⃗′_{1} to be empty.

Case 2: b ∈ H and is non algebraic over a⃗. 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 ∈ acl(Ha⃗). Add a tuple h⃗_{1} ∈ H such that a⃗bh⃗_{1} is H-independent, and use Cases 1 and/or 2.

Case 4: b ∉ acl(Ha⃗). By the extension property, there is b′ ∈ N such that b′ ∉ acl(Ha′) 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 2.9. All H-structures are elementarily equivalent.

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

To axiomatize T^{ind} we follow the ideas of [27, Prop 2.15]. Here we use for the first time the fact that T eliminates ∃^{∞}. Recall that whenever T eliminates ∃^{∞} the expression the formula φ(x, b⃗) is nonalgebraic is first order.

Proposition 2.10. The theory T^{dim} is axiomatized by:

(1) T′.

(2) For all L-formulas φ(x, y⃗)

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

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

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

The second scheme of axioms corresponds to the density property and the third scheme to the extension property. The first axiom says that H is a collection of independent elements. The proof is the same on as the one in [3, Thm 2.10].

Example 2.11. Let T be the theory of infinite dimensional vector spaces over a fixed finite field, say F_{2}. Note that T is strongly minimal so T is geometric. Let V ⊧ T be countable and let H = {v_{j} : j ∈ ω} be its basis. Then (V, H) ⊧ T^{ind} but it is NOT an H-structure since it does not satisfy the extension property.

Example 2.12. Let T = T h(ℝ, +, ×, 0, 1, <), T is o-minimal extending DLO so T is geometric. Let H = {e_{i} : i ∈ I} be a dense trascendence basis, then (ℝ, H) ⊧ T^{ind}. Note that (ℝ, H) is not an H-structure, since it does not satisfy the extension property.

3. Definable sets in H-structures

Fix T a geometric 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}. We will also address the question of the elimination of ∃^{∞} in T^{ind}.

Notation 3.1. Let (M, H(M)) be an H-structures. 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 3.2. 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(b⃗/H) = dim(a⃗/H).

Let h⃗ = (h_{1}, …, h_{l}) ∈ H(M) be such k := dim(a⃗/h⃗) = dim(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}) ∈ acl(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 algebraic in z⃗ and M ⊧ ψ(h⃗, a^{1}, a^{2}). Since etp_{H}(a⃗) = etp_{H}(b⃗) we get that dim(b⃗/H) ≤ k. A similar argument shows that dim(a⃗/H(M)) ≤ dim(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(a⃗/h⃗) = dim(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 2.8. □

Corollary 3.3. Let (M, H(M)) be a sufficiently saturated H-structure, assume that T = T h(M) is trivial and that dcl = acl in T. Then every L_{H} formula φ(x⃗) in (M, H(M)) is equivalent to a boolean combination of L formulas and formulas of the form H(f(x⃗)), where f is an L-definable function.

Proof. It suffices to check that types of tuples in (M, H(M)) are isolated by the the L-formulas that they satisfy and the values of expressions of the form H(f(x⃗)), where f is an L-definable function..

Let a⃗, b⃗ be tuples of the same arity from M and assume that they satisfy the same L-type and that for every L-definable function f(x⃗) we have that H(f(a⃗)) holds if and only if H(f(b⃗)) holds. We will prove that tp_{H}(a⃗) = tp_{H}(b⃗).

Claim dim(b⃗/H) = dim(a⃗/H).

Let h⃗ = (h_{1}, …, h_{l}) ∈ H(M) be such k := dim(a⃗/h⃗) = dim(a⃗/H(M)) and assume that h⃗ is a minimal such tuple. Then since T is trivial, for each i ≤ k we have that h_{i} = f_{i}(a_{j_{i}}) for some j_{i} and some L-definable function f_{i}. Let h′_{i} = f_{i}(b_{j_{i}}) and let h⃗′ = (h′_{1}, …, h′_{l}). Then H(h′_{i}) holds for each i ≤ l and dim(b⃗/H) ≤ l = dim(a⃗/H). The other inequality follows in the same way.

Note that for h⃗ and h⃗′ defined as above, we have that a⃗h⃗ and b⃗h⃗′ are Hindependent and thus by Lemma 2.8 we have that tp(a⃗) = tp(b⃗) as desired. □

Now we are interested in the L_{H}-definable subsets of H(M). This material is very similar to the results presented in [14, Theorem 2].

Lemma 3.4. Let (M_{0}, H(M_{0})) ⪯ (M_{1}, H(M_{1})) and assume that (M_{1}, H(M_{1})) is |M_{0}|-saturated. Then 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} ∈ acl(a_{1}, …, a_{n−1}, H(M_{1})) and a_{n} ∉ acl(a_{1}, …, a_{n−1}, H(M_{0})). Let φ(x, y⃗, z⃗) be a formula and b⃗ ∈ H(M_{1})z⃗ be a tuple such that

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

holds. Since (M_{0}, H(M_{0})) ⪯ (M_{1}, H(M_{1})) there is b⃗′ ∈ H(M_{0})y⃗ such that

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

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

Proposition 3.5. 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 3.4 that Ma⃗, Mb⃗ are H-independent sets and thus by Lemma 2.8 we get tp_{H}(a⃗/M) = tp_{H}(b⃗/M). □

Remark 3.6. A small warning is due here. In the previous proof, we may need extra parameters in the small model to define an L-formula equivalent to the original L_{H}-formula.

Definition 3.7. Let (M, H) ⊧ T^{ind} be saturated. We say that an L_{H} formula ψ(x, c⃗) defines a large subset of M is there is b ⊧ ψ(x, c⃗) such that b ∉ scl(c⃗). This is equivalent as requiring that there are infinitely many realizations of ψ(x, c⃗) that are algebraically independent over H(M)c⃗.

Definition 3.8. 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 small closure of A if b⃗ ∈ acl(AH(M)) and write b⃗ ∈ scl(A). Let X ⊂ M^{n} be A-definable. We say that X is small if X ⊂ scl(A).

Since T is geometric, scl satisfies the exchange property and thus it is a closure operator.

Next, we introduce the notion of the H-basis, which first appeared in [13] in the o-minimal context.

Proposition 3.9. 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. Clearly there is a 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 over H(M) and a⃗_{2} ∈ scl(a⃗_{1}). If a⃗_{2} = ∅, then h⃗ = ∅ and the result follows. So we may assume that a⃗_{2} ≠ ∅.

Then a⃗_{2} ∈ acl(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. Since a⃗_{2} ∈ acl(a⃗_{1}, h⃗_{1}, h⃗_{2})\acl(a⃗_{1}, h⃗_{1}) and a⃗_{2} ∈ acl(a⃗_{1}, h⃗_{1}, h⃗′_{2})\ acl(a⃗_{1}, h⃗_{1}) then by the exchange property dim(h⃗′_{2}/a⃗_{1}h⃗_{1}h⃗_{2}) < dim(h⃗′_{2}/a⃗_{1}h⃗_{1}). Since a⃗_{1} is independent over H we get that dim(h⃗′_{2}/h⃗_{1}h⃗_{2}) < dim(h⃗′_{2}/h⃗_{1}) and thus since H is independent, h⃗_{2} has a common element with h⃗′_{2}, a contradiction. □

Remark 3.10. Let (M, H(M)) be an H-structure. Let a⃗ = (a_{1}, …, a_{n}) ∈ M and let C ⊂ M be such that C = acl(C) and C is H-independent. As before, there is a unique smallest tuple h⃗ ∈ H(M) such that a⃗ | ⌣_{h⃗C} H.

Definition 3.11. 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 = acl(C) and 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.

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

Proposition 3.12. 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 small, where △ stands for a boolean connective for the symmetric difference.

Proof. If Y is small or cosmall, the result is clear, so we may assume that both Y and M \ Y are large. Assume that Y is definable over a⃗ and that a⃗ = a⃗HB(a⃗). Let b ∈ Y be such that b ∉ scl(a⃗) and let c ∈ M \ Y be such that c ∉ scl(a⃗). 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 first assume that X_{bc} only depends on tp(b/a⃗) and applying compactness again we may assume that X_{bc} does no depend on tp(b/a⃗) and we will call it simply X. Thus for b′ ∈ Y and c′ ∈ M \ Y not in the small closure of a⃗, we have b′ ∈ X and c′ ∈ M \ X. This shows that Y △X is small. □

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

Lemma 3.13. Let T be a geometric theory, M ⊧ T, (M, H(M)) 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 ∉ scl(A), 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 2.8, each a_{i} realizes tp_{H}(a/A), and thus a ∉ acl_{H}(A).

If a ∈ scl(A), take a minimal tuple b⃗ ∈ H(M) such that a ∈ acl(Ab⃗). Using the coheir property of H-structures, we can find b⃗_{i} ∈ H(M), i ∈ ω, such that b⃗_{i} are acl-independent over A and realize tp(b⃗/A). Take a_{i} ∈ acl(Ab⃗_{i}) such that tp(a_{i}b⃗_{i}/A) = tp(ab⃗/A). Then {a_{i} : i ∈ ω} are acl-independent over A. On the other hand, for any i ∈ ω, Ab⃗_{i}a_{i} is a H-independent set and thus by Lemma 2.8 tp_{H}(a_{i}b⃗_{i}/A) = tp_{H}(ab⃗/A) and in particular tp_{H}(a_{i}/A) = tp_{H}(a/A). □

Corollary 3.14. Let T be a geometric theory, M ⊧ T, (M, H(M)) an H-structure, and let A ⊂ M. Then acl_{H}(A) = acl(A, HB(A)).

Proof. By Proposition 3.9, 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)) □

It is interesting to check which properties of T are preserved in T^{ind}.

Question 3.15. Does T^{ind} eliminate the quantifier ∃^{∞}?

We give a partial answer. Namely, we will show the elimination of the ∃^{∞}x for L_{H}-formulas ϕ(x, z⃗) implying H(z⃗). Note that if a ∈ H(M) and h⃗ ∈ H(M) then a ∈ acl_{H}(h⃗) exactly when a is a part of h⃗. Thus we may assume that ϕ(x, z⃗) implies ¬H(x) ∧ H(z⃗). We will be working in a sufficiently saturated H-structure (M, H) of a geometric theory T.

First, note that if h⃗ ∈ H(M) and |ϕ(M, h⃗)| = n < ω, then for any a ∈ M with ⊧ ϕ(a, h⃗), we have a ∈ acl(h⃗), and there is an L-formula ϕ_{n}(x, z⃗) such that ϕ_{n}(M, h⃗) = ϕ(M, h⃗). By compactness, ϕ_{n}(x, z⃗) does not depend on the choice of h⃗ ∈ H(M), but it may still depend on n (unless T is ω-categorical). Thus, this approach does not seem to work. Instead, we will take a closer look at the L_{H}-formula ϕ(x, z⃗).

