CONSTRUCTING THE HYPERDEFINABLE GROUP FROM THE GROUP CONFIGURATION

TRISTRAM DE PIRO, BYUNGHAN KIM AND JESSICA MILLAR

Abstract. Under P(4)^{−}-amalgamation, we obtain the canonical hyperdefinable group from the group configuration.

The group configuration theorem for stable theories given by Hrushovski [5], which extends Zilber’s result for ω-categorical theories [18], plays a central role in producing deep results in geometric stability theory (For a complete exposition of it, see [15]). For example, it is pivotal in the proof of the dichotomy theorem for Zariski’ structures (See [9]). It is fair to say the group configuration theorem is one of the foundational theorems in geometric stability theory and its applications to algebraic geometry.

The theorem roughly says that one can get the canonical non-trivial type-definable group from the group configuration, a certain geometrical configuration, in stable theories. The complete generalization of the theorem into the context of simple theories seemed unreachable. In their topical paper [1], Ben-Yaacov, Tomasic and Wagner generalize the group configuration theorem by obtaining an invariant group from the group configuration in simple theories. However the group they produce does not completely fit into the first-order context.

On the other hand, Kolesnikov in his important thesis [13], categorizes simple theories by strengthening the type-amalgamation property (the independence theorem [11]), along the lines of early suggestions by Shelah [16] and Hrushovski [6]. These works suggest to us the possibility of using generalized amalgamation for the group configuration problem. This approach proves successful, and in this paper we succeed in getting the canonical hyperdefinable group from the group configuration under stronger type-amalgamation in simple theories. The element of the group is a hyperimaginary, an equivalence class of a type-definable equivalence relation, and the group operation is type-definable, hence the group belongs to the domain of the standard first-order logic.

We assume that the reader is familiar with basics of simplicity theory [17]. Throughout the paper, T is a complete simple theory. We work in a saturated model M of T with hyperimaginaries, and a, b, ... are (possibly infinitary) hyperimaginaries, M, N are small elementary submodels. (Note that tuples from M^{eq} are also hyperimaginaries). As usual, a ≡_{A} b (a ≡_{A}^{L} b) means a, b have the same type (Lascar strong type, resp.) over A. We point out that usually bdd(a) denotes the set of all countable hyperimaginaries definable over a [17, 3.1.7]. Here, depending on the context, it can be either a specific sequence which linearly orders the set bdd(a); or, since a sequence of hyperimaginaries is again a hyperimaginary (of a large arity), a fixed hyperimaginary interdefinable with the sequence.

2000 Mathematics Subject Classification. Primary: 03C45.

Tristram de Piro was supported by the Seggie Brown research fellowship. Byunghan Kim was supported by KOSEF grant R01-2006-000-10638-0. Jessica Millar was supported by NSF grant DMS-0102502.

We thank Frank O. Wagner for valuable correspondence on our earlier note which improves the presentation.

1. Around the generalized amalgamation property

The usual amalgamation property (or the independence theorem) for Lascar strong types in simple theories is stated as follows: For B ⌣^{|} _{A} C with A ⊆ B, C, if p is a Lascar strong type over A and p_{B}, p_{C} are nonforking Lascar strong extensions of p over B,C, respectively, then there is d ⊧ p_{A} ∪ p_{B} such that d ⌣^{|} _{A} BC. We call it ‘3-amalgamation’ [8] (rather than 2-amalgamation [14]) which shall be compatible with Definition 1.3. Note that we can think of B, C (after naming A) as two vertices of a base edge of a triangle and d a top vertex, and p_{B} = Lstp(d/B), p_{C} = Lstp(d/C) are the 2 types to be amalgamated. One would expect generalized amalgamation to be a natural generalization of 3-amalgamation, using a tetrahedron and higher dimensional simplices instead of a triangle. Indeed, this is the case, but the following example draws attention to why we need extra care in defining the general n-amalgamation property.

Example 1.1. In the random graph M in L = {R}, choose distinct a_{i}, b_{i}, c_{i} ∈ M and imaginary elements d_{i} = {a_{i}, b_{i}} (i = 0, 1, 2). We can additionally assume that R(a_{0}, c_{0}) ∧ R(b_{0}, c_{1}) ∧ ¬R(a_{0}, c_{1}) ∧ ¬R(b_{0}, c_{0}) and tp(a_{0}b_{0}; c_{0}c_{1}) = tp(a_{1}b_{1}; c_{1}c_{2}) = tp(a_{2}b_{2}; c_{2}c_{0}). Now it follows that Lstp(d_{2}/c_{0}) = Lstp(d_{0}/c_{0}), Lstp(d_{0}/c_{1}) = Lstp(d_{1}/c_{1}) and Lstp(d_{1}/c_{2}) = Lstp(d_{2}/c_{2}). However it is easy to see that Lstp(d_{0}/c_{0}c_{1}), Lstp(d_{1}/c_{1}c_{2}), Lstp(d_{2}/c_{2}c_{0}) have no common realization.

In above example, {c_{0}, c_{1}, c_{2}} can be considered as a base triangle, and Lstp(d_{0}/c_{0}c_{1}), Lstp(d_{1}/c_{1}c_{2}), Lstp(d_{2}/c_{2}c_{0}) form other 3 triangles attached to the base triangle. The example shows that even if the edges of the 3 triangles are compatible over the base vertices, there is no common vertex joining the 3 triangles. On the other hand, due to the nature of the random graph if we only work in the home-sort, then any desired 3 types attached on a base triangle with compatible edges will be realized. As we want the notion of generalized amalgamation to be preserved in interpreted theories, Kolesinkov suggests, in his revised works [13] [14], the following as generalized amalgamation which we call here K(n)-amalgamation. We briefly explain the notation. In this paper, strong type indeed means Lascar strong type. Likewise, p ∈ S_{L}(A) means p is a Lascar strong type over A, and for B ⊆ A, p⌈_{L}B (or simply p⌈B) denotes Lstp(a/B) for any (some) a ⊧ p. Note that for q ∈ S_{L}(B), q ⊆ p means p⌈B = q or equivalently p ⊢ q.

Definition 1.2. • We say strong types p_{i} ∈ S_{L}(A_{i}) are compatible over A(⊆ A_{i}) if each p_{i} does not fork over A and for i, j, p_{i}⌈_{L}A_{i} ∩ A_{j} = p_{j}⌈_{L}A_{i} ∩ A_{j}. (Hence p_{i}⌈_{L}A = p_{j}⌈_{L}A). We say these A-compatible strong types p_{i} are (generically) amalgamated if there is q ∈ S_{L}(⋃_{i} A_{i}) nonforking over A such that ∪_{i}p_{i} ⊆ q (i.e. q ⊢ p_{i}).

• We say T has K(n)-amalgamation over B if for B-independent A = {a_{1}, ..., a_{n}} and any B-compatible p_{i} ∈ S_{L}(BA_{i}) where A_{i} = A ∖ {a_{i}} for i = 1, .., n, whenever bdd(aB) ⊆ dcl(aB) for any a ⊧ p_{1}⌈_{L}B(= p_{i}⌈_{L}B), then p_{1} ∪ ... ∪ p_{n} is generically amalgamated. We say T has K(n)-amalgamation if it has K(n)-amalgamation over an arbitrary set.

The modification is that the realizations of strong types are required to be boundedly closed over the parameter set, i.e. in above bdd(aB) ⊆ dcl(aB). Note that K(2)-amalgamation is equivalent to 3-amalgamation (usual amalgamation), and due to weak elimination of imaginaries, it can now be seen that the random graph has K(n)-amalgamation for all n. Each stable theory has K(n)-amalgamation as well, by stationarity.

However when we use inductive arguments for example, we often have to consider not only the bounded closures of vertices of amalgamated types but also those of higher dimensional surfaces as well, since after naming parameters the surface dimension is increasing. Indeed, there exists in the literature another notion of amalgamation, called P(n)^{−}-amalgamation, introduced by Hrushovski [6] prior to Kolesnikov’s work, and which already takes care of this concern. Moreover, in contrast to K(n)-amalgamation (or the statement of the independence theorem), the base simplex is not regarded as an embedded parameter, but another type to be amalgamated. We think this is conceptually more correct and we shall take it as our definition of n-amalgamation.

Definition 1.3. Let I = P(n)^{−}(= P(n)\{n}), ordered by inclusion. Let ({A_{i}}_{i∈I}, {π_{j}^{i}}_{i≤j∈I}) be a directed family: Namely, each π_{j}^{i} : A_{i} → A_{j} is an elementary map between the two sets, π_{i}^{i} = id_{A_{i}}, and π_{k}^{j} ∘ π_{j}^{i} = π_{k}^{i} for i ≤ j ≤ k ∈ I. We say T has P(n)^{−}-amalgamation, or simply n-amalgamation if whenever

(1) {π_{u}^{{i}}(A_{{i}}) : i ∈ u} is π_{u}^{∅}(A_{∅})-independent,

(2) A_{u} = bdd(⋃_{i∈u} π_{u}^{{i}}(A_{{i}})), for any u ∈ I, then we can extend the direct family to one indexed by P(n) (by finding A_{n} and π_{n}^{j}) so that (1) and (2) hold for n, too. We say T has P(n)^{−}-amalgamation (namalgamation) over A, if A_{∅} = bdd(A).

