Another stable group

Andreas Baudisch^{1} 2. Mathematisches Institut Freie Universität Berlin

Arnimallee 3 14195 Berlin

baudisch@math.fu-berlin.de

May 17, 1995

1Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft

Abstract

In [3] an uncountably categorical group is constructed that has a non-locally-modular geometry and does not allow the interpretation of a field. We consider a system Δ of elementary axioms fulfilled by some special subgroups of the group above. We show that Δ is complete, stable, but not superstable. It is even not a R-group in the sense of [10].

1 Introduction

A structure with a stable elementary theory is called a stable group if there is a ∅-definable binary function that gives a group operation on the underlying set. If we restrict us to the pure group structure, then we have essentially the following examples. Note that direct sums of stable groups are stable again.

1. Abelian groups Also groups Abelian by finite are stable. Note that every stable onebased group is Abelian by finite, as proved by E.Hrushovski and A.Pillay [6].

2. Algebraic groups over algebraically closed fields Every superstable field is algebraically closed as proved by A.Macintyre for the ω-stable case ( [7]) and by G.Cherlin and S.Shelah for the general case ([4]). Algebraic groups over algebraically closed fields are ω-stable of finite Morley rank. There is a conjecture of G.Cherlin and B.Zil’ber that the converse is true for simple groups.

3. Algebraic groups over separably closed fields Separably closed fields of characteristic p ≠ 0 are the only known examples of stable fields that are not superstable (see C.Wood [11]).

4. Mekler-groups For every stable structure of finite signature A.Mekler [8] has constructed a group such that for the elementary theories T of the both structures the function κ → max{|S_{n}(M)| : M ⊧ T, |M| = κ} is the same. These groups are nilpotent of class 2 and of exponent p where p is a prime greater than 2. Mekler groups cannot have finite Morley rank (U-rank).

5. Free groups F_{c}(p^{n}, κ) in the variety of nilpotent groups of class c of exponent p^{n} with κ many free generators where p is a prime greater than c and κ is infinite. These groups are ω-stable with c dimensions. They have infinite Morley rank. See [1], [2].

6. A new uncountably categorical group In [3] an axiom system Σ for a new uncountably categorical group is given. Σ has a non-locally-modular geometry and does not allow the interpretation of a field.

B.Zil’ber conjectured that every uncountably categorical theory with a nonlocally-modular geometry allows the interpretation of a field. To refute this conjecture E.Hrushovski [5] constructed an uncountably categorical (even strongly minimal) relational structure with a non-locally-modular geometry that does not allow the interpretation of a group. The group in 6) above is a further counterexample to Zil’bers Conjecture in the language of group theory. It is not known whether Σ can be an algebraic group over an algebraically closed field. We conjecture that for algebraic groups over algebraically closed fields Zil’bers Conjecture is true. If this was true, it would imply that Σ is not an algebraic group over an algebraically closed field.

Let p be a prime greater than 2. Let G be a model of Σ. We use Z(G) or short Z to denote the center. Let [x, y] = x^{−1}y^{−1}xy and [X, Y] = {[x, y] : x ∈ X, y ∈ Y }. ⟨X⟩ is used to denote the subgroup generated by X in G. Then G′ = ⟨[G, G]⟩ is the commutator subgroup of G. If X = {a_{1}, …, a_{n}} or X is a tuple (a_{1}, …, a_{n}), then we write ⟨a_{1}, …, a_{n}⟩ only. The first two axioms in Σ say:

(Σ1) G is a nilpotent group of class 2. G has exponent p.

(Σ2) ∀x ∉ Z(G)∀z ∈ Z(G)∃y([x, y] = z).

A weaker but not elementary version of (Σ2) is the following:

(Σ2)^{w} Z(G) = G′.