We say that an L-formula ψ(x, y⃗) has a bounded finite number of realizations in x, if there exists n < ω such that for any b⃗, |ψ(M, b⃗)| < n. Thus, ϕ(x, b⃗) is either inconsistent or witnesses x ∈ acl(b⃗).

Lemma 3.16. For any h⃗ ∈ H(M) and a ∈ M with a ∉ H(M), tp_{H}(a, h⃗) is axiomatized by ¬H(x), H(z⃗), L-formulas, and the formulas of the form

∃y⃗ ∈ H θ(x, y⃗, z⃗)

or

¬∃y⃗ ∈ H θ(x, y⃗, z⃗),

where θ(x, y⃗, z⃗) is an L-formula having a bounded finite number of realizations in x.

Proof. Assuming a ∉ H(M) and h⃗ ∈ M, the L_{H}-type of the tuple ah⃗ is determined by its L-type, and either the fact that a ∉ scl(h⃗) or the L-type of some k⃗ ∈ H(M) over h⃗, such that a ∈ acl(k⃗, h⃗). All these properties can be expressed with the given types of formulas. □

Note that ϕ(x, z⃗) is a conjunction of H(z⃗), ¬H(x), Boolean combination of Lformulas and formulas of the form ∃y⃗ ∈ H θ(x, y⃗, z⃗), where θ(x, y⃗, z⃗) is an L-formula having a bounded finite number of realizations in x. Note that elimination of ∃^{∞}x is preserved under disjunction. Thus, we may assume that ϕ(x, z⃗) is a conjunction of H(z⃗), ¬H(x), an L-formula Γ(x, z⃗), and/ or formulas of the form ∃y⃗ ∈ H θ(x, y⃗, z⃗) or ¬∃y⃗ ∈ H θ(x, y⃗, z⃗), where θ(x, y⃗, z⃗) is an L-formula having a bounded finite number of realizations in x. Note that the class of L-formulas θ(x, y⃗, z⃗) having a bounded finite number of realization in x is closed under conjunction and disjunction. Thus, we may assume that ϕ(x, z⃗) has one of the four forms:

(1) ¬H(x) ∧ H(z⃗) ∧ Γ(x, z⃗),

(2) ¬H(x) ∧ H(z⃗) ∧ Γ(x, z⃗) ∧ ∃y⃗ ∈ H θ(x, y⃗, z⃗),

(3) ¬H(x) ∧ H(z⃗) ∧ Γ(x, z⃗) ∧ ¬∃y⃗ ∈ H θ(x, y⃗, z⃗),

(4) ¬H(x) ∧ H(z⃗) ∧ Γ(x, z⃗) ∧ ∃y⃗ ∈ H θ_{1}(x, y⃗, z⃗) ∧ ¬∃y′ ∈ H θ_{2}(x, y′, z⃗), where Γ(x, z⃗) is an L-formula, and θ(x, y⃗, z⃗), θ_{1}(x, y⃗, z⃗) and θ_{2}(x, y′, z⃗) are L-

formulas having a bounded finite number of realizations in x.

Clearly, in cases (1) and (3), the algebraicity of ϕ(x, h⃗) is determined by algebraicity of Γ(x, z⃗). Indeed, in (3), if Γ(x, h⃗) is infinite, it defines a large set, and clearly has an infinite number of realizations that do not satisfy the small formula ∃y⃗ ∈ H θ(x, y⃗, h⃗). Also in cases (2) and (4) we can absorb Γ(x, z⃗) in θ(x, y⃗, z⃗) or θ_{1}(x, y⃗, z⃗).

We will now reduce case (4) to case (2). Note that we can assume that θ_{1}(x, y⃗, z⃗) implies that y⃗ is a tuple of distinct elements and is disjoint from z⃗, and θ_{2}(x, y′, z⃗) implies the same about y′.

The idea of the proof is the following. If we assume that θ(a, k⃗, h⃗) holds for some k⃗ ∈ H(M), then to analyze ∃y′ ∈ H θ_{2}(a, y′, h⃗), we will look at the relationship between k⃗ and y′, namely, how much of an overlap do we have between y′ and k⃗. For each subtuple k⃗^{∗} of k⃗, we consider all the tuples y′ ∈ H(M) such that y′∩k⃗ = k⃗^{∗}. It will turn out that the existence of such y′ in H(M) with ⊧ θ_{2}(a, y′, h⃗) is (uniformly) L-definable in xy⃗z⃗. Then we take the disjunction over all the subtuples of y⃗.

Claim 1: Let h⃗ ∈ H(M), a ∉ H(M). Suppose k⃗ ∈ H(M) is such that ⊧ θ_{1}(a, k⃗, h⃗). Let k⃗′ be a subtuple of k⃗. Suppose b⃗ ∈ H(M) is such that b⃗ ∩ k⃗ = k⃗′ and ⊧ θ_{2}(a, b⃗, h⃗). Then a ∈ acl(k⃗′, h⃗).

Proof of Claim 1: Since ⊧ θ_{2}(a, b⃗, h⃗), we have a ∈ acl(b⃗, h⃗). On the other hand, a ∈ acl(k⃗, h⃗). Since b⃗h⃗ is independent from k⃗h⃗ over k⃗′h⃗, we have a ∈ acl(k⃗′, h⃗).

Claim 2: Suppose y^{∗} is a subtuple of y⃗. Then the formula

∃y′ ∈ H (y′ ∩ y⃗ = y^{∗} ∧ θ_{2}(x, y′, z⃗))

is equivalent to an L-formula Δ(x, y⃗, z⃗) modulo

H(z⃗) ∧ ¬H(x) ∧ H(y⃗) ∧ θ_{1}(x, y⃗, z⃗).

Proof of Claim 2: By Claim 1 and compactness, there exists an L-formula ψ(x, y^{∗}, z⃗) having a bounded finite number of realizations in x, such that

⊧ (H(z⃗) ∧ ¬H(x) ∧ H(y⃗) ∧ θ_{1}(x, y⃗, z⃗) ∧ y′ ∩ y⃗ = y^{∗} ∧ θ_{2}(x, y′, z⃗)) → ψ(x, y^{∗}, z⃗).

Let y^{∗∗} be such that y′ = y^{∗}y^{∗∗} (permute the variables if needed). Then modulo H(z⃗) ∧ ¬H(x) ∧ H(y⃗) ∧ θ_{1}(x, y⃗, z⃗),

∃y′ ∈ H (y′ ∩ y⃗ = y^{∗} ∧ θ_{2}(x, y′, z⃗))

is equivalent to

∃y^{∗∗} ∈ H (θ_{2}(x, y^{∗}y^{∗∗}, z⃗) ∧ ψ(x, y^{∗}, z⃗)).

The latter is equivalent, modulo H(z⃗) ∧ ¬H(x) ∧ H(y^{∗}) ∧ y^{∗∗} ∩ y^{∗}z⃗ = ∅ and the statement that y^{∗∗} is a tuple of distinct elements, to the existence of a tuple y^{∗∗}, acl-independent over xy⃗z⃗, such that θ_{2}(x, y^{∗}y^{∗∗}, z⃗) ∧ ψ(x, y^{∗}, z⃗) holds true. Indeed, suppose h⃗, k⃗^{∗} ∈ H(M), a ∉ H(M). Note that

θ_{2}(a, k⃗^{∗}y^{∗∗}, h⃗) ∧ ψ(a, k⃗^{∗}, h⃗)

implies that a ∈ acl(k⃗^{∗}, h⃗). Then there exists a tuple of distinct elements k⃗^{∗∗} ∈ H(M), disjoint from k⃗^{∗}h⃗, such that

⊧ θ_{2}(a, k⃗^{∗}k⃗^{∗∗}, h⃗) ∧ ψ(a, k⃗^{∗}, h⃗)

exactly when there exists a tuple k⃗^{∗∗} ∈ M, acl-independent over ak⃗^{∗}h⃗, such that

⊧ θ_{2}(a, k⃗^{∗}k⃗^{∗∗}, h⃗) ∧ ψ(a, k⃗^{∗}, h⃗).

This condition is L-definable in xy⃗z⃗, as needed, which proves Claim 2.

Next, rewrite

¬H(x) ∧ H(z⃗) ∧ ∃y⃗ ∈ H θ_{1}(x, y⃗, z⃗) ∧ ¬∃y′ ∈ H θ_{2}(x, y′, z⃗)

as

¬H(x) ∧ H(z⃗) ∧ ∃y⃗ ∈ H (θ_{1}(x, y⃗, z⃗) ∧ ¬∃y′ ∈ H θ_{2}(x, y′, z⃗)),

and note that it is equivalent to

¬H(x) ∧ H(z⃗) ∧ ∃y⃗ ∈ H (θ_{1}(x, y⃗, z⃗) ∧ ¬ ⋁_{y^{∗}⊂y⃗} ∃y′ ∈ H (y′ ∩ y⃗ = y^{∗} ∧ θ_{2}(x, y′, z⃗))).

By Claim 2, the disjunction above can be replaced with an L-formula in xy⃗z⃗, and therefore can be absorbed in θ_{1}(x, y⃗, z⃗). This reduces case (4) to case (2). Thus, we may assume that ϕ(x, z⃗) has form

¬H(x) ∧ H(z⃗) ∧ ∃y⃗ ∈ H θ(x, y⃗, z⃗),

where θ(x, y⃗, z⃗) is an L-formula having a bounded finite number of realizations in x.

Lemma 3.17. Suppose θ(x, y, z⃗) is an L-formula having a bounded finite number of realizations in x. Then for any h⃗ ∈ H(M), ∃y ∈ H θ(x, y, h⃗) is infinite if and only if M ⊧ ∃^{∞}x∃y θ(x, y, h⃗).

Proof. Left to right is clear. For the other direction, suppose M ⊧ ∃^{∞}x∃y θ(x, y, h⃗). Then we can find a sequence (a_{i} : i ∈ ω) of realizations of ∃y θ(x, y, h⃗), aclindependent over h⃗. For each a_{i} there exists b_{i} ∈ M such that M ⊧ θ(a_{i}, b_{i}, h⃗). We have a_{i} ∈ acl(b_{i}h⃗). Thus, b_{i} ∉ acl(h⃗), and acl(a_{i}h⃗) = acl(b_{i}h⃗). Then the sequence (b_{i} : i ∈ ω) is also acl-independent over h⃗, and thus we may assume that b_{i} ∈ H(M). Then a_{i} all realize ∃y ∈ H θ(x, y, h⃗), as needed.