Since the definition is not transparent to conceptualize with the above notation, we give a rewritten definition as in [2] or [7]. Recall that when we say a hyperimaginary b = ā/E realizes a type r over d = c̄/F, we mean r = r(x̄) is a (real) type such that i) r(ā); ii) whenever E(ē, ē′), then r(ē) iff r(ē′); iii) r(ā′) if ā′/E = f(b) for some d-automorphism f. If additionally the converse of iii) holds, we call r a complete type of b over d.

Definition 1.4. We say T has n-complete amalgamation over a set B if the following holds: Let W be a collection of subsets of {1, ..., n} = u_{n}, closed under subsets. For each w ∈ W, complete type r_{w}(x_{w}) over B is given where x_{w} is possibly an infinite set of variables. Suppose that

(1) for w ⊆ w′, x_{w} ⊆ x_{w′} and r_{w} ⊆ r_{w′}. Moreover for any a_{w} ⊧ r_{w},

(2) {a_{{i}}|i ∈ w} is B-independent,

(3) a_{w} is as a set bdd(⋃_{i∈w} a_{{i}}B) (and the map a_{w} → x_{w} is a bijection). Then there is a complete type r_{u_{n}}(x_{u_{n}}) over B such that (1),(2),(3) hold for all w ∈ W ∪{u_{n}}. We say T has n-complete amalgamation (n-CA) if it has n-complete amalgamation over any set.

We leave the reader to show that T has n-CA over B iff T has m-amalgamation over B for all m ≤ n. The following can be freely used: For B-independent A = {a_{1}, ..., a_{n}}, {A_{w}|w ∈ P(u_{n})} is a partition of bdd(BA), where A_{w} = bdd(∪a_{i}B|i ∈ w) \ ⋃_{v∈P(w)^{−}} bdd(∪a_{i}B|i ∈ v). For example n = 2, {bdd(B), bdd(a_{1}B) \ bdd(B), bdd(a_{2}B) \ bdd(B), bdd(Ba_{1}a_{2}) \ (bdd(a_{1}B) ∪ bdd(a_{2}B))} is a partition of bdd(Ba_{1}a_{2}), since using the fact that a_{1} ⌣^{|} _{B} a_{2}, we have bdd(a_{1}B) ∩ bdd(a_{2}B) = bdd(B). It also follows in 1.4, for v, w ∈ W, x_{v} ∩ x_{w} = x_{v∩w}.

Any simple T has P(3)^{−}-amalgamation due to usual amalgamation (the independence theorem), and we shall see that 4-amalgamation implies K(3)-amalgamation (1.8). For each n > 2, there is a simple theory having n-CA but not having (n+1)-CA over any set [14]. (The example also shows n-amalgamation does not necessarily imply k-amalgamation for k < n.) All stable theories have n-amalgamation over a model (1.6) for all n. Many important simple structures also have n-CA for all n such as the random graph (1.6), every PAC-structure (over some parameter) [7], and ACFA [2].

As now we know n-CA is the correct notion of generalized amalgamation, in recent work [12], corresponding modifications of terminology in [13][14] in regard to n-CA are made.^{1}

The lemma 1.5 and 1.6.1,2 below essentially come from the proof of the generalized independence theorem [2]. We thank Zoe Chatzidakis for her explanation.

Lemma 1.5. Let T be stable.

(1) Suppose that for a set C, whenever a ⌣^{|} _{C} b with b = b_{1} ∪ ... ∪ b_{n}, then

dcl(acl(ab_{1}C)... acl(ab_{n}C)) ∩ acl(bC) = dcl(acl(b_{1}C)... acl(b_{n}C)).  (♯)

Then the following are satisfied.

(a) tp(acl(ab_{1}C)... acl(ab_{n}C)/ acl(b_{1}C)... acl(b_{n}C)) is stationary.

(b) Let A = {a_{1}, ...a_{n}}, B = {c_{1}, ..., c_{n}} be C-independent, respectively. For 1 ≤ i ≤ n, let v_{i} = {1, ..., n} \ {i}. Now given k ≤ n, assume there is a bijective map

h : ⋃_{1≤i≤k} acl(a_{v_{i}}C) → ⋃_{1≤i≤k} acl(c_{v_{i}}C)

where a_{v_{i}} = {a_{j}|j ∈ v_{i}} such that h(a_{i}) = c_{i}, h⌈C = id and, for each v_{i}, h⌈acl(a_{v_{i}}C) is elementary. Then h is C-elementary.

(2) In fact, the condition (♯) holds when the set C is a universe of a model M. (Hence (1)(a),(b) also are true over a model.)

Proof. We can safely assume a, a_{i}, b_{i}, c_{i} are finite tuples from M^{eq} = M.

(1)(a) is immediate from (♯) using Cb(c/d) ⊆ dcl(cd) ∩ acl(d).

(1)(b) For 1 ≤ i ≤ k, let h_{i} = h⌈acl(a_{v_{i}}C). Then put h^{j} = h_{1} ∪ ... ∪ h_{j} (where h^{k} = h) and D_{a}^{j} = dom(h^{j}) = ∪_{i=1,...,j} acl(a_{v_{i}}C) and D_{c}^{j} = ran(h^{j}) = ∪_{i=1,...,j} acl(c_{v_{i}}C). For induction, assume h^{j−1} is elementary for j > 1. We shall show that h^{j} is elementary too. Now for each i < j let w_{i} = v_{j} ∩ v_{i}. Then a_{v_{i}} = {a_{j}} ∪ a_{w_{i}}. Now since h_{j} is elementary, there is an automorphism ĥ_{j} extending h_{j}. Then by induction h^{j−1} ∘ ĥ^{−1} maps ĥ(D_{a}^{j−1}) to D_{c}^{j−1}, so they have the same type, in particular, over the set ⋃_{i=1,...,j−1} acl(c_{w_{i}}C) fixed by h^{j−1} ∘ ĥ^{−1}. Note now that c_{j} ⌣^{|} _{C} c_{w_{1}}...c_{w_{j−1}} and c_{v_{i}} = {c_{j}} ∪ c_{w_{i}}. Hence we can apply (1)(a) to conclude that ĥ(D_{a}^{j−1}) and D_{c}^{j−1} = ⋃_{i=1,...,j−1} acl(c_{v_{i}}C) also have the same type over acl(c_{v_{j}}C), i.e. there is an elementary map g sending ĥ(D_{a}^{j−1}) to D_{c}^{j−1} fixing acl(c_{v_{j}}C) = ran(h_{j}). Therefore it follows h^{j}(⊆ g ∘ ĥ) is elementary.

1For instance, the notion of K(n)-simplicity is introduced in terms of an infinite Morley sequence. This notion is quite natural as showing the equivalence of K(1)-simplicity and 3-amalgamation is exactly the way of proving the independence theorem [11]. Kolesnikov’s ideas in [13] go through to obtain the equivalence of K(2)-simplicity and 4-amalgamation. However, counterexamples are constructed witnessing that the equivalence no longer holds for n > 2. Then, it is shown that an altered concept of n-simplicity (implying K(n)-simplicity) defined via a finite Morley sequence is now equivalent to (n + 2)-CA for every n.

(2) It suffices to show that e ∈ dcl(acl(b_{1}M)... acl(b_{n}M)) for e ∈ dcl(acl(ab_{1}M)... acl(ab_{n}M))∩ acl(bM). Since e ∈ dcl(acl(ab_{1}M)... acl(ab_{n}M)), there are e_{1}...e_{n} and L(M)-formulas φ(x; y_{1}...y_{n}), ψ_{i}(y_{i}, zw_{i}) with ⊧ φ(e; e_{1}...e_{n}), ⊧ ψ_{i}(e_{i}, ab_{i}) such that ⊧ φ(u; v) implies that u is definable over vM, and ψ_{i}(u′, v′) implies that u′ is algebraic over v′M. Therefore

⊧ ∃y_{1}...y_{n}(φ(e, y_{1}...y_{n}) ∧ ⋀_{i} ψ_{i}(y_{i}, ab_{i})).

Now since e ∈ acl(bM), a ⌣^{|} _{M} eb and so tp(a/Meb) is a coheir extension of tp(a/M). Thus we have m ∈ M such that

⊧ ∃y_{1}...y_{n}(φ(e, y_{1}...y_{n}) ∧ ⋀_{i} ψ_{i}(y_{i}, mb_{i})).

Hence e ∈ dcl(acl(b_{1}M)... acl(b_{k}M)). □

Proposition 1.6. (1) Let T be stable. If a set C satisfies (♯) in 1.5.1, then for each n,

T has n-CA over C.

(2) All stable theories have n-CA over a model.

(3) The random graph has n-CA over any set.