Let G be the category of groups that satisfy (Σ1) and (Σ2)^{w}. In [3] an auxiliary category S is introduced. The objects are vector spaces V over the finite field F_{p} with p elements together with a subspace of relations N(V) of the exterior square ∧^{2}V. There is a functorial correspondence between G and S that is 1 - 1 at the level of isomorphism types of objects. In S morphisms are lost. An S-structure corresponding to a countable ω-saturated model of Σ can be constructed by successive amalgamation of finite objects in S that satisfy some further conditions. This process starts with some finite (V, N(V)) where N(V) = ⟨0⟩. We are interested in the subgroup that is constructed using the simplest steps only. That means: If (V_{n}, N(V_{n})) is construced in the nth round, then V_{n+1} = V_{n} ⊕ ⟨a⟩ and N(V_{n+1}) = N(V_{n}) or N(V_{n+1}) is generated by one relation Φ + (c ∧ a) where c ∈ V_{n}, Φ ∈ ∧^{2}V_{n}, and there is no Φ + (c ∧ x) in N(V_{n}). We assume that all possible problems as described in the step from (V_{n}, N(V_{n})) to (V_{n+1}, N(V_{n+1})) are solved at the end of the construction. The group obtained in this way satisfies the following set Δ of axioms:

(Δ0) There are three elements linearly independent modulo Z.

(Σ1) The group is nilpotent of class 2 and of exponent p.

(Σ2) ∀x ∉ Z∀z ∈ Z∃y([x, y] = z).

(Δ3) Every three elements that are linearly independent modulo the center generate a subgroup isomorphic to F_{2}(p, 3).

(Δ4) For every finite subgroup H with (Σ2)^{w} and dim(H/Z(H)) > 3 there is a subgroup H_{0} with (Σ2)^{w}, dim(H_{0}/Z(H_{0})) + 1 = dim(H/Z(H)),

and dim(Z(H)) ≥ dim(Z(H_{0})) + dim(H_{0}/Z(H_{0})) − 1.

In (Δ4) H/Z(H), H_{0}/Z(H_{0}), Z(H), and Z(H_{0}) are considered as vectorspaces over F_{p}. Furthermore note that if (Σ2)^{w} is true for H and H_{0} and dim(H/Z(H)) = dim(H_{0}/Z(H_{0})) + 1, then dim(Z(H)) ≤ dim(Z(H_{0})) + dim(H_{0}/Z(H_{0})).

We show that Δ is a complete theory. It is stable but not superstable in a very strong sense. If G is a model of Δ and \overline{G} = G/Z(G), then every three elements of \overline{G} are in the algebraic closure of two elements of \overline{G}. Hence Δ is not the theory of a R-group in the sense of F.Wagner [10]. Δ implies all axioms of Σ except (Σ5). Hence all models G of Δ fulfil (Σ3). Therefore all \overline{G} carry a closure operation cl that gives us a Hrushovski-geometry. cl is part of the algebraic closure but the converse is not true. Let A ⊆ B be finite subspaces of \overline{G}. Then B is part of the past of A if there is no subspace B_{0} of B such that A ⊆ B_{0}, dim(B) = dim(B_{0}) + 1, and dim(B′) ≥ dim(B′_{0}) + dim(B_{0}) − 1. B′ is used to denote the subspace ⟨[B, B]⟩ of Z(G). Then past(A) is the essential part of acl(A). The past(A) determines the type of every preimage of A in G.

The paper is organized in the following way. Section 2 is a short summary of the results concerning the new uncountably categorical group in [3]. Hrushovski geometries for groups in G are considered. In section 3 Δ is introduced. We construct the standard models. In section 4 we start the investigation of the algebraic closure in \overline{H} for models H of Δ. In section 5 it is 1 INTRODUCTION 4 shown that the past is contained in the algebraic closure. Non-superstability follows. In section 7 completeness is shown. In section 8 we prove stability.

2 The new uncountably categorical group

E. Hrushovski has introduced a geometry on structures with few relations. In this section we will explain how this can be done for groups that satisfy (Σ1), (Σ2)^{w}, and a property (Σ3), that guarantees that the groups have few relations only. We explain the results of [3].

Let G be a group that satisfies (Σ1) and (Σ2)^{w}. G is the category of these groups. We denote G/Z(G) by \overline{G}. Then G is uniquely determined by the alternating bilinear map [x, y] of \overline{G} × \overline{G} into Z(G), where the image of this map generates Z(G). Every such map defines a group with (Σ1) and (Σ2)^{w} (see [3]). We consider the category B of alternating bilinear maps β : V × V → W, where V and W are vectorspaces over the field with p elements and the image of β generates W. The last condition corresponds to (Σ2)^{w}. The morphisms from (V_{1}, W_{1}; β_{1}) to (V_{2}, W_{2}; β_{2}) are pairs (f,g) with f : V_{1} → V_{2}, g : W_{1} → W_{2} such that β_{2}f = gβ_{1}. Working in this category we lose some morphisms but we have a 1 - 1 - correspondence between the bilinear maps and the groups. For every vectorspace V there exists a free alternating bilinear map ∧ into a vectorspace ∧^{2}V. ∧^{2}V is called the exterior square of V.