□

Lemma 3.18. Suppose h⃗ ∈ H(M), θ(x, y⃗, z⃗) an L-formula having a bounded finite number of realizations in x, and implying that y⃗ = y_{1} … y_{n} is a tuple of distinct elements and y⃗ ∩ z⃗ = ∅. Then the formula ∃y⃗ ∈ H θ(x, y⃗, h⃗) is infinite if and only if there exists 1 ≤ i ≤ n and b_{1}, … b_{i−1}, b_{i+1}, …, b_{n} ∈ H(M) such that

∃y_{i} ∈ H θ(x, b_{1}, …, b_{i−1}, y_{i}, b_{i+1}, …, b_{n}, h⃗)

is infinite.

Proof. Right to left is clear. For the other direction, suppose ∃y⃗ ∈ H θ(x, y⃗, h⃗) is infinite. Then it has infinitely many realizations a ∉ acl(h⃗). For each such a there exists b⃗ ∈ H such that

⊧ θ(a, b⃗, h⃗).

Then a ∈ acl(b⃗, h⃗). Note that the tuple b⃗ acl-independent over h⃗. There is a nonempty minimal subtuple b⃗′ of b⃗ such that a ∈ acl(b⃗′, h⃗). Take any b_{i} contained in b⃗′. Then clearly a is interalgebraic (in terms of acl) with b_{i} over b_{1} … b_{i−1}b_{i+1} … b_{n}h⃗. Taking infinitely many acl-independent L-conjugates a′b′_{i} of ab_{i} over b_{1} … b_{i−1}b_{i+1} … b_{n}h⃗, with b_{i} ∈ H(M), we get infinitely many realizations of

∃y_{i} ∈ H θ(x, b_{1}, …, b_{i−1}, y_{i}, b_{i+1}, …, b_{n}, h⃗).
□

Proposition 3.19. Suppose ϕ(x, z⃗) is an L_{H}-formula implying H(z⃗). Then ∃^{∞}x ϕ(x, z⃗) is first order.

Proof. We may assume that ϕ(x, z⃗) has form

¬H(x) ∧ H(z⃗) ∧ ∃y⃗ ∈ H θ(x, y⃗, z⃗),

where θ(x, y⃗, z⃗) is an L-formula having a bounded finite number of realizations in x, and θ(x, y⃗, z⃗) implies that y⃗ = y_{1} … y_{n} is a tuple of distinct elements, disjoint form z⃗.

Then by Lemma 3.18, ∃^{∞}x ϕ(x, z⃗) is equivalent (modulo ¬H(x) ∧ H(a⃗)) to

⋁_{1≤i≤n} ∃y_{1} ∈ H … y_{i−1} ∈ H y_{i+1} ∈ H … y_{n} ∈ H ∃^{∞}x∃y_{i} ∈ H θ(x, y⃗, z⃗).

By Lemma 3.17, ∃^{∞}x∃y ∈ H θ(x, y⃗, z⃗) is a first order formula. □

We finish this section with a property of non-trivial geometric theories that we will use in the next sections.

Definition 3.20. Let T be a geometric theory, let M ⊧ T and let a⃗ = (a_{1}, …, a_{n−1}, a_{n}) ∈ M^{n} be such that dim(a⃗) = n − 1 but any n − 1 subset of {a_{1}, …, a_{n−1}, a_{n})} is independent. We call such a tuple an algebraic n-gon.

Proposition 3.21. Let T be a non-trivial geometric theory and let M ⊧ T be saturated. Then for every n there is m ≥ n and an algebraic m-gon.

Proof. Working over a finite independent tuple, if necessary, me may assume that T has an algebraic triangle, i.e. an algebraic 3-gon (triangle) abc. Let a′ ⊧ tp(a/b) be independent from ac over b. Note that then aca′ is an independent tuple. Let c′ be such that tp(a′c′/b) = tp(ac/b). Then aca′c′ is an algebraic 4-gon (quadrangle). Then take a′′ such that tp(a′′c′) = tp(ac) and a′′ is independent from abca′c′ over c′. Then aca′a′′ is an independent tuple. Let b′′ be such that tp(a′′b′′c′) = tp(abc). Then aca′a′′b′′ is an algebraic 5-gon. Continuing in this way, we can generate algebraic n-gons for an arbitrarily large n. □

4. Lovely pairs, iterated H-structures, and externally definable sets

In this section we will explore topics motivated by analogies between H-structures and lovely pairs. First, we take a closer look at the connections between the two constructions. Then we look at the iterated version of the H-structures (similar to “tuples” of structures and “double pairs”). Finally, we look at the expansion of H(M) with traces of externally definable sets.

4.1. Independent subsets and lovely pairs. In this subsection we study the connections between H-structures and lovely pairs. Let T be a geometric theory in a language L and let N ⪯ M ⊧ T. We say that the pair (M, N) is a lovely pair of models of T if

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

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

Note that the properties characterizing lovely pairs are very similar to the ones of H-structures, the role of the independent set H is played by the elementary substructure N. In this section we will only use the definition of lovely pairs. More information on lovely pairs of geometric structures can be found in [3].

Proposition 4.1. Let T be a geometric theory and let (M, H) be an H-structure. Let N = acl(H). Then (M, N) is a lovely pair of models of T.

Proof. Let T, (M, H) and N be as above.

Claim N ⪯ M.

We apply the Tarski-Vaught test. Let a⃗ ∈ N, let b ∈ M and assume that M ⊧ φ(b, a⃗). If b ∈ acl(a⃗) then b ∈ N and N ⊧ φ(b, a⃗). If b ∉ acl(a⃗) let p(x) = tp(b/a⃗). By the coheir property for H-structures there is b′ ∈ H such that tp(b′/a⃗) = tp(b/a⃗)

Now we check that (M, N) satisfies the coheir property. Let A ⊂ M be finite dimensional and let q ∈ S_{1}(A) be non-algebraic. By the coheir property for Hstructures, there is b ∈ H such that b ⊧ q. Since N = acl(H) we have b ∈ N.

Now we check that (M, N) satisfies the extension property. Let A ⊂ M be finite dimensional and let q ∈ S_{1}(A) be non-algebraic. By the extension property for Hstructures, there is b ∈ M such that b ⊧ q and b ∉ acl(A ∪ H). Since N = acl(H) then b ∉ acl(A ∪ N) as desired. □

Let P be a new predicate that does not appear in L and let L_{P} = L ∪ {P } be the old language extended with a new predicate symbol. If (M, N) is a lovely pair of models of T, we can study (M, N) as an L_{P} structure by interpreting P as N. In [3] it is shown that if (M, N) and (M′, N′) are lovely pairs of models of T, then T h((M, N)) = T h((M′, N′)) (seen as L_{P} structures). Note that Corollary 2.9 is the analogous result for H-structures. We denote by T_{P} this common theory in the language L_{P}.

It is shown in [3] that when T is geometric, weakly 1-based, and ω-categorical, then the associated theory T_{P} of lovely pairs is also ω-categorical. This is not the case for the associated theory T^{ind}:

Example 4.2. Let T be the theory of infinite dimensional vector spaces over a fixed finite field, say F_{2}. Note that T is strongly minimal, ω-categorical and 1-based. Let V ⊧ T be countable and let H = {v_{j} : j ∈ ω} be an enumeration of a basis. Let i < ω and let H_{i} = {v_{j} : j ∈ ω, j > i}. Then (V, H_{i}) ⊧ T^{ind} for every i and the models (V, H_{i}), (V, H_{j}) are not isomorphic for i < j. Thus the theory T^{ind} is not ω-categorical.

Now let H_{even} = {v_{2j} : j ∈ ω}, then as before (V, H_{even}) ⊧ T^{ind} and it is not isomorphic to any of the pairs (V, H_{i}). Also note that (V, H_{even}) is an H-structure, but for every i ∈ ω the pair (V, H_{i}) is NOT an H-structure.

If we take algebraic closures, then we see that for every i < ω, (V, acl(H_{i})) is not a model of T_{p}, since it does not satisfy the axiom corresponding to the extension property (see the third scheme of axioms in 2.10). On the other hand, (V, acl(H_{even})) is a model of T_{p} and it is the unique model up to isomorphism.

The previous example shows:

Remark 4.3. Let T be geometric and let (M, H) ⊧ T^{ind}. Then (M, acl(H)) may not be a model of T_{P}. The pair (M, acl(H)) will satisfy the scheme of axioms corresponding to the density property, but it may fail to satisfy the scheme of axioms corresponding to the extension property.

4.2. Iterating the construction: H-tuples. In this subsection we show how to iterate the process of expanding by H-structures. We will do the details for an expansion with two extra predicates but this procedure can be easily generalized to n tuples of predicates. As before, we start with T a geometric theory in a language L and we consider H_{1}, H_{2} two new predicate symbols. Let L_{2H} = L∪{H_{1}}∪{H_{2}}. Let T′ be the L_{H}-theory of all structures (M, H_{1}, H_{2}), where M ⊧ T and H_{1}(M)∪ H_{2}(M) is an L-algebraically independent subset of M and H_{1}(M) ∩ H_{2}(M) = ∅.

Definition 4.4. We say that (M, H_{1}(M), H_{2}(M)) is an H-triple if (1) (M, H(M)) ⊧ T′

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

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

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

As before, if (M, H_{1}(M), H_{2}(M)), (N, H_{1}(N), H_{2}(N)) are H-triples, then T h(M, H_{1}(M), H_{2}(M)) = T h(N, H_{1}(N), H_{2}(N)), we denote the common theory by T^{tri}.

We will now follow the approach from Fornasiero [16] and consider an H-structure associated to the small closure in (M, H_{1}). Fornasiero [16] considers lovely pairs in a general framework of a closure operator associated to an existential matroid. In this paper we will only consider the special case of the small closure.

Let T_{2} be the L_{H}-theory of all structures (M, H_{1}, H_{2}), where (M, H_{1}(M)) is an H-structure and H_{2}(M) is an algebraically independent subset of M over H_{1}(M) = scl(∅).

Definition 4.5. We say that (M, H(M), H_{2}(M)) is an scl-structure if (1) (M, H_{1}(M), H_{2}(M)) ⊧ T_{2}

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

(3) (Extension property) If A ⊂ M is finite dimensional and q ∈ S_{1}^{ind}(A) is non-small, there is a ∈ M, a ⊧ q and a ∉ scl(A ∪ H_{2}(M)).