Proof. (1) In a stable theory T we can work in M^{eq} and substitute algebraic closures for bounded closures. We use the notation in 1.4. It suffices to show the case W = P(u_{n})^{−} with corresponding types r_{w}(x_{w}) (for w ∈ W). Again for 1 ≤ i < k ≤ n, let v_{i} = {1, ..., n} \ {i} and w_{i} = v_{k} ∩ v_{i}. We shall show that ⋃_{1≤i≤n} r_{v_{i}} is consistent and realized by ⋃_{1≤i≤n} a_{v_{i}} such that {a_{{1}}, ..., a_{{n}}} is B-independent. (Then the type of its algebraic closure over B extending ∪_{1≤i≤n}r_{v_{i}} is the desired r_{u_{n}}(x_{u_{n}}).) Now due to usual amalgamation there is a_{v_{1}}a_{v_{2}} ⊧ r_{v_{1}} ∪ r_{v_{2}} such that {a_{{1}}, ..., a_{{n}}} is B-independent. Then for induction, assume that there is a_{v_{1}}...a_{v_{k−1}} ⊧ r_{v_{1}} ∪...∪r_{v_{k−1}} (for 2 < k) such that a_{v_{1}}...a_{v_{k−1}} extends a_{{1}}, ..., a_{{n}}. Now let b_{v_{k}} ⊧ r_{v_{k}}. Then there is a map h : ⋃_{1≤i<k} b_{w_{i}} → ⋃_{1≤i<k} a_{w_{i}} such that h sends b_{w_{i}} to a_{w_{i}}. Hence h is elementary by 1.5.1(b), and extends to an automorphism ĥ. Then we have a_{v_{k}} = ĥ(b_{v_{k}}) ⊧ r_{v_{k}}. Now a_{v_{1}}...a_{v_{k}} realizes r_{v_{1}} ∪ ... ∪ r_{v_{k}} if for y = x_{v_{k}} ∩ (x_{v_{1}} ∪ ... ∪ x_{v_{k−1}}), a_{v_{1}}...a_{v_{k−1}}⌈y = a_{v_{k}}⌈y. But this clearly holds since by independence x_{v_{k}} ∩ x_{v_{i}} = x_{w_{i}}, whence y = x_{w_{1}} ∪ ... ∪ x_{w_{k−1}}. This finishes the proof of (1).

(2) It follows from 1.5.2 and (1) above.

(3) Note that for the random graph M = (M̄, R) we can work in M^{eq} and substitute algebraic closures for bounded closures. Now since the random graph has weak elimination of imaginaries, for any A there is A′ in the home sort M̄ such that acl(A) = dcl(A′). Hence when we check n-CA, we can assume each r_{w} is a type of a set in M̄. Then in M̄, since tp(A/B) is determined by equality and R relations of pairs in A ∪ B, due to randomness of R we clearly have the generalized amalgamation. □

However, there is a stable theory which does not even have 4-amalgamation over an algebraically closed set (in M^{eq}). We thank Ehud Hrushovski for supplying us with this example.

Example 1.7. Let A be an infinite set with [A]^{2} = {{a, b}|a, b ∈ A, a ≠ b}, and let B = [A]^{2} ×{0, 1} where {0, 1} = ℤ/2ℤ. Also let E ⊆ A×[A]^{2} be a membership relation, and let P be a subset of B^{3} such that ((w_{1}, δ_{1})(w_{2}, δ_{2})(w_{3}, δ_{3})) ∈ P iff there are distinct a_{1}, a_{2}, a_{3} ∈ A such that for {i, j, k} = {1, 2, 3}, w_{i} = {a_{j}, a_{k}}, and δ_{1} + δ_{2} + δ_{3} = 0. Now let M be a model with the 3-sorted universe A, [A]^{2}, B equipped with relations E, P and the projection f : B → [A]^{2}. Then since M is a reduct of (A, ℤ/2ℤ)^{eq}, M is stable. We work in M^{eq} and show that it does not have P(4)^{−}-amalgamation. Note first that dcl(∅) = acl(∅), and for a ∈ A, dcl(a) = acl(a). Now choose distinct a_{1}, a_{2}, a_{3}, a_{4} ∈ A. For {i, j, k} ⊆ {1, 2, 3, 4}, fix an enumeration \overline{a_{i}a_{j}} = (b_{ij}, ...) of acl(a_{i}a_{j}) where b_{ij} = (\{a_{i}, a_{j}\}, δ) ∈ B = [A]^{2} × \{0, 1\}. Let r_{ij}(x_{ij}) = tp(\overline{a_{i}a_{j}}), and let x_{ij}^{1} be the variable for b_{ij}. Note that b_{ij} = (\{a_{i}, a_{j}\}, δ) and b′_{ij} = (\{a_{i}, a_{j}\}, δ+ 1) have the same type over a_{i}a_{j}. Hence there is (\overline{a_{i}a_{j}})′ = (b′_{ij}, ...) also realizing r_{ij}(x_{ij}). Therefore we have complete types r_{ijk}(x_{ijk}), r′_{ijk}(x′_{ijk}) both extending r_{ij}(x_{ij}) ∪ r_{ik}(x_{ik}) ∪ r_{jk}(x_{jk}) realized by some enumerations of acl(a_{i}a_{j}a_{k}) such that P (x_{ij}^{1}x_{ik}^{1}x_{jk}^{1}) ∈ r_{ijk} whereas ¬P (x_{ij}^{1}x_{ik}^{1}x_{jk}^{1}) ∈ r′_{ijk}. Then it is easy to see that r_{123} ∪ r_{124} ∪ r_{134} ∪ r′_{234} is inconsistent.

In the example, ({a_{2}, a_{3}}, 0) ∈ dcl(acl(a_{1}a_{2}) ∪ acl(a_{1}a_{3})), since ({a_{2}, a_{3}}, 0) is a unique solution to P (({a_{1}, a_{2}}, 0), ({a_{1}, a_{3}}, 0), x). But ({a_{2}, a_{3}}, 0) ∉ dcl(acl(a_{2}) ∪ acl(a_{3})), i.e. 1.5.1(♯) does not hold over an algebraically closed set. In [8], Hrushovski shows that if a stable T eliminates generalized finite imaginaries then T has 4-amalgamation.

Proposition 1.8. If T has 4-amalgamation over B, then it has K(3)-amalgamation over B.

Proof. Assume T has 4-amalgamation. Now suppose that B-independent A = {a_{1}, a_{2}, a_{3}} and B-compatible p_{i} ∈ S_{L}(BA_{i}) where A_{i} = A ∖ {a_{i}} (i = 1, 2, 3) are given. Also for d_{i} ⊧ p_{i}⌈_{L}B, assume

bdd(d_{i}B) ⊆ dcl(d_{i}B) (*).

Now let r_{∅}(x_{∅}) = tp(bdd(B)/B). Let r_{i}(x_{i}) = tp(bdd(a_{i}B)/B) for i = 1, 2, 3, and r_{4}(x_{4}) = tp(bdd(d_{i}B)/B) extend r_{∅}(x_{∅}). Now we have r_{14}(x_{14}) = tp(bdd(a_{1}d_{2}B)/B) extends r_{1} ∪ r_{4}, since a_{1} ⌣^{|} _{B} d_{2} implies x_{1} ∩ x_{4} = x_{∅}. Let f_{1} : bdd(a_{1}d_{2}B) → x_{14} be the realization map. Now note that due to compatibility of types p_{i}, for i ∈ ℤ/3ℤ, there is an automorphism h_{i} sending d_{i+2} to d_{i+1} fixing bdd(a_{i}B). Then due to (*),

h_{i+2} ∘ h_{i}⌈bdd(d_{i+2}B) = h_{i+1}^{−1}⌈bdd(d_{i+2}B) (**).

Now via f_{1} ∘ h_{1} : bdd(a_{1}d_{3}B) → x_{14}, bdd(a_{1}d_{3}B) ⊧ r_{14}(x_{14}). Similarly there is r_{24}(x_{24}) = tp(bdd(a_{2}d_{3}B)/B) extending r_{2}(x_{2}) ∪ r_{4}(x_{4}), since also x_{14} ∩ x_{2} = x_{∅}. Thus by the map f_{2} ∘ h_{2} : bdd(a_{2}d_{1}B) → x_{24} where f_{2} : bdd(a_{2}d_{3}B) → x_{24}, bdd(a_{2}d_{1}B) ⊧ r_{24}(x_{24}). Note that f_{2}⌈bdd(d_{3}B) = f_{1} ∘ h_{1}⌈bdd(d_{3}B). Moreover, r_{34}(x_{34}) = tp(bdd(a_{3}d_{1}B)/B) extends r_{3}(x_{3})∪r_{4}(x_{4}). Let f_{3} : bdd(a_{3}d_{1}B) → x_{34}. Note again that f_{3}⌈bdd(d_{1}B) = f_{2}∘h_{2}⌈bdd(d_{1}B). Now f_{3} ∘ h_{3} : bdd(a_{3}d_{2}B) → x_{34} extends f_{1}⌈bdd(d_{2}B) : bdd(d_{2}B) → x_{4} since from (**), on bdd(d_{2}B), f_{3} ∘ h_{3} = (f_{2} ∘ h_{2}) ∘ h_{3} = (f_{1} ∘ h_{1} ∘ h_{2}) ∘ h_{3} = f_{1}. Therefore f_{1} ∪ f_{3} ∘ h_{3} : bdd(a_{1}d_{2}B) ∪ bdd(a_{3}d_{2}B) → r_{14}(x_{14}) ∪ r_{34}(x_{34}) is a well-defined realization map extending the realizations of r_{j}(x_{j}) (j = 1, 3, 4). It now is easy to find additional types r_{w} so that they satisfy (1),(2),(3) of 1.4 for n = 4. Therefore by 4-amalgamation we have d(≡_{B}^{L} d_{i}) such that {a_{1}, a_{2}, a_{3}, d} is B-independent and the type of bdd(a_{1}a_{2}a_{3}dB/B) extends the types r_{w}. Obviously, d is the generic realization of p_{1} ∪ p_{2} ∪ p_{3}. □

The main object of this paper is a generalization of the group configuration theorem for stable theories to the simple context. We succeed in obtaining a hyperdefinable group from the group configuration under 4-amalgamation. In fact, what we are using is 4-amalgamation over a parameter properly containing a model (see the proof of 2.6). But as indicated even a stable theory need not have such a property. Hence to make it work in a more general context, we introduce the notion of model-n-CA, a small variation of n-CA, which is n-CA over sets containing a model such that none of the types concerned fork over the model. The formal definition is as follows.