Definition 2.1 ∧ is defined by the following property: For every alternating bilinear map β : V × V → W in B there exists some vectorspace homomorphism f_{β} from ∧^{2}V into W, such that β = f_{β}∧.

Let N(β) ⊆ ∧^{2}V be the kernel of f_{β}. Often we write N(V) only. Then (V, W; β) is isomorphic to (V, (∧^{2}V)/N(V)); π) where π is the composition of the canonical maps V × V → ∧^{2}V → ∧^{2}/N(V). Hence we can work in the category S of pairs (V, N(V)) where N(V) is a subspace of ∧^{2}V. The morphisms are maps f : (V_{1}, N_{1}) → (V_{2}, N_{2}) such that the induced map ∧^{2}f between the exterior squares satisfies: ∧^{2}f(N_{1}) = ∧^{2}f(V_{1}) ∩ N_{2}. The categories B and S are equivalent. For an element (M, N(M)) in S we write often M only. If H is a subspace of M, then we consider ∧^{2}H as canonically identified with a subspace of ∧^{2}M. Then N(H) = ∧^{2}H ∩ N(M). We need the following lemma from [3].

Lemma 2.2 Let G and H be groups in G. Assume there is an isomorphism f_{0} of a subgroup G_{0} ∈ G of G onto a subgroup H_{0} ∈ G of H. Let M and N be S-structures corresponding to G and H respectively. Let M_{0} be the substructure of M corresponding to G_{0} and let N_{0} be the substructure of N corresponding to H_{0}. Assume there is a S-isomorphism g of M onto N such that the restriction of g to M_{0} is the S-isomorphism induced by f_{0}. Then there is an isomorphism f of G onto H that extends f_{0} and induces the given S-isomorphism g of M onto N.

An essential function in [5] is

δ(A) = number of points of A − number of links in A,

where A is a finite subset of the considered structure. Now let A be a finite subspace of M where (M, N(M)) is a structure in S. We define analogously

δ(A) = dim(A) − dim(N(A)),

where dim is the linear dimension of the corresponding vectorspaces. To obtain the desired geometries, we assume that all considered S - structures satisfy

(Σ3) For all finite subspaces A ⊆ M we have δ(A) = dim(A), if dim(A) ≤ 3, and δ(A) ≥ 3 otherwise.

We say that a group with Σ1) and (Σ2)^{w} satisfies (Σ3), if the corresponding S-structure does. (Σ3) for groups can be written down as a set of elementary sentences.

Note that N(A) ∩ N(B) = N(A ∩ B) for A and B in M. It follows easily:

Lemma 2.3 For all finite subspaces A and B of M

δ(A + B) ≤ δ(A) + δ(B) − δ(A ∩ B).

We say, that a finite subspace A is selfsufficient in M, if δ(A) ≤ δ(B) for all finite B with A ⊆ B ⊆ M. By (Σ3) every finite A is contained in a finite selfsufficient subspace. Using Lemma 2.3 it is possible to show:

Lemma 2.4 The intersection of two selfsufficient finite subspaces is selfsufficient .

Hence there exists a smallest selfsufficient subspace CSS(A) containing A. It is the intersection of all selfsufficient finite subspaces of M that contain A. We call δ(CSS(A)) = d(A) the geometrical dimension of A and define

a ∈ cl(⟨a_{1}, …, a_{n}⟩) iff d(⟨a_{1}, …, a_{n}⟩) = d(⟨a, a_{1}, …, a_{n}⟩).

We say that B and C are independent over A if d(B + C) − d(C) = d(B) − d(A).

Theorem 2.5 cl defines a pregeometry on M.