Now we will show that considering H-triples is equivalent as considering sclstructures

Proposition 4.6. Let T be a geometric structure, 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 scl-structure if and only if (M, H_{1}(M), H_{2}(M)) is an H-triple.

Proof. Assume first that (M, H_{1}(M), H_{2}(M)) is a scl-structure. Then the pair (M, H_{1}(M)) is an H-structure and thus (M, H_{1}(M), H_{2}(M)) satisfies the density/coheir axiom for H_{1}. Now let A ⊂ M be finite dimensional and let q ∈ S_{1}(A) be non-algebraic. Let q̂ ∈ S_{1}^{ind}(A) be an extension of q that contains no small formula with parameters in A. Then by the Density/coheir property for scl it follows that there is a ∈ H_{2}(M) such that a ⊧ q̂. In particular, a ⊧ q and a ∉ acl(A ∪ H_{1}(M)) and it follows the density/coheir property for H_{2}/H_{1}. Finally, since the same q̂ is not small, there is c ∈ M, c ⊧ q̂ and c ∉ scl(A∪H_{2}(M)) = acl(A∪H_{1}(M)∪H_{2}(M)). Thus the extension property H-triples holds.

Now assume that (M, H_{1}(M), H_{2}(M)) is an H-triple. By the density property for H_{1} and the extension property it follows that (M, H_{1}(M)) is an H-structure and that (M, H_{1}(M), H_{2}(M)) ⊧ T_{2}. Now let A ⊂ M be finite dimensional and let q̂ ∈ S_{1}^{ind}(A) be non-small. We may enlarge A and assume that A = A ∪ HB(A), so that A is H_{1}-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/coheir property for H_{2}/H_{1} there is a ∈ H_{2}(M) such that a ⊧ q, a ∉ acl(H_{1}A) and thus a ⊧ q̂. Finally the extension property follows from the extension property for H-triples. □

We will now show that the class of scl-structures is "first order", that is, that there is a set of axioms whose |T |^{+}-saturated models are the scl-structures. For this we consider H-triples.

Proposition 4.7. Assume T eliminates ∃^{∞}. Then the theory T^{tri} is axiomatized by:

(1) T′.

(2) For all L-formulas φ(x, y⃗)

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

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

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

Furthermore, if (M, H_{1}(M), H_{2}(M)) ⊧ T^{tri} is |T |^{+}-saturated, then (M, H_{1}(M), H_{2}(M)) is an H-triple.

The proof is the same one as for H-structures and we leave it for the reader. Note that since T eliminates the quantifier ∃^{∞}, then T^{ind} eliminates the quan-

tifier ∃^{large}. This is the main reason why the theory of H-triples is axiomatizable.

4.3. Elimination of ∃y ∈ H. In this subsection we will look at elimination of quantifiers in the structure obtained by naming all the externally definable relations on H(M) in an H-structure (note we have already shown any L_{H}-definable relation on H(M) is L-definable). This problem is known as elimination of ∃y ∈ P, where P is a unary predicate symbol. Such an elimination is known to hold in the case when P is an elementary submodel of a model of a stable theory (by definability of types), or an elementary submodel of a sufficiently saturated model of an NIP theory (established by Shelah [25]). The case when P is the smaller model in a lovely pair of models of a simple theory has been considered in [22], where the elimination of ∃y ∈ P has been shown to be equivalent to the property called weak lowness. In the case of lovely pairs of geometric structures, the elimination of ∃y ∈ P was shown in [3]. Here we will show that any H-structure of a geometric theory eliminates ∃y ∈ H.

We will follow the Definition 1.1 from [22].

Definition 4.8. Let T be a first order theory in a language L, and let (M, H) be an expansion of M with a new unary predicate. We say that (M, H) eliminates the quantifier ∃y ∈ H, if for any L-formula ϕ(x⃗, y, z⃗) and a⃗ ∈ M, there exists an L-formula ψ(x⃗, w⃗) and b⃗ ∈ M, such that for any c⃗ ∈ H(M),

(M, H) ⊧ ∃y ∈ H ϕ(c⃗, y, a⃗) ⟺ M ⊧ ψ(c⃗, b⃗).

If the choice of ψ(x⃗, w⃗) does not depend on the choice of a⃗ ∈ M (i.e. depends only of the formula ϕ(x⃗, y, y⃗)), we say that the elimination is uniform.

Proposition 4.9. Let T be a geometric theory, and let (M, H) be an H-structure of T. Then (M, H) eliminates the quantifier ∃y ∈ H.

Proof. Let ϕ(x⃗, y, z⃗) be an L-formula, and let a⃗ ∈ M. Let c⃗ ∈ H(M).

If the formula ϕ(c⃗, y, a⃗) is non-algebraic, then clearly , it is realized in H(M). Now, suppose the formula ϕ(c⃗, y, a⃗) is algebraic and is realized by e ∈ H(M),

where e is not a part of the tuple c⃗. Let d⃗ = HB(a⃗), viewed as a tuple. If e is not a part of d⃗, then e is not algebraic over c⃗d⃗, and thus c⃗e |̸ ⌣d⃗ a⃗. This contradicts the definition of HB(a⃗). Thus e is a part of d⃗.

Thus for any c ∈ H(M), we have (M, H) ⊧ ∃y ∈ H ϕ(c, y, a⃗) if and only if either ϕ(c⃗, y, a⃗) is non-algebraic, or M ⊧ ϕ(c⃗, e, a⃗) where e is a part of c⃗HB(a⃗). Both conditions on c⃗ are L-definable over the elements of a⃗HB(a⃗).

□

Question 4.10. Is this elimination uniform? Note that the L-definition of ∃y ∈ H ϕ(c⃗, y, a⃗) involves HB(a⃗), and this tuple could be arbitrarily long.

5. Strongly minimal, SU-rank 1 and thorn rank 1 cases

In this section we study four special cases of geometric theories, when the underlying theory T is strongly minimal, SU-rank 1, thorn rank 1 or strongly dependent of finite dp-rank. In these cases, we show that the theory T^{ind} becomes ω-stable, supersimple of SU-rank less than or equal to ω, super-rosy of thorn-rank less than or equal to ω or strongly dependent respectively. We also characterize in each of these cases when T is trivial in terms of the rank of T^{ind}.

5.1. Strongly minimal case. Let T be a strongly minimal theory (in particular it is a geometric theory). In this section we prove that T^{ind} is ω-stable and has Morley rank less than or equal to ω.

Proposition 5.1. Let T be strongly minimal. Then T^{ind} is ω-stable.

Proof. Suppose (M, H(M)) is a sufficiently saturated model of T^{ind}, and A ⊂ M is a countable set. We may assume that A is H-independent. We will count the number of types of the form tp_{H}(b/A) where b ∈ M.

Case 1: b ∈ H(M). Then bA is H-independent, and tp_{H}(b/A) is determined by tp(b/A) and the fact that b ∈ H(M). By strong minimality of T, there are at most countably many such types.

Case 2: b ∈ scl(A). Then there are h_{1}, …, h_{l} ∈ H(M) such that b ∈ acl(h_{1} … h_{l}A). By Case 1, there are at most countably many types of the form tp_{H}(h_{1}, …, h_{l}/A) where h_{i} ∈ H(M), and thus at most countably many types of the form tp_{H}(b/A) for b as above.

Case 3: b ∉ scl(A). Note that bA is H-independent, and thus tp_{H}(b/A) is determined by tp_{H}(b/A) and the fact that b ∉ scl(A). There is a unique such type. □

In the setting of lovely pairs of strongly minimal theories, there is a strong connection between the underlying geometry of the theory T and the Morley rank of the associated theory of lovely pairs T_{P}. Buechler [9] showed that T is trivial iff MR(T_{P}) = 1, T is locally modular non-trivial iff MR(T_{P}) = 2 and T is not locally modular iff MR(T_{P}) = ω. He used this result to prove that pseudolinearity implies linearity. We will now show that the Morley rank of the theory T^{ind} “detects” trivial theories, in the sense that T is trivial iff MR(T^{ind}) ≤ 2 and non-trivial iff MR(T^{ind}) = ω.

Proposition 5.2. Let T be strongly minimal and trivial (i.e acl(A) = ⋃_{a∈A} acl(a)). Let (M, H) ⊧ T^{ind}. Then T^{ind} has Morley rank 1 iff for all a ∈ M, a ∉ acl(∅), acl(a)\ acl(∅) is finite. Otherwise, T has Morley rank 2.

Proof. Let (M, H) be a sufficiently saturated model of T^{ind}. Note that because of triviality, acl_{H} = acl in (M, H).

Let b ∈ H(M). Note for any (small) A ⊂ M, there is a unique non-algebraic L_{H}-type of an element of H(M) over A. Thus MR(b) = 1. This shows that MR(H(x)) = 1.

Now, let b ∉ H(M). Let A be a small subset of M. If b ∈ scl(A), by triviality of T, either b ∈ acl(A), in which case, MR(b/A) = 0, or b ∈ acl(h)\ acl(A) for some h ∈ H(M). Note also that h ∈ acl(b). Then MR(b/A) = MR(h/A) = MR(h) = 1. This shows that Morley rank of any small definable set in (M, H) is ≤ 1 (=1 if the set is infinite).

Note that any two large definable sets in (M, H) have a large intersection, so there is a unique large type. It follows that T^{ind} has Morley rank ≤ 2.

Suppose acl(a)\ acl(∅) is finite for all non-algebraic a ∈ M, say of size n. Let θ(x) be the first order formula expressing “x ∈ acl(h)\ acl(∅) for some h ∈ H(M)”. Then θ(x) has n non-algebraic extensions over any small A ⊂ M. Since there is a unique large type over A, there are only finitely many non-algebraic types over A. Thus, in this case T^{ind} has Morley rank 1.

Suppose acl(a)\ acl(∅) is infinite for all non-algebraic a ∈ M. Then we can find L-formulas ϕ_{n}(x, y), n ∈ ω, such that for a ∈ M\ acl(∅), we have ϕ_{n}(M, a) ⊂ acl(a)\ acl(∅), and ϕ_{n}(M, a) are finite, disjoint and non-empty. Let ψ_{n}(x) = ∃y ∈ H ϕ_{n}(x, y). Then ψ_{n}(M) are infinite and small. From the disjointness of ϕ_{n}(M, a) for a fixed a and independence of H(M) it follows that ψ_{n}(M) are disjoint. Thus, in this case, T^{ind} has Morley rank 2.

□

Proposition 5.3. Let T be strongly minimal and non-trivial. Then T^{ind} has Morley rank ω.