Definition 1.9. We say T has model-n-complete amalgamation if the following holds: Let u_{n} = {1, ..., n}, and W_{n} = P(u_{n+1})^{−} \ {u_{n}}. Let W be a subset of W_{n}, closed under subsets. For each w ∈ W, complete type r_{w}(x_{w}) over a model M is given where x_{w} is possibly an infinite set of variables. Suppose that

(1) for w ⊆ w′, x_{w} ⊆ x_{w′} and r_{w} ⊆ r_{w′}. Moreover for any a_{w} ⊧ r_{w},

(2) {a_{{i}}|i ∈ w} is M-independent,

(3) a_{w} is as a set bdd(∪_{i∈w}a_{{i}}M) (and the map a_{w} → x_{w} is a bijection). Then there is a complete type r_{u_{n+1}}(x_{u_{n+1}}) over M such that (1),(2),(3) hold for all w ∈ W ∪ {u_{n+1}}.

For example, if T has model-4-amalgamation, then by choosing W = P({1, 2, 3, 5})^{−}, we see that T has 4-amalgamation over models. Moreover, for

w ∈ W_{4} = {{1, 2, 3; 5}, {1, 2, 4; 5}, {1, 3, 4; 5}, {2, 3, 4; 5}} ∪ ⋃_{1≤i<j<k≤5} P({i, j, k}),

and the types r_{w} over M satisfying (1)(2)(3), r_{{123;5}} ∪ r_{{124;5}} ∪ r_{{134;5}} ∪ r_{{234;5}} can be generically amalgamated over M. This is 4-amalgamation over a parameter containing the model. Indeed, 4-amalgamation over sets containing models implies model-4-amalgamation. But some stable structure, which must satisfy the latter by 1.6.2, need not satisfy the former.

More generally, each of stability, (n + 1)-CA over models, or n-CA implies model-n-CA for every n; and model-n-CA implies n-CA over models. Model-n-CA also holds in aforementioned algebraic examples such as ACFA and PAC-structures. Model-4-CA is the property we shall use, hence covers the case that T is stable.

2. The group configuration

Definition 2.1. By a group configuration we mean a 6-tuple of hyperimaginaries C = (f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3}) over a hyperimaginary e such that, for {i, j, k} = {1, 2, 3},

(1) f_{i} ∈ bdd(f_{j}, f_{k}; e),

(2) x_{i} ∈ bdd(f_{j}, x_{k}; e),

(3) all other triples and all pairs from C are independent over e.

If the group configuration C = (f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3}) over e has the property that bdd(f_{i}; e) = bdd(Cb(x_{j}x_{k}/f_{i}e); e), we call such C a bounded quadrangle. If additionally x_{i}, x_{j} are interdefinable over f_{k}e, then we call C a definable quadrangle over e.

f_{1} f_{2} f_{3}  x_{2}  x_{3} x_{1}

Fact 2.2. (1) If C = (f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3}) is a group configuration/bounded quadrangle over e and bdd(f_{i}e) = bdd(f′_{i}e), bdd(x_{i}e) = bdd(x′_{i}e), then C′ = (f′_{1}, f′_{2}, f′_{3}, x′_{1}, x′_{2}, x′_{3}) is also a group configuration/bounded quadrangle over e. In this case, we say C and C′ are (boundedly) equivalent over e.

(2) For C a group configuration/bounded quadrangle over e and e′ ⊇ e, if C ⌣^{|} _{e} e′ then C also is a group configuration/bounded quadrangle over e′.

(3) In a group configuration (f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3}) over e even if we replace f_{i} by f′_{i} = Cb(x_{j}x_{k}/ef_{i}) for {i, j, k} = {1, 2, 3}, (f′_{1}, f′_{2}, f′_{3}, x_{1}, x_{2}, x_{3}) is still a group configuration (hence a bounded quadrangle) over e.

Proof. We sketch the proof. (1) Obvious for a group configuration. For a bounded quadrangle notice that in general Cb(a_{1}/a_{2}) and Cb(b_{1}/b_{2}) are interbounded as long as a_{i}, b_{i} are interbounded. (2) Easy. (3) Since x_{i}x_{j} ⌣^{|} _{f′_{k}} f_{k}e and x_{i} ⌣^{|} _{x_{j}f′_{k}} f_{k}e, x_{i}, x_{j} are interbounded over f′_{k} (*). On the other hand, x_{j} ⌣^{|} _{f_{k}e} f_{i} implies x_{i}x_{j} ⌣^{|} _{f_{k}e} f_{i}f_{j} (**), x_{i}x_{j} ⌣^{|} _{f_{k}f_{i}e} f_{j} and thus x_{i}x_{j}x_{k} ⌣^{|} _{f_{k}f_{i}e} f_{j} and x_{i}x_{j}x_{k} ⌣^{|} _{f_{k}f_{i}f_{j}f′_{j}e} f_{j}. Then from (**), it follows x_{i}x_{k} ⌣^{|} _{f′_{j}} f_{i}f_{j}f_{k}e and from (*) x_{i}x_{j} ⌣^{|} _{f′_{i}f′_{j}} f_{i}f_{j}e. Therefore f′_{k} = Cb(x_{i}x_{j}/f_{k}e) = Cb(x_{i}x_{j}/f_{i}f_{j}e) ∈ bdd(f′_{i}f′_{j}). The Other independences over e follow easily. □

¿From now on, assume that a group configuration over ê = A/Ē is given. We shall produce a non-trivial canonical hyperdefinable group from it. By 2.2.3 above, we can replace it by a bounded quadrangle C = (f̂, ĝ, ĥ, â, b̂, ĉ) over a model M containing A. After naming M, we freely assume that ∅ = bdd(∅). We further suppose f̂, ĝ, ĥ, â, b̂, ĉ are all boundedly closed (by extending each to its bounded closure, if necessary.) Clearly C still is a bounded quadrangle over ∅. Let p = tp(f̂)(= Lstp(f̂)), q = tp(ĝ), r = tp(ĥ) and let Γ_{q}(uv) = q(u) ∧ q(v) ∧ u ⌣^{|} v. (Later we shall omit q in Γ_{q}.) Now we can think of ĥ as a multi-valued function such that dom(ĥ) = tp(â/ĥ) = Lstp(â/ĥ) and rag(ĥ) = Lstp(b̂/ĥ). More precisely b ∈ k_{r}(a) means k_{r} ⊧ r, k_{r}ab ≡ ĥâb̂. Similarly we write a ∈ h_{q}(c), b ∈ g_{p}(c) for h_{q}ca ≡ ĝâĉ, g_{p}cb ≡ f̂b̂ĉ, respectively. In the same manner, b ∈ dom(f_{p}) ≡ ∃c(f_{p}bc ⊧ tp(f̂b̂ĉ)), and so on.

We say a set A is n-independent if any subset of A having n elements is independent. Now we define R = R^{q} to be a symmetric type-definable relation over ∅ on the set of independent realizations of q such that

R(fg; f′g′) iff f, g, f′g′ ⊧ q, {f, g, f′, g′} 3-independent, and there are b and

a ⌣^{|} fgf′g′ such that f(a) ∩ g(b) ≠ ∅, f′(a) ∩ g′(b) ≠ ∅.

It is easy to see that a ⌣^{|} fgf′g′ above can be replaced by b ⌣^{|} fgf′g′. Similarly, one can define R^{p}, R^{pq} by replacing f, g, f′g′ ⊧ q by f, g, f′g′ ⊧ p or f, f′ ⊧ p, g, g′ ⊧ q, respectively.

Lemma 2.3. (1) If fg ⊧ Γ_{q}, and c ∈ g(a) ∩ f(b) with c ⌣^{|} fg, then (a) f, g are interbounded over e := Cb(ba/fg), (b) e ⌣^{|} f, e ⌣^{|} g, and

(c) (f, g, e, a, b, c) forms a bounded quadrangle.

(2) (1) still holds when we replace f, g ⊧ q by f, g ⊧ p, or f ⊧ p, g ⊧ q.

(3) If (fg, f′g′) ⊧ R (or R^{p}, R^{pq}), then any element in {f, g, f′, g′} is in the bounded closure of the other 3 elements.