As a result of the construction in the new uncountably categorical group cl is the algebraic closure on \overline{G}. An important notion to work in a S - structure M with the property (Σ3) is minimality. Assume A and B are finite subspaces of M with A ⊆ B and A is selfsufficient. B is in cl(A) if and only if δ(A) = δ(B). In this case B is selfsufficient too, and there is a sequence A = B_{0} ⊆ B_{1} ⊆ … ⊆ B_{n} = B with B_{i} ≠ B_{i+1} and δ(B_{i}) = δ(B_{i+1}), that cannot be confined. Therefore we define B is a minimal extension of A (also for non-selfsufficient A), if A ⊆ B, δ(A) = δ(B), and δ(C) > δ(A) for every C with A ⊆ C ⊆ B and A ≠ C ≠ B.

A lot of efforts are necessary to obtain the desired uncountably categorical group G via a special amalgamation process of finite S - structures with selfsufficient embeddings (see [3]). G is a countable saturated model of the following set Σ of axioms that determine a complete theory. Σ is the union of the already known sets of Axioms (Σ1), (Σ2), and (Σ3) with further sets of axioms (Σ 4) and (Σ5). (Σ4) is very complicated (see [3]). It is introduced because of the following important consequence:

Corollary 2.6 Assume that G is a model of Σ and A ⊆ B are finite subspaces in \overline{G}, such that A is selfsufficient in \overline{G} and B is a minimal extension of A. Then there are at most 2(dim(B)) + 1 many realizations of B over A, that are linearly independent over A.

Corollary 2.6 is the key for the proof of acl = cl in \overline{G}. Also axiom (Σ5) we present by a consequence:

Corollary 2.7 Assume that G is a saturated model of Σ. Let A ⊆ B be finite S - structures, that satisfy (Σ3) and (Σ4). A is selfsufficient in B. Then every selfsufficient embedding of A in \overline{G} can be extended to a selfsufficient embedding of B in \overline{G}.

(Σ5) is a set of formulas that describe the statement of Corollary 2.7 for all possible A ⊆ B, where the selfsufficient embeddings of A and B into \overline{G} are replaced by suitable relativations. In the last theorem of this section we summarize the results of [3].

Theorem 2.8 Σ is a consistent complete uncountably categorical theory in the elementary language of group theory. Σ is not countably categorical. It has Morley - rank 2. The geometry of Σ is not locally modular. \overline{G} and Z(G) are strongly minimal sets. In \overline{G} acl defines exactely the introduced Hrushovski-geometry cl. Σ is not CM-trivial. Hence it is not possible to interprete an infinite field in Σ.

3 The standard models

In the introduction the elementary theory Δ is formulated in the elementary language of group theory. It is easier to understand the axioms (Δ3) and (Δ4) if we formulate their meaning for the corresponding S-structures M:

(Δ3) For all linearly independent elements a_{1}, a_{2}, a_{3} in M we have N(⟨a_{1}, a_{2}, a_{3}⟩) = 0.

(Δ4) For every finite subspace A of M with dim(A) > 3 we get some subspace A_{0} of A such that dim(A_{0}) + 1 = dim(A) and dim(N(A)) ≤

dim(N(A_{0})) + 1.

Note that (Δ3) is equivalent to

(Δ3) If a_{1} and a_{2} in M are linearly independent, then a_{1} ∧ a_{2} ∉ N(M).

A S-structure that satisfies (Δ3) and (Δ4) is called a Δ-structure. A Sstructure M is a Δ-structure iff every finite substructure is a Δ-structure.

Lemma 3.1 A finite S-structure A with dim(A) = n + 2 > 2 is a Δstructure if and only if A = ∪_{1≤i≤n}A_{i} where the A_{i} are subspaces, A_{i} ⊆ A_{i+1}, dim(A_{1}) = 3, dim(A_{i+1}) = dim(A_{i}) + 1, N(A_{1}) = ⟨0⟩, and if dim(N(A_{i})) < dim(N(A_{i+1})), then there is some generator b∧a+Φ of N(A_{i+1}) over N(A_{i}), where b generates A_{i+1} over A_{i}, a ∈ A_{i}, Φ ∈ ∧^{2}A_{i} \ N(A_{i}), and there is no x ∈ A_{i} with x ∧ a + Φ ∈ N(A_{i}).

Proof: (→) Successive application of (Δ4) gives the chain A_{1} ⊆ A_{2} ⊆ … ⊆ A_{n}. (Δ3) implies N(A_{1}) = ⟨0⟩ and the uniqueness of solutions x for x ∧ a + Φ ∈ N(A). (←) Use induction on n. □

Definition 3.2 Let M be a Δ-structure.