Proof. Suppose (M, H(M)) is sufficiently saturated model of T^{ind}, and A ⊂ M is a countable set. We may assume that A is H-independent. Let b ∈ M.

Case 1: b ∈ H(M). Then bA is H-independent, and tp_{H}(b/A) is determined by tp(b/A) and the fact that b ∈ H(M). In this case MR(b/A) ≤ 1.

Case 2: b ∈ scl(A). Then there are h_{1}, …, h_{l} ∈ H(M) such that b ∈ acl(h_{1} … h_{l}A). We may assume that l is minimal. Then b is L_{H}-interalgebraic with h_{1} … h_{l} over A. Thus MR(b/B) = MR(h_{1} … h_{l}/A) = l. Since T is not trivial, by Proposition 3.21 for every n there exists an algebraic n-gon a_{1}, …, a_{n−1}, a_{n}, and we can assume that a_{1}, …, a_{n−1} ∈ H(M) (and thus a_{n} ∉ H). We may also assume that a_{1} … a_{n} is independent from A over ∅. Thus for any b ∈ scl(A), MR(b/A) < ω but can have arbitrarily large finite values.

Case 3: b ∉ scl(A). As noted in the proof of Proposition 5.1, there a unique such 1-type over A. Then MR(b/A) ≤ ω. Since T is not trivial, for every n there exists an algebraic n+2-gon a_{1}, …, a_{n+2}, where a_{n+2} = b, a_{n+1} ∉ H(M), a_{1}, …, a_{n} ∈ H(M) and a_{1} … a_{n+2} is independent from A over ∅. Then tp_{H}(b/a_{1} … a_{n+1}A) has Morley rank n. Therefore MR(b/A) ≥ ω. Thus MR(b/A) = ω.

□

Next we will take a look at the geometric properties of T^{ind}. It is well-known that in case of lovely pairs (or belles paires, in the stable case), if T is one-based, then so is the pair theory T_{P}. This is no longer the case for T^{ind}, as the following example illustrates.

Example 5.4. Let (V, +, 0, H) be a vector space over ℚ, where H(V) = {v_{i} : i ∈ ω} consists of linearly independent vectors over ℚ. Furthermore assume that V ≠ span({v_{i} : i ∈ ω}). Then (V, +, 0, {v_{i} : i ∈ ω}) ⊧ T^{ind} where T is the theory of vector spaces over ℚ. Let u ∈ V \ span({v_{i} : i ∈ ω}). Note that u being generic is H-independent and that acl_{H}(u) = span(u). Let t = u + v_{1}.

Claim T^{ind} is not 1-based.

Note that t is small over u, t is interdefinable with v_{1} over u and that MR(tp(t/u)) = 1. Let t′ = u+v_{2}, then tp_{H}(v_{1}/u) = tp_{H}(v_{2}/u) (since they are H-independent) and thus tp_{H}(t/u) = tp_{H}(t′/u). Note that t | ⌣_{u}^{ind} t′. Note also that t − t′ = v_{1} − v_{2} so t is interdefinable with {v_{1}, v_{2}} over t′. Thus MR(tp_{H}(t/t′)) = 2 and thus t |̸ ⌣_{t′}^{ind} u. Hence T^{ind} is not 1-based.

Carmona showed in [10] that when T is linear, T^{ind} is CM-trivial.

5.2. SU-rank one theories. Let T be an SU-rank one theory.

Theorem 5.5. 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 be small sets 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 "geometric conditions" for the type of a⃗ over C and D that guarantee that tp_{H}(a⃗/D) does not divide over C.

We may write a⃗ = (a⃗_{1}, a⃗_{2}, a⃗_{3}) ∈ M so that a⃗_{1} is an independent tuple over CH, a⃗_{2} is an independent tuple over Ca⃗_{1}, a⃗_{2} ∈ acl(HCa⃗_{1}) and a⃗_{3} ∈ acl(Ca⃗_{1}a⃗_{2}). Assume that a⃗_{1} is an independent tuple over DH and that HB(a⃗/D) = HB(a⃗/C).

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 independent 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 independent 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, a⃗_{1}D_{i} is also H-independent for any i ∈ ω. So by Lemma 2.8 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 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}. 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 2.8 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 and since a⃗ ∈ acl(a⃗_{1}, h⃗C) we get that tp(a⃗/D) does not divide over C.

Since for any D and a⃗ we can always choose a countable set C with the properties described above, T is simple.

Given any D = acl_{H}(D) and a tuple a⃗ ∈ M, write a⃗ = (a⃗_{1}, a⃗_{2}, a⃗_{3}) ∈ M so that a⃗_{1} is an independent tuple over DH, a⃗_{2} is an independent tuple over Da⃗_{1}, a⃗_{2} ∈ acl(HDa⃗_{1}) and a⃗_{3} ∈ acl(Da⃗_{1}a⃗_{2}). We can always choose a finite B ⊂ D such that for C = acl_{H}(B) we have HB(a⃗/C) = HB(a⃗/D) and a⃗_{3} ∈ acl(C, a⃗_{1}, a⃗_{2}). Then tp(a⃗/D) does not divide over B and T is supersimple. □

Proposition 5.6. Let (M, H) ⊧ T^{ind} be saturated, let C ⊂ D ⊂ M be small and 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 ∈ scl(D) \ scl(C) or HB(a/C) ≠ HB(a/D).

Proof. In the proof of Theorem 5.5 we showed that if a ∈ C or if HB(a/C) = HB(a/D) 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 became algebraic over D and tp(a/D) forks over C.

Case 2: Assume that a ∈ scl(D) \ scl(C). Let d⃗ ∈ D and let c⃗ ∈ C be such that a ∈ acl(c⃗d⃗H). We can choose d⃗ independent over HC. Let h⃗ ∈ H be such that a ∈ acl(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 2.8 {d⃗_{i} : i ∈ ω} is an L_{H}-indiscernible sequence 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′ ∈ acl(d⃗_{i}, c⃗, h⃗_{i}) for every i. But a′ ∉ acl(CH), so d⃗_{0} |̸ ⌣_{CH} d⃗_{1}, a contradiction.

Case 3: Assume that HB(a/D) ≠ HB(a/C). 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} ∈ D is an independent tuple over C.

Let p(x, y⃗) = tp_{H}(a, h⃗_{E}/C). Let {h⃗_{E}^{i} : i ∈ ω} be an 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 2.8, the sequence {h⃗_{E}^{i} : i ∈ ω} is 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 h_{E}^{i} are independent, we get that the H-basis of a′ over C is not unique, a contradiction. □

Corollary 5.7. Let (M, H) ⊧ T^{ind} be saturated, let C ⊂ M be small and such that C = acl_{H}(C) and let a ∈ H(M). Then SU(a/C) ≤ 1.

Proof. Clearly SU(a/C) = 0 iff a ∈ C. If a ∉ C, then HB(a/C) = {a} so if it forks over some superset D of C by Proposition 5.6 we must have that HB(a/ acl_{H}(D)) = ∅ and that means that a ∈ acl_{H}(D). □

Corollary 5.8. Let T be non-trivial and let (M, H) ⊧ T^{ind} be saturated, let C ⊂ M be small such that C = acl_{H}(C) and let a ∈ M. Then

(1) a ∈ C iff SU(tp_{H}(a/C)) = 0

(2) a ∈ scl(C) iff SU(tp_{H}(a/C)) < ω and SU(tp_{H}(a/C)) = |HB(a/C)|.
(3) a ∉ scl(C) iff SU(tp_{H}(a/C)) = ω.

Proof. If a ∈ scl(C), then a is interalgebraic with HB(a/C) over C. By Corollary 5.7, SU(tp_{H}(HB(a/C)/C)) = |HB(a/C)| and thus SU(tp_{H}(a/C)) = |HB(a/C)|. Since T is not trivial, there are algebraic n-gons for n large enough and thus we can get arbitrarily large values for SU(a/C). If a ∉ scl(C), then by Proposition 5.6 and the existence of algebraic n-gons for n large enough shows that SU(tp(a/C)) = ω. The other statements are clear. □

Corollary 5.9. Let T be trivial and let (M, H) ⊧ T^{ind} be saturated, let C ⊂ M be small such that C = acl_{H}(C) and let a ∈ M. Then

(1) a ∈ C iff SU(a/C) = 0

(2) If a ∈ scl(C) \ C then SU(tp_{H}(a/C)) = 1.
(3) If a ∉ scl(C) then SU(tp_{H}(a/C)) = 1.

Proof. The first statement is clear. If a ∈ scl(C) \ C then by triviality of T there is a single h ∈ H such that a ∈ acl(h) and by Corollary 5.7 SU(a/C) = SU(h/C) = 1. If a ∉ scl(C) and D is a superset of C such that tp_{H}(a/D) forks over C, then by Proposition 5.6 and triviality we must have that a ∈ acl_{H}(D). □

Remark 5.10. Note that in the case when T is strongly minimal, the behavior of Morley rank maybe different form that of the SU-rank (U-rank). Namely, as we showed in Proposition 5.3, for a trivial strongly minimal theory T where acl(a)\ acl(∅) is infinite for a ∉ acl(∅), the theory T^{ind} has Morley rank 2 (while its U-rank is 1).

Corollary 5.11. (Coordinatization) Let (M, H) ⊧ T^{ind} be κ-saturated, let C ⊂ M be such that C = acl_{H}(C), |C| < κ and let a⃗ ∈ M^{n}. Write a⃗ = a⃗_{1}a⃗_{2}a⃗_{3} where a⃗_{1} is algebraically independent over HC, a⃗_{2} is algebraically independent over Ca⃗_{1} and a⃗_{2} ∈ scl(Ca⃗_{1}) and a⃗_{3} ∈ acl(a⃗_{1}a⃗_{2}C). Then for every e ∈ a⃗_{1}, tp_{H}(e/C) is regular, a⃗_{2} is interalgebraic with HB(a⃗/C) over Ca⃗_{1} and for each h ∈ HB(a⃗/C), tp_{H}(h/Ca⃗_{1}) is regular. So there is an explicit coordinatization in T^{ind} in terms of types of real elements.

Our next goal is to describe canonical bases in T^{ind}, for any SU-rank 1 theory T. Note that since T^{ind} is supersimple, it eliminates hyperimaginaries, so canonical bases exist as imaginaries, both in T and T^{ind}. Let Cb(a⃗/B) denote Cb(stp(a⃗/B)), and Cb_{H}(a⃗/B) denote Cb(stp_{H}(a⃗/B)).