Proof. (1)(a) Note that from ab ⌣^{|} _{e} fg and that a, b are interbounded over fg, it follows that a, b are interbounded over e, too (*). Now from c ⌣^{|} _{g} f, we have ca ⌣^{|} _{g} fe. Moreover from c ⌣^{|} _{f} g, it follows cb ⌣^{|} _{fe} g, and then by (*), ca ⌣^{|} _{fe} g. Hence g ∈ bdd(Cb(ca/g)) ⊆ bdd(fe). By a similar argument, f ∈ bdd(ge) can be shown too.

(1)(b) There are h_{1}u_{1}, h_{2}u_{2} such that h_{1}fbu_{1}c, h_{2}gau_{2}c ⊧ f̂ĝâb̂ĉ. Then since c is boundedly closed, by amalgamation we have

hu ⊧ tp(h_{1}u_{1}/cbf) ∪ tp(h_{2}u_{2}/cag).

such that {u, c, f, g} is independent. Then we have k, k′ such that hgkauc, hfk′buc ⊧ f̂ĝĥâb̂ĉ. From c ⌣^{|} fgh, we have ba ⌣^{|} _{fg} kk′. From b ⌣^{|} fgh and u, a ∈ bdd(kk′b), it follows that ba ⌣^{|} _{kk′} fg and thus e ∈ bdd(kk′) (†). On the other hand, f ⌣^{|} _{h} g implies k′ ⌣^{|} _{h} gk and k′ ⌣^{|} gk. Hence {g, k, k′} is independent. Similarly {f, k, k′} is independent. Then from (†), e ⌣^{|} f, e ⌣^{|} g.

(1)(c) This easily follows from (a),(b), and (*).

(2) Similar to (1).

(3) There are c, c′, b and a ⌣^{|} fgf′g′ such that c ∈ f(a) ∩ g(b), c′ ∈ f′(a) ∩ g′(b). Hence ab ⌣^{|} _{fg} f′g′ and ab ⌣^{|} _{f′g′} fg. Therefore e = Cb(ab/fg) and e′ = Cb(ab/f′g′) are interbounded (**). From (1)(a), f, g are interbounded over e, and so are f′, g′ over e′. Hence it follows from (**), g′ ∈ bdd(f′e′) = bdd(f′e) ⊆ bdd(f′fg) and similarly for the other relations. □

The proof of 2.3.1(b) above is essentially due to Frank O. Wagner.

Lemma 2.4. (1) For fg, f′g′ ⊧ Γ_{q}, R(fg, f′g′) iff there are b and a ⌣^{|} fgf′g′ such that

f(a) ∩ g(b) ≠ ∅, f′(a) ∩ g′(b) ≠ ∅ and fg ⌣^{|} _{e} f′g′ where e = Cb(ba/fg).

(2) Given independent f, g ⊧ q, there are f′, g′ such that R(fg, f′g′).

(3) Above (1)(2) hold for R^{p}, R^{pq}, as well.

Proof. (1) (⇒) Note that since a ⌣^{|} fgf′g′, ba ⌣^{|} _{fg} f′g′, ba ⌣^{|} _{f′g′} fg. Hence e, e′ = Cb(ba/f′g′) are interbounded. Then due to 2.3.1(a), f′, g′ are interbounded over e. Now from fg ⌣^{|} f′, fg ⌣^{|} _{e} f′, thus fg ⌣^{|} _{e} f′g′.

(⇐) Again since a ⌣^{|} fgf′g′, e, e′ are interbounded. Then from fg ⌣^{|} _{e} f′g′, equivalently fg ⌣^{|} _{e′} f′g′, and 2.3.1(b), {f, g, f′, g′} is 3-independent.

(2) By amalgamation there is c ∈ ran(f)∩ran(g) such that c ⌣^{|} fg. Choose a ∈ g^{−1}(c), b ∈ f^{−1}(c). Then, by the extension axiom, we have f′g′ such that {ab, fg, f′g′} is e-independent where e = Cb(ba/fg) and f′g′ ≡_{abe} fg. Then the right hand side of (1) follows easily.

(3) Clear. □

The following lemma is crucial to our argument.

Lemma 2.5. Let R(fg, f′g′) hold. Namely, {f, g, f′, g′} is 3-independent, and we can find c, c′, d and a ⌣^{|} fgf′g′ such that c ∈ f(a)∩g(d), c′ ∈ f′(a)∩g′(d). Then there are h, h′ ⊧ p and b such that c ∈ h(b), c′ ∈ h′(b), b ⌣^{|} hh′ff′gg′, {f, h, f′, h′} is 3-independent, and hh′ ⌣^{|} _{ff′} gg′. It follows that R^{pq}(hf, h′f′) and R^{pq}(hg, h′g′).

f  g h  f′  g′ h′ c c′  d a b

Proof. Note since fac, f′ac′ ⊧ ĝâĉ, there are k_{0}, b_{0}; k_{1}, b_{1} such that f′k_{0}ab_{0}c′, fk_{1}ab_{1}c ⊧ ĝĥâb̂ĉ. Then by amalgamation, we have kb, hh′ such that

kb ⊧ tp(k_{0}b_{0}/af′) ∪ tp(k_{1}b_{1}/af), kb ⌣^{|} _{a} ff′ and h′f′kabc′, hfkabc ⊧ f̂ĝĥâb̂ĉ.

We can further assume that hh′ ⌣^{|} _{ff′a} gg′ (*). Now from kb ⌣^{|} _{a} ff′ and a ⌣^{|} fgf′g′, it follows ab ⌣^{|} _{k} ff′, b ⌣^{|} _{k} ff′hh′, and b ⌣^{|} ff′hh′. Then from (*), we have a ⌣^{|} _{hh′ff′} gg′, ab ⌣^{|} _{hh′ff′} gg′, and b ⌣^{|} ff′gg′hh′. Moreover from f ⌣^{|} _{k} f′, we have that hf ⌣^{|} _{k} h′f′, and thus {f, h, f′, h′} is 3-independent. Then since hh′ ⌣^{|} _{ff′} gg′, {g, h, g′, h′} is 3-independent as well. Therefore R^{pq}(hf, h′f′) and R^{pq}(hg, h′g′). □

Now define R′ = (R′)^{q} by R′(fg; f′g′) iff f, g, f′g′ ⊧ q, {f, g, f′, g′} 3-independent, and for any a, b such

that f(a) ∩ g(b) ≠ ∅ and a ⌣^{|} fg, there are a′b′ ≡_{fg}^{L} ab such that a′ ⌣^{|} fgf′g′,
f′(a′) ∩ g′(b′) ≠ ∅.

Again we also define (R′)^{p}, (R′)^{pq} by substituting f, g, f′g′ ⊧ p or f, f ⊧ p, gg′ ⊧ q, respectively.

Assumption For the rest of this paper, we assume that T has 4-CA, or more weakly, model-4-CA. Recall that both implies 4-CA over models. Notation For bounded closed sequences a_{1}, a_{2}, ..., we use \overline{a_{1}a_{2}...} to denote some sequence of bdd(a_{1}a_{2}...) extending the orderings of a_{1}, a_{2}, .... More precisely, if x_{i} ↦ a_{i} is a bijective indexing of a_{i} by the variable x_{i}, then \overline{a_{1}a_{2}...} is an indexing x(⊇ x_{1} ∪ x_{2} ∪ ...) ↦ bdd(a_{1}a_{2}...) extending each x_{i} ↦ a_{i}.

We shall prove that R and R′ are equivalent. Note that clearly R′ implies R.

Theorem 2.6. R and R′ are equivalent.

Proof. We shall show that R implies R′. Let R(fg, f′g′) hold. We use Lemma 2.5 with the same notation. We then have cc′, abd, hh′ such that c ∈ f(a) ∩ h(b) ∩ g(d); c′ ∈ f′(a) ∩ h′(b) ∩ g′(d); b ⌣^{|} fghf′g′h′; f′ ∈ bdd(fhh′), g′ ∈ bdd(hgh′); and {f, g, h, h′} is independent. Now let \overline{hf}, \overline{hg}, \overline{fg}, \overline{hbc} = bdd(hb) = bdd(hc), \overline{fac} = bdd(fc) = bdd(fa), \overline{gdc} = bdd(gc) = bdd(gd) be sequences of boundedly closed sets extending the boundedly closed sequences f, g, h, c, a, b, d (See Notation above 2.6). Since hbc ≡ h′bc′ we also have \overline{h′bc′} ≡ \overline{hbc}.

Now, to show R′(fg, f′g′), assume there are a_{1}, d_{1}, c_{1} such that c_{1} ∈ f(a_{1}) ∩ g(d_{1}) and c_{1} ⌣^{|} fg. We also have \overline{fa_{1}c_{1}} ≡ \overline{fac} ≡ \overline{gd_{1}c_{1}} ≡ \overline{gdc}. Then by 4-amalgamation (over a model which is named), we can find c_{2} such that

tp(\overline{fgh}) ∪ tp(\overline{fhc}) ∪ tp(\overline{ghc}) ∪ tp(\overline{fgc_{1}}) ⊆ tp(\overline{fghc_{2}}),

where \overline{fa_{1}c_{1}}, \overline{gd_{1}c_{1}} ⊂ \overline{fgc_{1}}; and \overline{fac}, \overline{hbc} ⊂ \overline{fhc}. Hence there are b_{2}, a_{2}, d_{2} such that

c_{2} ∈ f(a_{2}) ∩ g(d_{2}) ∩ h(c_{2}).