1. If {a_{1}, …, a_{m}} is a basis of a subspace A of M, then it is called a Δbasis if m < 3 or m = n + 2 and A_{1} = ⟨a_{1}, a_{2}, a_{3}⟩ and for 1 < i ≤ n A_{i} = ⟨a_{1}, …, a_{i+2}⟩ fulfil the conditions in Lemma 3.1.

2. If I is an ordering and X = {x_{α} : α ∈ I} is a basis of the subspace A of M, then X is a Δ-basis of A if every finite subset X_{0} of X is a Δ-basis of ⟨X_{0}⟩.

3. If X = {x_{α} : α ∈ I} is a Δ-basis of A ⊆ M, then for a ∈ A o_{X}(a) = α if a = ∑_{β≤α} r_{β}x_{β} where r_{α} ≠ 0 and r_{β} = 0 for almost all β ≤ α. o_{X}(a) is called the order of a. Often we write o(a) only. I = o(X) = o(⟨X⟩) we call the order of X or of ⟨X⟩

Let X be a subset of a Δ-structure M as in Definition 3.2. Then let X_{<α} be {x_{β} : β < α} and X_{≤α} be {x_{β} : β ≤ α}. Note that X is a Δ-basis of ⟨X⟩ iff for α ∈ I dim(N(X_{≤α})/N(X_{<α})) ≤ 1. From these remarks follows:

Lemma 3.3 Let M be a Δ-structure and X = {x_{α} : α ∈ I} be a Δ-basis of a subspace A of M. Let Y = {y_{α} : α ∈ I} where y_{α} = r_{α}^{α}x_{α} + ∑_{β<α} r_{β}^{α}x_{β} with 0 < r_{α}^{α} < p and almost all r_{β}^{α} = 0. Then Y is a Δ-basis.

If we have the situation described in Lemma 3.3, then we say that Y is a Δ-transformation of X. In a Δ-basis X we can replace every element x ∈ X by every y ∈ ⟨X⟩ with o(x) = o(y).

Lemma 3.4 Let M be a S-structure and let X be a Δ-basis of a subspace of M. If A is a finite subspace of ⟨X⟩, then there is a Δ-transformation Y of X such that Y contains a Δ-basis for A.

Proof: Every element of A is a linear combination of finitely many elements of X. Hence we can assume w.l.o.g. that X is finite. Assume that |X| = m. We prove by induction on j (1 ≤ j ≤ m) that there is a Δ-transformation Y_{≤j} of X_{≤j} that contains a Δ-basis of A ∩ ⟨X_{≤j}⟩. We start with y_{1} = x_{1}. If A ∩ ⟨X_{≤j+1}⟩ = A ∩ ⟨X_{≤j}⟩, then let Y_{≤j+1} be Y_{≤j} ∪ {x_{j+1}}.Otherwise we find the desired y_{j+1} in (A ∩ ⟨X_{≤j+1}⟩) \ (A ∩ ⟨X_{≤j}⟩). □

Such a Δ-basis Y ∩ A for A as obtained in Lemma 3.4 we call a Δ-basis for A induced by X. A Δ-basis Z of A ⊆ M is induced by the Δ-basis X of M iff o_{Z} for A is a restriction of o_{X} for M. Now we show the existence of a Δ-basis for countable structures.

Lemma 3.5 Every countable Δ-structure M has a Δ-basis.

Proof: Since M is countable we can write M as a union of a chain of finite subspaces: M = ∪_{1≤i<ω}M_{i} where M_{i} ⊆ M_{i+1}, dim(M_{1}) = 3, and dim(M_{i+1}) = dim(M_{i}) + 1. We define a tree in the following way. The Δbases of M_{i} are the points of level i of the tree. A point X of level i + 1 is connected with a point Y of level i iff Y = X ∩ M_{i}. There are points on each level but finitely many only. By Lemma 3.4 for every point of the tree there is a path to level 1. Hence by König’s Lemma there is an infinite path. It represents the desired Δ-basis of M. □

Definition 3.6 Let A ⊆ B be two S-structures.

1. If dim(B) = dim(A) + 1 and dim(N(B)) ≤ dim(N(A)) + 1, then B is called an one-extension of A. It is transcendental if N(B) = N(A) and algebraic otherwise.