Lemma 5.12. 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}a⃗_{3}, where a⃗_{1} acl-independent over B ∪ H(M), a⃗_{2} ∈ acl(H(M)Ba⃗_{1})\ acl(Ba⃗_{1}), a⃗_{3} ∈ acl(Ba⃗_{1}a⃗_{2}). Note that a⃗_{2} ∈ acl(a⃗_{1}Bh), so a⃗h | ⌣_{e} B implies that a⃗_{2} ∈ acl(a⃗_{1}eh) and thus HB(a⃗/B) = HB(a⃗/e). Since a⃗h | ⌣_{e} B, we also have a⃗_{3} ∈ acl(ea⃗_{1}a⃗_{2}). 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. □

Proposition 5.13. 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). Now 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). Note that 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}), and hence 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. In particular HB(a⃗_{n}/B) = HB(a⃗_{n}/a⃗_{<N}) and thus h_{n} ∈ dcl_{H}(a⃗_{i} : i < ω) for every n. We then get e ∈ acl_{H}^{eq}({a⃗_{i} : i < ω}). Now, since {a⃗_{i} : i < ω} is a Morley sequence in tp_{H}(a⃗/ acl_{H}^{eq}(B)), we have

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

and thus also

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

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

□

Remark 5.14. Note that Proposition 5.13 implies geometric elimination of imaginaries in T^{ind} down to imaginaries of T, when T is a supersimple SU-rank 1 structure.

Question 5.15. If T is a geometric theory, does T^{ind} have (geometric) elimination of imaginaries down to imaginaries of T?

Example 5.16. Let (V, +, 0, H) = (V, +, 0, {v_{i} : i ∈ ω}) be the structure from Example 5.4. We will give another proof of non-1-basedness of T^{ind} = T h(V, +, 0, H), using Lemma 5.13. Take t, u, v_{1} as in Example 5.4, so u, t are generic and t = u+v_{1}.

First note that HB(t/u) = {v_{1}}. Now, by Lemma 5.13, Cb_{H}(t/u) is interalgebraic (in (T^{ind})^{eq}) with Cb(tv_{1}/u). Note that Cb(tv_{1}/u) is interdefinable with u. On the other hand, u ∉ acl_{H}(t) = acl(t) = span(t). Thus, Cb_{H}(t/u) is not algebraic over t, and therefore T^{ind} is not 1-based.

Let t′ = u + v_{2}, then t, t′ are the first two elements in a Morley sequence in tp_{H}(t/u). Note that t − t′ = v_{1} − v_{2}, so v_{1}, v_{2} ∈ acl_{H}(t, t′) and thus u ∈ acl_{H}(t, t′). We will show below that when T is 1-based, T^{ind} is 2-based: we need two elements in a Morley sequence in T^{ind} to recover the canonical base.

Carmona [10] proved that when T is linear SU-rank one theory, T^{ind} is CMtrivial. We will show below that if T is 1-based, then T^{ind} is 2-based:

Proposition 5.17. Let T be a simple theory of SU-rank 1 and assume that T is 1-based. Let (M, H) ⊧ T^{ind} be saturated, let A ⊂ M be small and let p ∈ S_{k}^{H}(A). Then whenever {a^{i} : i ∈ ω} is an L_{H}-Morley sequence in p over A we have that a^{2} | ⌣_{a^{0}a^{1}}^{ind} A.

Proof. Let {a^{i} : i ∈ ω} be an L_{H}-Morley sequence in p over A. We can write a^{i} = a_{1}^{i}a_{2}^{i}a_{3}^{i} where a_{1}^{i} is an independent tuple over AH(M), a_{2}^{i} is an independent tuple over Aa_{1}^{i} and a_{2}^{i} ∈ scl(Aa_{1}^{i}) and a_{3}^{i} ∈ acl(Aa_{1}^{i}a_{2}^{i}). Let h⃗^{i} = HB(a^{i}/A) seen as a tuple. We may choose the ordering of h⃗^{i} so that {a^{i}h⃗^{i} : i ∈ ω} is an L_{H}-Morley sequence in tp(a^{0}h⃗^{0}/A). Note that both {a^{i} : i ∈ ω} and {a^{i}h⃗^{i} : i ∈ ω} are L-Morley sequences over A. Indeed, since a^{i}h⃗^{i} | ⌣_{A}^{ind} a^{0}h⃗^{0} … a^{i−1}h⃗^{i−1}, we have that a_{1}^{i}h⃗^{i} is an acl-independent tuple over Aa^{0}h⃗^{0} … a^{i−1}h⃗^{i−1}. Since a^{i}h⃗^{i} ∈ acl(Aa_{1}^{i}h⃗^{i}), it follows that a^{i}h⃗^{i} | ⌣_{A} a^{0}h⃗^{0} … a^{i−1}h⃗^{i−1}, and thus, {a^{i}h⃗^{i} : i ∈ ω} is an L-Morley sequence over A. Then clearly {a^{i} : i ∈ ω} is also L-Morley over A.

Since T is 1-based and {a^{i}h⃗^{i} : i ∈ ω} is an L-Morley sequence, we have that a^{1}h⃗^{1} | ⌣_{_{a^{0}}h⃗^{0}} A, in particular a_{2}^{1} ∈ acl(a_{1}^{1}h⃗^{1}a_{1}^{0}a_{2}^{0}h⃗^{0}). Since T is 1-based and {a^{i} : i ∈ ω} is a L-Morley sequence, we also get a_{3}^{1} ∈ acl(a_{1}^{1}a_{2}^{1}a_{1}^{0}a_{2}^{0}a_{3}^{0}). Thus HB(a^{1}/a^{0}) ⊂ {h⃗^{0}h⃗^{1}}. Similarly, HB(a^{2}/a^{0}) ⊂ {h⃗^{0}h⃗^{2}} and HB(a^{2}/a^{1}) ⊂ {h⃗^{1}h⃗^{2}}.

Note that h⃗_{2} = HB(a^{2}/A) = HB(a^{2}/Aa^{0}a^{1}) ⊂ HB(a^{2}/a^{0}a^{1}). We want to show that a^{2} | ⌣_{a^{0}a^{1}}^{ind} A, so it suffices to show that h⃗_{2} = HB(a^{2}/a^{0}a^{1}) and to show this it suffices to prove that HB(a^{2}/a^{0}a^{1}) ⊂ h⃗_{2}. Note that HB(a^{2}/a^{0}a^{1}) ⊂ HB(a^{2}/a^{0}) ∩ HB(a^{2}/a^{1}) = {h⃗^{0}h⃗^{2}} ∩ {h⃗^{1}h⃗^{2}}. Since h⃗^{i} = HB(a^{i}/A) are disjoint from acl(A), and {a^{i}h⃗^{i} : i ∈ ω} is an L-Morley sequence, the tuples h⃗^{i} are disjoint. Thus, {h⃗^{0}h⃗^{2}} ∩ {h⃗^{1}h⃗^{2}} = h⃗^{2}. Hence HB(a^{2}/a^{0}a^{1}) ⊂ h⃗_{2}, as needed.

□

Remark 5.18. Note that a 2-based SU-rank 1 theory is 4-pseudolinear, meaning that canonical bases of plane curves have SU-rank ≤ 4. Indeed, suppose SU(ab/A) = 1, and let {a_{i}b_{i} : i ∈ ω} be a Morley sequence in tp(ab/A). Then 2-basedness implies Cb(ab/A) ⊂ acl^{eq}(a_{0}b_{0}, a_{1}b_{1}), and therefore SU(Cb(ab/A)) ≤ 4. In [9], it is shown that pseudolinear strongly minimal theories are locally modular (1-based). In [26], it is shown that a pseudolinear ω-categorical SU-rank 1 theory is 1-based. In [19], Hrushovski gives an example of an ω-categorical SU-rank 1 theory which is not 1based. By the above, this theory is not 2-based (or even finitely based), but it is known to be CM-trivial. Thus, CM-triviality does not imply 2-basedness.

5.3. Thorn rank one. Assume that T is a thorn rank one theory. The goal of this subsection is to show that T^{ind} is super-rosy of thorn-rank ≤ ω. Our proof follows the proof of super-rosyness given by Boxall for lovely pairs of thorn rank one theories.

Theorem 5.19. The theory T^{ind} is super-rosy of thorn rank less that or equal to ω.

Proof. Let (M, H) ⊧ T^{ind} be highly saturated. In order to show that T^{ind} is super-rosy, we need to understand two steps:

Claim 1 Let φ(x, c⃗) define an infinite subset of H(M). Then φ(x, c⃗) does not thorn divide over ∅.

The proof is word by word the same one as the one presented in [6].

Claim 2 Let θ(x, a⃗) be an L_{H} formula defining a large subset of M. Then θ(x, a⃗) does not thorn divide over ∅.

The proof is again very similar to the one presented by Boxall in [6] for lovely pairs of thorn rank one theories, but we will do some small changes to see how the arguments adapt to the new setting.

Suppose θ(x, a⃗) thorn divides. Let â be the canonical parameter of θ(x, a⃗), we will also write the definable set as θ(x, â). Let D be a finite set such that â ∉ acl_{H}^{eq}(D) and such that {θ(x, â′) : â′ ⊧ tp(â/D)} is k -inconsistent. We may assume that D ⊂ M, that is, it contains only real elements. By noticing that HB(D) ∈ acl_{H}(D) and exchanging D for D ∪ HB(D) we may also assume that D is H-independent.

Let b ∈ θ(x, â), since the family {θ(x, â′) : â′ ⊧ tp(â/D)} is k -inconsistent, there are at most k − 1 conjugates of â over bD, so â ∈ acl_{H}(bD). Since θ(x, â) defines an infinite large set, we may assume that b ∉ scl_{H}(âD). Let φ(ŷ, x) be an algebraic formula in the variable ŷ such that (M, H) ⊧ φ(â, b). Let â^{∗} be the canonical parameter of φ(â, x). Note that â^{∗} ∈ dcl(â).

Claim 3 â ∈ acl(â^{∗}D).

Let n be the multiplicity of φ(ŷ, x) (in the variable ŷ). Let â_{1},…, â_{n+1} be realizations of tp(â/â^{∗}D). Then for any b′ with φ(â^{∗}, b′), we also have φ(â_{1}, b′), …, φ(â_{n+1}, b′) and thus there are i < j ≤ n + 1 such that â_{i} = â_{j}.

Thus, â, â^{∗} be interalgebraic over D. By Proposition 3.12 there is an L-definable set ψ(x, c⃗), where c⃗ is a real tuple, such that ψ(x, c⃗)△φ(â^{∗}, x) is small, where △ is a boolean connective for the symmetric difference. Note that we can choose c⃗ to be a real base of â^{∗}.