In particular,

\overline{fa_{2}c_{2}} \overline{gd_{2}c_{2}} \overline{fg} ≡ \overline{fa_{1}c_{1}} \overline{gd_{1}c_{1}} \overline{fg}; \overline{fh} \overline{hb_{2}c_{2}} ≡ \overline{fh} \overline{hbc}; and \overline{gh} \overline{hb_{2}c_{2}} ≡ \overline{gh} \overline{hbc} (*).

What we are going to amalgamate next are the following strong types:

Lstp(\overline{hb_{2}c_{2}}/fg; h), Lstp(\overline{hbc}/h′f; h) and Lstp(\overline{hbc}/h′g; h).

(Equivalently tp(\overline{c_{2}fg; h}), tp(\overline{ch′f; h}), tp(\overline{ch′g; h}) and tp(\overline{h′fg; h}), where h, b_{2}, c_{2} ⊂ \overline{c_{2}h}, and h, b, c ⊂ \overline{ch}.) The base parameter is h (here is the place where we need model-4-CA, since indeed the parameter is Mh; also see the example after 1.9), and each realization is boundedly closed over the parameter. Each type does not fork over h. Additionally due to (*), it can be seen that the 3 strong types are h-compatible. Hence we have \overline{hb_{3}c_{3}}, a generic solution of the types.

Now we analyze consequences obtained from the generic solution. Note first that we have \overline{fhabc} = bdd(f, bc; h) extending \overline{fac}, \overline{hbc}. Also since c′ ∈ bdd(h′, bc; h), we have \overline{hh′bcc′} = bdd(h′, bc; h) extending \overline{hbc}, \overline{h′bc′}. Similarly there is \overline{hgbdc} = bdd(g, bc; h) extending \overline{hbc}, \overline{gdc}. Note here that the existence of the generic solution indeed says that there exist elementary maps h̃_{1}, h̃_{2}, h̃_{3} with

dom(h̃_{1}) = bdd(f, bc; h) bdd(h′, bc; h) bdd(fh′; h),
dom(h̃_{2}) = bdd(f, b_{2}c_{2}; h) bdd(g, b_{2}c_{2}; h) bdd(fg; h),
dom(h̃_{3}) = bdd(h′, bc; h) bdd(g, bc; h) bdd(h′g; h),

fixing bdd(fh′; h), bdd(fg; h) bdd(gh′; h), respectively, such that h̃_{1}(\overline{hbc}) = h̃_{2}(\overline{hb_{2}c_{2}}) = h̃_{3}(\overline{hbc}) = \overline{hb_{3}c_{3}}, the generic solution. Moreover they are compatible with elementary maps

sending \overline{fhabc} → \overline{fha_{2}b_{2}c_{2}}, \overline{hgbdc} → \overline{hgb_{2}d_{2}c_{2}} and \overline{hh′bcc′} ⟶^{id} \overline{hh′bcc′}. (In particular, maps

h̃_{1}⌈bdd(h′, bc; h) = h̃_{3}⌈bdd(h′, bc; h).) Hence there are a_{3}, c′_{3}, d_{3} such that a_{3} = h̃_{1}(a) = h̃_{2}(a_{2}), c′_{3} = h̃_{1}(c′) = h̃_{3}(c′), d_{3} = h̃_{2}(d_{2}) = h̃_{3}(d) and

(1) \overline{fha_{3}b_{3}c_{3}} \overline{hh′b_{3}c_{3}c′_{3}} ≡_{bdd(fh′;h)} \overline{fhabc} \overline{hh′bcc′};

(2) \overline{fha_{3}b_{3}c_{3}} \overline{hgb_{3}d_{3}c_{3}} ≡_{bdd(fg;h)} \overline{fha_{2}b_{2}c_{2}} \overline{hgb_{2}d_{2}c_{2}};

(3) \overline{hh′b_{3}c_{3}c′_{3}} \overline{hgb_{3}d_{3}c_{3}} ≡_{bdd(gh′;h)} \overline{hh′bcc′} \overline{hgbdc}.

Then by (*) and (2), a_{1}d_{1} ≡_{fg}^{L} a_{2}d_{2} ≡_{fg}^{L} a_{3}d_{3}. Also since f′ ∈ bdd(fh′h), from (1), c′_{3} ∈ f′(a_{3}) ∩ h′(b_{3}) and c_{3} ∈ f(a_{3}). Note that g′ ∈ bdd(gh′h), since R(hg, h′g′) holds. Then, similarly from (3), c′_{3} ∈ g′(d_{3}) and c_{3} ∈ g(d_{3}). Therefore R′(fg, f′g′). □

3. Type-definability of the transitive closure of R

In this section, we use R ≡ R′ (Theorem 2.6) to prove that, the transitive closure of R is type-definable. The proof is similar to the proof in [4] that the transitive closure of the relation ∼_{1} forms a hyperimaginary canonical base, (or the improvement of this proof in [17, 3.3.1]).

Let R̃ be the transitive closure R. We remark that if both {a, b, c, d}, {a′, b′, c, d} are 3-independent and ab ⌣^{|} _{cd} a′b′, then {a, b, a′, b′} is also 3-independent.

Lemma 3.1. Suppose that R(fg, hk), R(hk, f′g′) and fg ⌣^{|} _{hk} f′g′. Then R(fg, f′g′) and fg ⌣^{|} _{f′g′} hk.

Proof. By the previous remark, {f, g, f′, g′} is 3-independent (*). Now since R(fg, hk), there are b and a ⌣^{|} fghk such that f(a) ∩ g(b) ≠ ∅, h(a) ∩ k(b) ≠ ∅ (so ab ⌣^{|} _{hk} fg). Then since R′(hk, f′g′), there are a′b′ ≡_{hk}^{L} ab such that a′ ⌣^{|} hkf′g′, f′(a′) ∩ g′(b′) ≠ ∅. Hence by amalgamation, we have

a′′b′′ ⊧ Lstp(ab/hk, fg) ∪ Lstp(a′b′/hk, f′g′), and a′′b′′ ⌣^{|} _{hk} fgf′g′.

It follows then a′′ ⌣^{|} fgf′g′ and f(a′′) ∩ g(b′′) ≠ ∅, f′(a′′) ∩ g′(b′′) ≠ ∅. This with (*) says R(fg, f′g′). It remains to show fg ⌣^{|} _{f′g′} hk. Since g ⌣^{|} hkf′g′, g ⌣^{|} _{f′g′} hk. Now by 2.3.3, f ∈ bdd(gf′g′), and therefore fg ⌣^{|} _{f′g′} hk. The proof is finished. □

Theorem 3.2. The following are equivalent.

(1) R̃(f̄, ḡ).

(2) For some h̄, R(h̄, f̄) and R(h̄, ḡ).

(3) For some h̄ with h̄ ⌣^{|} f̄ ḡ and h̄ ⌣^{|} ḡ f̄, R(h̄, f̄) and R(h̄, ḡ).

Proof. It suffices to show (1) implies (3). We prove this by induction on the length of an R-chain. Note that 2.4.2 gives the induction step for length 0. Now assume that there are f̄, f̄_{n}, h̄′ such that R̃(f̄, f̄_{n}) with the R-chain length n and R(f̄_{n}, h̄′). By the induction hypothesis for n, there is h̄ such h̄ ⌣^{|} f̄ f̄_{n} and h̄ ⌣^{|} f̄_{n} f̄(*), R(h̄, f̄) and R(h̄, f̄_{n}). By extension, we can assume h̄ ⌣^{|} f̄ ; f̄_{n} h̄′ (**). Then by (*), h̄ ⌣^{|} f̄_{n} h̄′f̄ (***). In particular, h̄ ⌣^{|} f̄_{n} h̄′. Hence from the lemma 3.1, R(h̄, h̄′) and h̄ ⌣^{|} h̄′ f̄_{n}. Then it follows from (***), h̄ ⌣^{|} h̄′ f̄. Moreover again by (*)(**), we have h̄ ⌣^{|} f̄ h̄′. Hence the induction step for n + 1 is shown. □

4. The generic group operation on Γ/R̃

Recall that in section 2, we define Γ(xy) = Γ_{q}(xy) = q(x) ∧ q(y) ∧ x ⌣^{|} y. Now since R is symmetric, clearly R̃ is an equivalence relation on Γ. By putting (R̃(x̄, ȳ)∧Γ(x̄)∧Γ(ȳ))∨x̄ = ȳ, we can extend R̃ to a type-definable equivalence relation on the whole universe. We shall find the canonical hyperdefinable group from the hyperdefinable generic group operation on Γ/R̃. First we state some more properties of R and R̃.

Lemma 4.1. Let f̄ = f_{1}f_{2}, ḡ ⊧ Γ, and let e = f̄/R̃.

(1) R̃(f̄, ḡ) and f̄⌣^{|} _{e} ḡ iff R(f̄, ḡ).