2. If a ∈ A and Φ ∈ ∧^{2}A\N(A) and there is no b in A such that a∧b+Φ ∈ N(A), then a ∧ x + Φ = 0 is called a problem over A.

If a∧x+Φ = 0 is a problem over A, then the following algebraic one-extension B contains a solution: B = A⊕⟨b⟩ for some b and N(B) = N(A)+⟨a∧b+Φ⟩. If A is a Δ-structure, then B is a Δ-structure. Furthermore we have:

Lemma 3.7 Let A and B be Δ-structures. Assume B is an algebraic oneextension of A. Then there are a ∈ A and Φ ∈ ∧^{2}A \ N(A) such that a problem over A has a solution in B iff it is of the form r(a ∧ x + Φ′) = 0 where Φ′ = Φ modulo N(A) + ⟨{a ∧ c : c ∈ A}⟩. □

The standard models of our theory Δ are the groups H_{st}(n) that correspond to the following Δ-structures M_{st}(n). Let us fix n with 3 ≤ n < ω:

M_{st}(n) = ∪_{1≤i<ω}M_{i}

where dim(M_{1}) = n, N(M_{1}) = ⟨0⟩, M_{i+1} is an algebraic one-extension of M_{i}, and every problem over M_{i} has a solution in some M_{j}. From Lemma 3.1 follows that (Δ3) and (Δ4) are true. (Σ2) is true since all problems are solved. It remains to show that all Δ-structures we obtain in this way are isomorphic. It follows by successive application of the following lemma.

Lemma 3.8 Let A be a Δ-structure and let a∧x+Φ = 0 and a′ ∧x+Φ′ = 0 be two different problems over A. That means a ≠ a′ or Φ ≠ Φ′ modulo N(A) + ⟨{a ∧ c : c ∈ A}⟩. Assume A ⊆ B_{1} ⊆ B_{2} where B_{1} is an algebraic one-extension of A given by a solution of a∧x+Φ = 0 and B_{2} is an algebraic one-extesion of B_{1} given by a solution of a′∧x+Φ′ = 0. Furthermore assume that C_{1} is an algebraic one-extension of A given by a solution of a′∧x+Φ′ = 0 and C_{2} is an algebraic one-extension of C_{1} given by a solution of a∧x+Φ = 0. Then B_{2} and C_{2} are isomorphic over A. □

In [3] it is proved that every finite Δ-structure satisfies the axions (Σ3) and (Σ4). Let G be a saturated model of Σ and M be the corresponding Δ-structure. Let M_{1} be a subspace of M generated by n geometrically independent elements of M. M_{1} is selfsufficient and can be identified with the M_{1} in the definition of M_{st}(n). Using axiom (Σ5) we obtain a selfsufficient embedding of every M_{i+1} in M extending the selfsufficient embedding of M_{i}. Hence M_{st}(n) is a selfsufficient subspace of M. It is the intersection of all selfsufficient subspaces of M that contain M_{1} and satisfy axiom (Σ2). We see there are several ways to embed H_{st}(n) in G.

4 The past

Every Δ-structure M fulfils (Σ3) and (Σ4) as shown in [3]. By (Σ3) we have the pregeometry cl on M as mentioned in section 2. If H is a model of Δ and M is the corresponding Δ-structure, then we can consider cl also on \overline{H}. If we consider the algebraic closure acl on \overline{H}, then we mean the algebraic closure on \overline{H} with respect to the full theory of H. As in [3] we get:

Lemma 4.1 In \overline{H} we have cl ⊆ acl.

Proof: Let A be a finite subspace of \overline{H} and let a be an element in cl(A). Since we have CSS(A) ⊆ acl(A) we can assume w.l.o.g. that A is selfsufficient . Then there is a sequence B_{1} = A ⊆ B_{2} ⊆ … ⊆ B_{m} such that B_{i+1} is a minimal extension of B_{i} and a ∈ B_{m}. Note that each B_{i} is selfsufficient . By (Σ4) it follows that B_{i+1} ⊆ acl(B_{i}). See [3]. □

Let A ⊆ B be finite subspaces of a Δ-structure M.

Definition 4.2 B is in the one-closure of A (B ⊆ cl_{1}(A)) if there is a chain A = B_{1} ⊆ B_{2} ⊆ … ⊆ B_{m} = B such that B_{i+1} is an algebraic one-extension of B_{i}.