Let E(u⃗, v⃗) be the equivalence relation ψ(x, u⃗)△ψ(x, v⃗) is finite. Since T eliminates ∃^{∞}, E(z⃗, w⃗) is a definable equivalence relation. Let e = c⃗/E.

Let ψ(x, c′) be such that ψ(x, c′)△ψ(x, c⃗) is small. It is were infinite, since ψ(x, c′)△ψ(x, c⃗) is an L definable set it would be large. Thus if ψ(x, c′)△ψ(x, c⃗) is small, then ψ(x, c′)△ψ(x, c⃗) is finite and E(c⃗, c′). Thus e = c⃗/E ∈ acl_{H}(âD).

Claim 4 â ∈ acl(eD).

Recall that n is the multiplicity of φ(ŷ, x) (in the variable ŷ). Let â_{1},…, â_{n+1} be realizations of tp(â/eD). Then there are c_{1}, …, c_{n+1} such that ψ(x, c_{i})△φ(â_{i}, x) is small for i ≤ n + 1. Since e = c_{i}/E for i ≤ n + 1, we have that ψ(x, c_{i})△ψ(x, c_{i}) is finite for i ≤ n + 1. Let b′′ ∈ ⋀_{i≤n+1} ψ(x, c_{i}) ∧ ⋀_{i≤n+1} φ(â_{i}, x). Then we have φ(â_{1}, b′′), …, φ(â_{n+1}, b′′) and thus there are i < j ≤ n + 1 such that â_{i} = â_{j}.

Thus e and â are interalgebraic over D. Note that e ∈ acl_{H}(bD) \ acl_{H}(D). Since b ∉ scl(D) the set bD is H-independent and thus e ∈ acl(bD) \ acl(D), but b ∉ acl(eD), a contradiction since T has thorn rank one. □

As with the supersimple case, when T is trivial, the thorn rank of T^{ind} is one and when T is not trivial, the thorn rank of T^{ind} is ω. The proof follows easily from the previous theorem and we leave the details to the reader.

Question 5.20. In [4] the authors developed a theory of weakly one-based geometric theories. A generalization of this notion appears in [7] in the setting of structures with a robust independence notion (for example rosy theories), where it is proved that when T is rosy of thorn rank one, weakly one-basedness coincides with linearity.

Find a reasonable notion of weak 2-basedness in the setting of rosy theories and explore its properties, in particular does Proposition 5.17 hold in this setting?

5.4. NIP theories. We finish this section by addressing the question of preservation of NIP.

Proposition 5.21. Let T be a geometric theory, (M, H) a sufficiently saturated H-structure of T, and suppose T has NIP. Then T h(M, H) also has NIP.

Proof. We apply the criterion from [12, Thm 2.4]. We begin by showing that every formula ϕ(x⃗, y⃗) has NIP over H(x⃗). Assume otherwise, so there is an L_{H}-formula ϕ(x⃗, y⃗), I = (b⃗_{i} : i ∈ ω) an indiscernible sequence of elements in H(M) and a⃗ ∈ M such that ϕ(b⃗_{i}, a⃗) holds iff i is even. Then by Proposition 3.5 we have that there is an L-formula ψ(x⃗, z⃗) and an element d⃗ such that ψ(x⃗, d⃗) ∧ H(x⃗) holds if and only if ϕ(x⃗, a⃗) ∧ H(x⃗) holds. Thus the L-formula ψ(x⃗, y⃗) has the IP, a contradiction. By Proposition 3.2 every formula in (M, H) is equivalent to a boolean combination of existential formulas over H. This fact together with Theorem 2.4 [12] shows that T h(M, H) also has NIP. □

Remark 5.22. The above result could also have been proved doing very small modifications on Theorem 2.8 [5]. Also, Theorem 2.11 [5] can be easily modified to show that if T is strongly dependent then T h(M, H) is strongly dependent.

Now we will study the special case when T is geometric and has finite dp-rank

Proposition 5.23. Let T be a geometric theory of dp-rank k < ω and let (M, H) be a sufficiently saturated H-structure of T. If T is trivial and dcl = acl, then T h(M, H) has dp-rank k.

Proof. Since T is trivial, every formula ψ(x, y⃗) in T^{ind} is a boolean combination of L-formulas and formulas of the form H(f(x, y⃗)) where f(x, y⃗) is an L-definable function over ∅.

Claim Let (a⃗_{i} : i ∈ ω) be an L_{H}-indiscernible sequence and let b ∈ M and let f(x, y⃗) be an definable function over ∅ in the language L. Then for all i ∈ ω either H(f(b, a⃗_{i})) or ¬H(f(b, a⃗_{i})).

Since T is trivial and f(x, y⃗) is an L-definable function, either there is an Ldefinable function h(x) such that for all i t(x, a⃗_{i}) = h(x) or there is L-definable function g(y⃗) such that for all i t(x, a⃗_{i}) = g(a⃗_{i}). The existence of h or g only depends on the type of a⃗_{i}. If t(x, a⃗_{i}) = g(a⃗_{i}) then since (a⃗_{i} : i ∈ ω) is L_{H}- indiscernible we have that the value of H(f(b, a⃗_{i})) agrees with the value of H(g(a⃗_{0})). If t(x, a⃗_{i}) = h(x) then the value of H(f(b, a⃗_{i})) agrees with the value of H(h(b)). In any case, the value of H(f(b, a⃗_{i})) does not depend on i.

Assume there is an ICT pattern of depth n in (M, H). Then there are L_{H} formulas ψ_{1}(x, y⃗_{1}), …, ψ_{n}(x, y⃗_{n}) and there are L_{H}-indiscernible sequences sequences {(a_{i}^{j} : i < ω) : j ≤ n} that form a ICT pattern of depth n. Let i_{1}, i_{2}, …, i_{n} < ω and let b realize ψ_{1}(x, a_{1}^{i_{1}}) ∧ ψ_{2}(x, a_{2}^{i_{2}}) ∧ ⋯ ∧ ψ_{n}(x, a_{n}^{i_{n}}) and the negation of all other formulas. Each formula ψ_{1}(x, y⃗_{1}) is a boolean combination of L-formulas and formulas of the form H(f(x, y⃗_{1})) where f(x, y⃗_{1}) is an L-definable function. Since the value of H(f(b, a⃗_{1}^{i})) does not depend on i we may replace each ψ_{i}(x, y⃗_{1}) just by the L-formulas inside it and obtain an ICT pattern of depth n in (M, H). □

Question 5.24. Does the result of the previous Proposition remain true if we remove the assumption acl = dcl?

Proposition 5.25. Let T be a geometric theory of dp-rank k < ω. Then T^{ind} has dp-rank greater than n for every n ∈ ℕ but bounded by ℵ_{0}.

Proof. Since T^{ind} is strongly dependent, then the dp-rank is bounded by ℵ_{0}. Let (M, H) be a sufficiently saturated H-structure. Let a_{1}, …, a_{n}, a_{n+1} ∈ M be an algebraic n + 1-gon. We may assume that a_{1}, …, a_{n} ∈ H(M), thus HB(a_{n+1}) = {a_{1}, …, a_{n}} and a_{1}, …, a_{n} ∈ acl(a_{n+1}). Since dp − rk(H(x_{1}) ∧ ⋯ ∧ H(x_{n})) ≥ n, we have dp-rank(x = x) ≥ n. □

Remark 5.26. Let T be a geometric theory which is dp-minimal and let (M, H) be a sufficiently saturated H-structure of T. Let C ⊂ M and let a⃗ ∈ M^{n} be such that a⃗ ∈ scl(C). Let h⃗ = HB(a⃗/C). Then a⃗ is interalgebraic with h⃗ over C. Each d ∈ h⃗ has a type of dp-rank one over C, so small tuples can be coordinatized in terms of dp-rank one types. In the stable setting, for a ∉ scl(C) we had that tp(a/C) was regular. What is the corresponding notion in the setting of strongly dependent theories? Does it have burden one?

6. Groups

In this section we study the special case where M is rosy. As before we consider the H-structure (M, H) and our aim in this section is to study definable groups in (M, H). We will show that there are no small definable groups. Then we consider the special case where M = G is a group with RM(G) = 1. We will show that the L_{H}-definable subgroups of G^{n} are L-definable. Finally we show that if M is stable of U-rank one, the connected component of any L_{H}-definable group is isomorphic to an L-type definable group.

We will use the following tool in the rosy setting:

Fact 6.1. Let T be rosy, let M ⊧ T and let G ⊂ M^{n} be a definable group. Then G has generics in the sense that there is g ∈ G such that for h ∈ G with g | ⌣ h we have gh | ⌣ h.

We start by showing that the generic elements in definable groups are independent from H:

Proposition 6.2. Let M be a model of a thorn rank one theory and assume that (M, H) is a ℵ_{0}-saturated H-structure. Let A ⊂ M be finite and let T ≤ M^{n} be a L_{H}-definable group over A. Let b⃗ ∈ T be a generic element of the group, then HB(b⃗/A) = ∅.

Proof. We may assume that A = A ∪ HB(A) and thus that A is H-independent. Let b⃗, c⃗ ∈ T be independent generics (over A) and let b⃗ · c⃗ stand for the product of b⃗, c⃗ in the group T. Since c⃗ | ⌣_{A}^{H} b⃗, HB(c⃗/A) = HB(c⃗/Ab⃗) and HB(c⃗b⃗/A) = HB(c⃗/A) ∪ HB(b⃗/A) and thus HB(c⃗ · b⃗/A) ⊂ HB(c⃗/A) ∪ HB(b⃗/A).

Since b⃗, c⃗ are independent generics, b⃗·c⃗ | ⌣_{A}^{H} b⃗ and thus HB(b⃗·c⃗/A) | ⌣_{A}^{H} HB(b⃗/A) and so HB(b⃗ · c⃗/A) ∩ HB(b⃗/A) = ∅. Similarly one has HB(b⃗ · c⃗/A) ∩ HB(c⃗/A) = ∅. This together with HB(c⃗·b⃗/A) ⊂ HB(c⃗/A)∪HB(b⃗/A) proves that HB(c⃗·b⃗/A) = ∅. Since c⃗ · b⃗ is generic, and c^{−1} is also generic and independent from c⃗ · b⃗ over A, we get that HB(b⃗/A) = HB(b⃗ · c⃗ · c^{−1}/A) = ∅. □

Corollary 6.3. Let M be a model of a thorn rank one theory and assume that (M, H) is a ℵ_{0}-saturated H-structure. Let A ⊂ M be finite and let T ≤ M^{n} be a L_{H}-definable group over A and assume that T is small. Then T is finite.