(2) For a, b such that f_{1}(a) ∩ f_{2}(b) ≠ ∅ and a ⌣^{|} f̄, e is interbounded with Cb(ab/f̄).

(3) f_{i} ⌣^{|} e, and f_{1}, f_{2} are interbounded over e.

Proof. (1) (⇒) By 3.2, there is h̄ such that h̄ ⌣^{|} f̄ ḡ, h̄ ⌣^{|} ḡ f̄, R(h̄, f̄) and R(h̄, ḡ). Then since f̄⌣^{|} _{e} ḡ, f̄⌣^{|} h̄ ḡ, and then by 3.1, R(f̄, ḡ).

(⇐) R(f̄, ḡ) implies R̃(f̄, ḡ). By the extension axiom, there is ḡ′ ⊧ tp(ḡ/e) such that ḡ′ ⌣^{|} _{e} ḡ (*). Hence R̃(ḡ, ḡ′) and by the proof of (⇒), R(ḡ, ḡ′). Again, by extension, we can assume that f̄⌣^{|} _{eḡ} ḡ′. Hence f̄⌣^{|} ḡ ḡ′ and then by 3.1, f̄⌣^{|} _{}_{ḡ′} ḡ. Therefore by (*), f̄⌣^{|} _{e} ḡ.

(2) Let e_{1} = Cb(ab/f̄). By extension, there is f̄′ ⊧ tp(f̄/abe_{1}) such that f̄⌣^{|} _{abe_{1}} f̄′. Hence {ab, f̄, f̄′} is e_{1}-independent and from 2.4.1, R(f̄, f̄′) and e_{1} = Cb(ab/f̄′). Then by (1), f̄⌣^{|} _{e} f̄′ (**). Let e_{2} = Cb(f̄/f̄′). Then due to (**), e_{2} ∈ bdd(e). Moreover since f̄⌣^{|} _{e_{2}} f̄′, e ⌣^{|} _{e_{2}} f̄′, and e ∈ dcl(f̄′), e_{2} ∈ bdd(f̄′), we have e ∈ bdd(e_{2}). Thus bdd(e) = bdd(e_{2}). Similarly since f̄⌣^{|} _{e_{1}} f̄′, it can be too seen bdd(e_{1}) = bdd(e_{2}). Therefore bdd(e_{1}) = bdd(e).

(3) follows from (2) and 2.3.1. □

The proof of the following lemma only uses 4-CA over a model.

Lemma 4.2. Let R(gh, vw). Then for any c ∈ g(a), d ∈ v(a) with a ⌣^{|} gv, there are c′a′d′ ≡_{gv}^{L} cad and b′ such that a′ ⌣^{|} ghvw, c′ ∈ h(b′), d′ ∈ w(b′).

Proof. By 2.5, there are a_{0}, b_{0}, c_{0}, d_{0}, t_{0} and f, u such that c_{0} ∈ g(a_{0}) ∩ f(t_{0}) ∩ h(b_{0}), d_{0} ∈ v(a_{0}) ∩ u(t_{0}) ∩ w(b_{0}), t_{0} ⌣^{|} fghuvw, {f, g, u, v} is 3-independent, and fu ⌣^{|} _{gv} hw. It follows R^{pq}(fh, uw), R^{pq}(fg, uv) and gv ⌣^{|} _{fu} hw (*). Let e = bdd(Cb(t_{0}a_{0}/fg)). Then from 2.4.3, e = bdd(Cb(t_{0}a_{0}/uv)) and fg ⌣^{|} _{e} uv. Similarly for k = bdd(Cb(t_{0}b_{0}/fh)) = bdd(Cb(t_{0}b_{0}/uw)), fh ⌣^{|} _{k} uw. We now let \overline{ea_{0}t_{0}} = bdd(ea_{0}t_{0}) = bdd(ea_{0}), \overline{evu} = bdd(ev), \overline{egf} = bdd(eg) \overline{ga_{0}c_{0}} = bdd(ga_{0}), \overline{va_{0}d_{0}} = bdd(va_{0}), \overline{gv} be sequences of bounded closed sets (see Notation above 2.6). Note that there are sequences \overline{gac}, \overline{vad} such that

\overline{ga_{0}c_{0}} ≡ \overline{gac} and \overline{va_{0}d_{0}} ≡ \overline{vad}.

Now by 4-amalgamation over the named model, there is a_{1} such that

tp(\overline{egv}) ∪ tp(\overline{ega_{0}}) ∪ tp(\overline{eva_{0}}) ∪ tp(\overline{gva}) ⊆ tp(\overline{egva_{1}}),

and {e, g, v, a_{1}} is independent, where

\overline{efg}, \overline{ga_{0}c_{0}}, \overline{ea_{0}t_{0}} ⊂ \overline{ega_{0}}; \overline{evu}, \overline{va_{0}d_{0}}, \overline{ea_{0}t_{0}} ⊂ \overline{eva_{0}}; and \overline{gac}, \overline{vad} ⊂ \overline{gva}.

In particular, there are c_{1}(⊂ \overline{ga_{1}}), d_{1}(⊂ \overline{va_{1}}), t_{1}(⊂ \overline{ea_{1}}) (all are in \overline{egva_{1}}) such that

\begin{matrix}(1) & \overline{ga_{1}c_{1}} \overline{ea_{1}t_{1}} ≡\overline{egf} \overline{ga_{0}c_{0}} \overline{ea_{0}t_{0}}; \\ (2) & \overline{va_{1}d_{1}} \overline{ea_{1}t_{1}} ≡\overline{evu} \overline{va_{0}d_{0}} \overline{ea_{0}t_{0}}; \\ (3) & \overline{ga_{1}c_{1}} \overline{va_{1}d_{1}} ≡\overline{gv} \overline{gac} \overline{vad}; \end{matrix}

and, from 2.3.1, {t_{1}, f, u, e} is independent (**). Due to (1)(2), ft_{0}c_{0} ≡ ft_{1}c_{1} and ut_{0}d_{0} ≡ ut_{1}d_{1}. Hence there are enumerations such that

\overline{ft_{0}c_{0}} ≡ \overline{ft_{1}c_{1}} and \overline{ut_{0}d_{0}} ≡ \overline{ut_{1}d_{1}}.

Similarly by 4-CA, we have t_{2}; c_{2}, d_{2}, b_{2} such that

tp(\overline{kfu}) ∪ tp(\overline{kft_{0}}) ∪ tp(\overline{kut_{0}}) ∪ tp(\overline{fut_{1}}) ⊆ tp(\overline{kfut_{2}}),

\begin{matrix}(4) & \overline{ft_{2}c_{2}} \overline{kb_{2}t_{2}} ≡\overline{khf} \overline{ft_{0}c_{0}} \overline{kb_{0}t_{0}}(⊂ \overline{kft_{0}}); \\ (5) & \overline{ut_{2}d_{2}} \overline{kb_{2}t_{2}} ≡\overline{kwu} \overline{ut_{0}d_{0}} \overline{kb_{0}t_{0}}(⊂ \overline{kut_{0}}); \\ (6) & \overline{ft_{2}c_{2}} \overline{ut_{2}d_{2}} ≡\overline{fu} \overline{ft_{1}c_{1}} \overline{ut_{1}d_{1}}(⊂ \overline{fut_{1}}); \end{matrix}

and {t_{2}, f, u, k} is independent. Hence due to (*),(**),2.3.1(a) and (6), we can apply amalgamation to have

d′c′t′ ⊧ Lstp(d_{1}c_{1}t_{1}/fu; gv) ∪ Lstp(d_{2}c_{2}t_{2}/fu; hw)

such that t′ ⌣^{|} _{fu} ghvw (†). Then there are the desired a′, b′ such that

(7) d′c′t′a′ ≡_{fguv}^{L} d_{1}c_{1}t_{1}a_{1}, and d′c′t′b′ ≡_{fhuw}^{L} d_{2}c_{2}t_{2}b_{2}.

Hence it follows from (†), a′ ⌣^{|} ghvw. Moreover by (3)(7), c′a′d′ ≡_{gv}^{L} cad; by (4)(7), c′ ∈ h(b′); and by (5)(7), d′ ∈ w(b′). The proof is finished. □

We are ready to define the promised generic operation on Γ/R̃. Let

▪(x_{1}y_{1}, x_{2}y_{2}; x_{3}y_{3}) := ∃xyz(R̃(x_{1}y_{1}; xy) ∧ R̃(x_{2}y_{2}; yz) ∧ R̃(xz; x_{3}y_{3}) ∧ ⋀_{i=1,2,3} Γ(x_{i}y_{i})
∧ x_{1}y_{1}/R̃ ⌣^{|} x_{2}y_{2}/R̃ ∧ {x, y, z} is independent).

Note that for x_{1}y_{1}, x_{2}y_{2} ⊧ Γ, x_{1}y_{1}/R̃ ⌣^{|} x_{2}y_{2}/R̃ iff ∃x′_{1}y′_{1}x′_{2}y′_{2}R̃(x_{1}y_{1}; x′_{1}y′_{1}) ∧ R̃(x_{2}y_{2}; x′_{2}y′_{2}) ∧ {x′_{1}, y′_{1}, x′_{2}, y′_{2}} independent. Hence ▪ is a partial type over ∅.

Claim 1. The relation ▪ is a hyperdefinable partial type over ∅ such that, for any independent e_{1} = f_{1}g_{1}/R̃, e_{2} = f_{2}g_{2}/R̃ ∈ Γ/R̃, there is e_{3} = fh/R̃ ∈ Γ/R̃ such that (e_{1}, e_{2}; e_{3}) realizes ▪(x_{1}y_{1}, x_{2}y_{2}; x_{3}y_{3}). (See the explanation above 1.4): It suffices to show there exist f, g, h such that e_{1} = fg/R̃, e_{2} = gh/R̃ and {f, g, h} is independent. Now by 4.1.3, e_{i} ⌣^{|} g_{i}, e_{i} ⌣^{|} f_{i} and f_{i}, g_{i} are interbounded over e_{i}. Then, by amalgamation, there exists g ⊧ tp(f_{2}/e_{1})∪tp(g_{1}/e_{2}) and {g, e_{1}, e_{2}} independent. We also have f, h such that fg ≡_{e_{1}} f_{1}g_{1}, gh ≡_{e_{1}} f_{2}g_{2}. Then, {f, g, h} is independent too.

Claim 2. e_{1} ▪ e_{2} = e_{3} = fh/R̃ does not depend on the choice of f, g, h, i.e. e_{1} ▪ e_{2} is unique: Suppose there are f′, g′, h′ such that e_{1} = f′g′/R̃, e_{2} = g′h′/R̃ and {f′, g′, h′} is independent. Then we can also find independent {u, v, w} such that u ⌣^{|} _{e_{1}e_{2}} fghf′g′h′ and e_{1} = uv/R̃, e_{2} = vw/R̃. Hence, from 4.1.2, uvw ⌣^{|} _{e_{1}e_{2}} fghf′g′h′, and {f, u, e_{1}, e_{2}} is independent (‡). We shall prove that R(fh, uw). (Then the by the same proof, R(f′h′, uw) and thus R̃(fh, f′h′).) Note from (‡) and 4.1.1, R(fg, uv) and R(gh, vw). Hence there are b, a ⌣^{|} fguv and c ∈ g(a) ∩ f(b), d ∈ v(a) ∩ u(b). Moreover, by 4.2, we have a′b′c′d′ such that c′a′d′ ≡_{gv}^{L} cad and a′ ⌣^{|} ghvw, c′ ∈ g(a′) ∩ h(b′), d′ ∈ v(a′) ∩ w(b′). Now again due to (‡) and 2.3.1, we have e_{1} ⌣^{|} _{gv} e_{2}, fu ⌣^{|} _{gv} hw (∗), and a ⌣^{|} _{gv} fu, cad ⌣^{|} _{gv} fu, c′a′d′ ⌣^{|} _{gv} hw. Hence, by amalgamation, we have

c_{1}a_{1}d_{1} ⊧ Lstp(cad/gv; fu) ∪ Lstp(c′a′d′/gv; hw),

such that a_{1} ⌣^{|} _{gv} fhuw (∗∗). Then we have b_{1}, b′_{1} such that

c_{1}a_{1}d_{1}b_{1} ≡_{fguv} cadb, c_{1}a_{1}d_{1}b′_{1} ≡_{ghvw} c′a′d′b′.

Thus, c_{1} ∈ f(b_{1}) ∩ h(b′_{1}), d_{1} ∈ u(b_{1}) ∩ w(b′_{1}) and from (∗∗), b_{1} ⌣^{|} fhuw. Moreover from (∗) and the remark above 3.1, {f, h, u, w} is 3-independent. Therefore R(fh, uw), as desired.

Claim 3. This generically given group satisfies the genericity properties in [17, 4.7.1]: Note that since {f, g, h} independent, it follows for i = 1, 2, e_{i} ⌣^{|} e_{1} ▪ e_{2}. For generic associativity, let {k_{1}, k_{2}, k_{3}} be independent realizations of Γ/R̃. Subclaim. There exists independent {h_{1}, h_{2}, h_{3}, h_{4}} such that h_{1}h_{2}/R̃ = k_{1}, h_{2}h_{3}/R̃ = k_{2} and h_{3}h_{4}/R̃ = k_{3}: As in the proof of Claim 1, we can find h_{2}, h_{3}, h_{4} independent such that h_{2}h_{3}/R̃ = k_{2} and h_{3}h_{4}/R̃ = k_{3}. Now, for h′_{1}h′_{2}/R̃ = k_{1}, amalgamation of Lstp(h′_{2}/k_{1}) and Lstp(h_{2}/k_{2}k_{3}) gives the subclaim. Now, by the subclaim, k_{1} ▪ k_{2} = h_{1}h_{3}/R̃ and k_{2} ▪ k_{3} = h_{2}h_{4}/R̃. Then k_{1} ▪ (k_{2} ▪ k_{3}) = h_{1}h_{4}/R̃ and (k_{1} ▪k_{2})▪k_{3} = h_{1}h_{4}/R̃ as well. Finally it can be easily seen that ▪ is generically surjective. Hence Claim 3 is verified.

Therefore we have the following;

Theorem 4.3. Given the group configuration, there exists a canonical hyperdefinable group and a definable bijection mapping Γ/R̃ to the generic types of the group such that ▪ is mapped to the group multiplication generically.

In recent work [10], the completion of the group configuration theorem is achieved. Namely, a hyperdefinable group action of the group on a hyperdefinable set is recovered, so that the induced bounded quadrangle from the hyperdefinable homogeneous space is equivalence to the original bounded quadrangle.

5. 1-based theories

One application of 4.3 is the following result. This extends the theorem [3, 3.23] that, in any 1-based non-trivial ω-categorical simple T, an infinite vector space over some finite field is definably recovered in M^{eq}. Recall that T is non-trivial if there are hyperimaginaries a_{1}, a_{2}, a_{3} and A such that for 1 ≤ i < j ≤ 3, a_{i}, a_{j} are independent over A whereas {a_{1}, a_{2}, a_{3}} is dependent over A.

Theorem 5.1. Suppose that T is 1-based, non-trivial, having model-4-CA. Then there is a hyperdefinable infinite bounded-by-Abelian group V over a model M of SU-rank 1 generic types. Moreover for the bounded subgroup V_{0} = V ∩ bdd(M), V/V_{0} forms a vector space over the division ring R of bdd(M)-endomorphisms of V such that for b, a_{1}, ..., a_{n} ∈ V, b ∈ bdd(a_{1}...a_{n}) iff b + V_{0} = α_{1}(a_{1} + V_{0}) + ... + α_{n}(a_{n} + V_{0}) for some α_{i} ∈ R.

Proof. By the proof of Lemma 3.22 in [3], there exists a non-trivial rank-1 Lstp p over some model M. For convenience, let M = ∅ after naming the model. As p is non-trivial, there exists {a, b, c} realizing p such that b, c is independent and a ∈ bdd(b, c) \ bdd(b) ∪ bdd(c). Let yx realize tp(ab/c) with yx ⌣^{|} _{c} ab. Then dim(ay/bx) = 1 as y ∈ bdd(abx) and a ⌣^{|} bx. Let z = Cb(Lstp(ay/bx)), then by 1-basedness, z ∈ bdd(ay) ∩ bdd(bx). Moreover, by a straightforward rank calculation, SU(z) = 1. This gives a bounded quadrangle (a, b, c, x, y, z). Now by Theorem 4.3, we obtain a hyperdefinable group G over ∅ such that the generic types all have SU-rank 1. The group G is 1-based since the underlying theory is 1-based. Now we use the following fact [17, 4.8.4],

Fact 5.2. Suppose G is an 1-based group hyperdefinable over ∅ in a simple theory. Then for the normal subgroup G_{∅}^{0}, the smallest ∅-hyperdefinable subgroup of bounded index, the commutator subgroup (G_{∅}^{0})′ of G_{∅}^{0} has boundedly many elements and contained in the center of G_{∅}^{0}.

Therefore, if we set G^{0} = V, then V is the desired bounded-by-Abelian hyperdefinable group. Note that by above V′ is contained in the normal subgroup V_{0} = V ∩ bdd(∅). Indeed again from [17, 4.8.18], the Abelian group V/V_{0} forms a vector space over a division ring R of bdd(∅)-endomorphisms of V, and dependence in V/V_{0} is given by linear dependence of the vector space. □

References

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[3] T. de Piro, B.Kim, ‘The geometry of 1-based minimal types’, Transactions of American Math. Soc., 355 (2003) 4241-4263.

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[10] B. Kim, ‘Recovering the hyperdefinable group action in the group configuration theorem’, submitted.

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[13] A. Kolesnikov, ‘Generalized amalgamation in simple theories and characterization of dependence in non-elementary classes’, Ph. D. Thesis, Carnegie Mellon University, 2004.

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[17] F. O. Wagner, Simple theories, Kluwer Academic Publishers, Dordrecht (2000).

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Dipartimento di Matematica e Informatica, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy

Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul 120-749, South Korea

MATX 1220, Mathematics Department, 1984 Mathematics Road, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

  E-mail address: tristam.depiro@unicam.it
  E-mail address: bkim@yonsei.ac.kr
  E-mail address: jessica@math.ubc.ca