We will now consider two settings, first the stable one and then topological one. Our goal is to see how close are L_{H}-definable groups from being L-definable.

6.1. Stable groups.

Notation 6.4. Let M be a structure of Morley rank one and assume that (M, H) is an H-structure. For an L-formula ψ, we denote by RM(ψ) the Morley rank and dM(ψ) the Morley degree calculated within T h(M). Similarly, for an L_{H}-formula ψ we denote by RM_{H}(ψ), dM_{H}(ψ) the Morley rank and the Morley degree inside the theory T h(M, H).

Assume now that M is a group. For a complete L-type p, we denote by Stab(p) the stabilizer of p in the language L and for a complete L_{H}-type q we write Stab_{H}(p) for its stabilizer.

Proposition 6.5. Let G be group with RM(G) = 1 and assume that (G, H) is a ℵ_{0}-saturated H-structure. Let A ⊂ G be finite and let T ≤ G^{n} be a L_{H}-definable subgroup over A. Then T is L-definable.

Proof. Let A and T be as above. By exchanging A for A ∪ HB(A) we may assume that A is H-independent. Let b⃗ ∈ T be generic, so HB(b⃗/A) = ∅.

Assume first that T = T^{0}, that is, T is connected. Since HB(b⃗/A) = ∅, we may assume that we can write b⃗ = (b_{1}, …, b_{l}, b_{l+1}, …, b_{n}), where (b_{1}, …, b_{l}) are H-independent and b_{l+1}, …, b_{n} ∈ acl(b_{1}, …, b_{l}). Let p = tp_{H}(b⃗/A) and note that RM_{H}(b⃗/A) = ωl. Since b⃗ is generic, Stab_{H}(p) = T. Let q = tp(b⃗/A), so q is the restriction of p to the old language.

Claim Let c⃗ ⊧ q be such that b⃗ | ⌣_{A} c⃗, then b⃗ · c⃗ ⊧ q.

We may assume that b⃗, c⃗ | ⌣_{A} H, so b⃗, c⃗ ⊧ p and b⃗ | ⌣_{A}^{H} c⃗. Since p is the unique generic type of T, we must have that b⃗ · c⃗ ⊧ p and thus b⃗ · c⃗ ⊧ q as we wanted.

Let D = Stab(q), where we now take the stabilizer in the L language. Since q is closed under generic multiplication and inverses, every member of D is a product of two realizations of q, we have that D = q · q. Since q is a generic for D, we obtain then that RM(D) = l, dM(D) = 1, RM_{H}(D) = ωl, dM_{H}(D) = 1. Also p · p ⊂ q · q = D, so T = p · p ≤ D. Since MR_{H}(T) = ωl and dM_{H}(D) = 1 we must have that D = T.

Now assume that T is not necessarily connected. Then T = T^{0}∪b⃗_{1}T^{0}∪⋯∪b⃗_{k}T^{0} for some finite b⃗_{1}, …, b⃗_{k} ∈ T. Since T^{0} is L-definable so is T. □

Proposition 6.6. Let M be a stable structure of U-rank one and let H be a subset of M such that (M, H) is a ℵ_{1}-saturated H-structure. Let A ⊂ M be countable and let T ⊂ M^{n} be a L_{H}-definable group over A. Let T^{0} be the connected component of T. Then T^{0} is definably isomorphic to an L-definable group.

Proof. We may assume by enlarging A that (A, H ∩ A) ⪯ (M, H) and that (M, H) is saturated and strongly homogeneous over A. Thus every complete L-type and every complete L_{H}-type over A is stationary. Also note that T^{0} is A-definable. Let b_{1}, b_{2}, a_{2} be independent generics in T^{0}. Let a_{3} = b_{1} · a_{2}, let a_{1} = b_{2}^{−1} · a_{3}, b_{3} = b_{2} ·b_{1}^{−1}. Note that since b_{1} and a_{2} are independent generics, HB(b_{1}, a_{2}/A) = ∅ and thus by Lemma 2.8 a_{3} ∈ acl(b_{1}, a_{2}, A). To simplify the argument we will assume that a_{3} ∈ dcl(b_{1}, a_{2}, A). Similarly a_{1} ∈ dcl(b_{2}, a_{3}, A), b_{3} ∈ dcl(b_{1}, b_{2}, A). Then {a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}} forms an algebraic quadrangle in the language L. Applying the group configuration construction to {a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}} (see for example [8, 23]) we obtain a connected L-type-definable group G. We will now follow the proof in [8] to understand how G is related to T^{0}. Let p = tp(b_{2}/A), note that p = tp(a_{1}/A), and that b_{2} defines a unique germ given by h_{b_{2}}(x) = b_{2} · x, and this L-function is defined for elements satisfying p that are independent from b_{2}. The group G is given by p × p/E, where for d_{1}, d_{2} ∈ p, c_{1}, c_{2} ∈ p, (d_{1}, d_{2})E(c_{1}, c_{2}) if for a ⊧ p independent from {d_{1}, d_{2}, c_{1}, c_{2}}, d_{1} · (d_{2} · a) = c_{1} · (c_{2} · a). In few words, every element in G is formed as the product of two generics (realizations of p) and we identify the product d_{1} · d_{2} with c_{1} · c_{2} if they agree generically. We will now build a definable isomorphism between T^{0} and G. For t ∈ T^{0}, let b ⊧ p be independent from t and define φ(t) = (b^{−1}, b · t)/E. Note that φ is A-definable and does not depend on the choice of b.

Claim The map φ is 1 − 1. Let t_{1}, t_{2} ∈ T and let b ⊧ p be independent from t_{1}, t_{2}. If (b^{−1}, b · t_{1})E(b^{−1}, b · t_{2}) then for a ⊧ p generic, b^{−1} · b · t_{1} · a = b^{−1} · b · t_{2} · a and thus t_{1} = t_{2}.

Claim The map φ is onto. Let d_{1}, d_{2} ⊧ p be generics. Let a ⊧ p be generic such that a | ⌣_{A} Hd_{1}d_{2}. Note that by stationarity a realizes the unique extension of p which is H-independent and thus a is a generic of the group T^{0}. Again by stationarity, d_{1} · a is generic in T^{0}, a^{−1} ·d_{2} is also generic in T^{0} and (d_{1}, d_{2})E(d_{1} ·a, a^{−1} ·d_{2}). Thus t = (d_{1} ·a)·(a^{−1} ·d_{2}) being a product of generics in T^{0} belongs to T^{0} and φ(t) = (d_{1}, d_{2})/E.

It is easy to see that φ is a homomorphism of groups and thus φ is an isomorphism between T^{0} and G. □

Remark 6.7. Let (M, H), T and A be as in the previous proposition. Note that the connected component T^{0} of T is isomorphic to an L-definable group and thus T is L-definable-by-bounded. In the case where M is ω-stable, the index of T^{0} in T is finite and thus T is L-definable-by-finite. In the later case, T can be written as a semidirect product of T^{0} and T/T^{0}, but the action of T^{0} is L_{H}-definable and it is not clear we can recover a L-definable copy of T.

References

[1] J. Baldwin and M. Benedikt, Stability theory, Permutations of Indiscernibles, and Embedded Finite Models, Trans. Amer. Math. Soc, 2000, vol 352, pp. 4937–4969

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

[3] A. Berenstein and E. Vassiliev, On lovely pairs of geometric structures, Annals of Pure and Applied Logic, Volume 161, Issue 7, April 2010, pp. 866-878.

[4] A. Berenstein and E. Vassiliev, Weakly one-based geometric theories , J. Symbolic Logic, 77 (2), 2012, pp. 392-422.

[5] A. Berenstein, A. Dolich and A. Onshuus, The independence property in generalized dense pairs of structures, Journal of Symbolic Logic, 76 (2), 2011, pp. 391-404.

[6] G Boxall, Lovely pairs and dense pairs of real closed fields, Ph.D. thesis, University of Leeds, 2009.

[7] Gareth Boxall, David Bradley-Williams, Charlotte Kestner, Alexandra Omar Aziz and Davide Penazzi, Weak one-basedness, preprint 2012.

[8] E Bouscaren, The group configuration-after E. Hrushovski, in The Model Theory of Groups, Nore Dame mathematical Lectures, Number 11, 1989.

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

[10] J. F. Carmona, Forking geometry on theories with an independent predicate, preprint 2012.

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

[12] A. Chernikov and P. Simon, Externally definable sets and dependent pairs, to be published in Israel Journal of Mathematics.

[13] A. Dolich, C. Miller and C. Steinhorn, Expansions of O-minimal Structures by Dense Independent Sets, preprint 2012.

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

[15] L. van den Dries and A. Gunaydin, The fields of real and complex numbers with a multiplicative subgroup, Proceedings of London Mathematical Society, 93, (2006), pp. 43–81.

[16] A. Fornasiero, Dimension, matroids, and dense pairs of first-order structures, APAL, Volume 162, No. 7, 2011, pp. 514-543.

[17] J. Gagelman, Stability in geometric theories, Annals of Pure and Applied Logic 132 (2005), 313-326.

[18] J. Gunaydin and P. Hyeronimi, The real field with the rational points of an elliptic curve, to be published in Fundamenta Mathematica.

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

[20] E. Hrushovski and A. Pillay, Groups definable in local fields and psedofinite fields, Israel Journal of Mathematics 85 (1994), no. 1-3, 203-262.

[21] A. Macintyre, Dense embeddings I: A Theorem of Robinson in a general setting, in Model Theory and Algebra, a memorial tribute to Abraham Robinson, Lecture Notes in Mathematics 498, 1975, pp. 200–219.

[22] A. Pillay, E. Vassiliev, On lovely pairs and the (∃y ∈ P) quantifier, Notre Dame Journal of Formal Logic, vol. 46, no. 4, 2005.

[23] A. Pillay, Geometric Stability Theory, Oxford Science publications, 1996.

[24] B. Poizat Paires de Structures Stables, Journal of Symbolic Logic, 48 (1983), pp 239–249.

[25] S.Shelah, Strongly dependent theories, arXiv:math.LO/0504197, 2005.

[26] I. Tomasic and F. Wagner, Applications of the group configuration theorem in simple theories, Journal of Math. Logic, 3(2), pp. 239–255, 2003.

[27] 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

   URL: www.matematicas.uniandes.edu.co/~aberenst

Sir Wilfred Grenfell College, Memorial University of Newfoundland, Corner Brook, NL A2H 6P9, Canada

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