We have cl_{1} ⊆ cl. The Δ-structure M_{st}(n) corresponding to the standard model H_{st}(n) is the one-closure of n elements. If we have A ⊆ B finite and we cannot find the B_{m−1} in Definition 4.2, then we have the following situation.

Definition 4.3 Let A ⊆ B be finite subspaces of a Δ-structure M. Then B is part of the past of A (short B ∈ pp(A)) if for every subspace B_{1} with A ⊆ B_{1} ⊆ B and dim(B_{1}) + 1 = dim(B) we get dim(N(B)) > dim(N(B_{1})) + 1. We say C ⊆ past(A) if there is some B ∈ pp(A) with C ⊆ B.

Note that B ∈ pp(A) and A ⊆ A_{1} implies B + A_{1} ∈ pp(A_{1}). We will show in Lemma 4.6 that B ∈ pp(A) and D ∈ pp(A) implies B + D ∈ pp(A). This justifies the definition of the past. It is a closure operation:

1. A ⊆ past(A).

2. If A ⊆ B, then past(A) ⊆ past(B).

3. past(past(A)) = past(A).

The last statement follows from Lemma 4.5 below. For infinite A let past(A) be the union of all past(A_{0}) where A_{0} is a finite subspace of A. Analogously we define pp(A) and past(A) for finite subspaces A of \overline{H} for models H of Δ. In the next section we will show that past(A) ⊆ acl(A) for subspaces A in \overline{H} where H is a model of Δ. Now we give examples.

Lemma 4.4 Let a_{1}, …, a_{n}, b_{1}, b_{2} be linearly independent elements in a Δstructure M that corresponds to a group that is model of Δ. Assume that n is odd. Furthermore suppose if A = ⟨b_{1}, b_{2}⟩ and B = ⟨a_{1}, a_{2}, …, a_{n}, b_{1}, b_{2}⟩, then N(A) = ⟨0⟩ and N(B) is generated by Φ_{1} = b_{2} ∧ a_{1} − ∑_{1≤i≤(n−1)/2} a_{2i} ∧ a_{2i+1} and Φ_{2} = b_{1} ∧ a_{2} − a_{1} ∧ a_{n} − ∑_{1≤i<(n−1)/2} a_{2i+1} ∧ a_{2i+2}. Then B ∈ pp(A).

Proof: Assume that A ⊆ B_{1} ⊆ B with dim(B_{1}) + 1 = dim(B). We have to show that N(B_{1}) = ⟨0⟩. We can choose a basis {b_{1}, b_{2}, c_{1}, …, c_{n−1}} of B_{1} where c_{i} = ∑_{1≤j≤n} r_{j}^{i}a_{j}. Using the usual basis transformations we can assume w.l.o.g. that there is some k such that c_{i} = a_{j_{i}} + r_{i}a_{k} where {j_{1}, j_{2}, …, j_{n−1}, k} = {1, 2, …, n} and 0 ≤ r_{i} < p. Then ∧^{2}B_{1} has a basis {b_{1}∧b_{2}}∪{b_{l}∧(a_{j_{i}}+r_{i}a_{k}) : l ∈ {1, 2}, 1 ≤ i < n}∪{(a_{j_{h}}+r_{h}a_{k})∧(a_{j_{i}}+r_{i}a_{k}) : 1 ≤ h < i < n}. It is easily seen that it is impossible to represent any nontrivial linear combination of Φ_{1} and Φ_{2} with respect to this basis. □

Note that for a_{1}, a_{2}, …, a_{n} geometrically independent you get A and B as in Lemma 4.4 by axiom (Σ2). Since (Σ3) is a consequence of Δ any three linearly independent elements are geometrically independent. So for any linearly independent a_{1}, a_{2}, a_{3} there are b_{1}, b_{2} such that a_{1}, a_{2}, a_{3} ∈ past({b_{1}, b_{2}}).

Lemma 4.5 Let A ⊆ B ⊆ C be finite subspaces of a Δ-structure M. If B ∈ pp(A) and C ∈ pp(B), then C ∈ pp(A).

Proof: Assume A ⊆ C_{1} with dim(C) = dim(C_{1}) + 1. If B ⊆ C_{1}, then by assumption dim(N(C_{1})) + 1 < dim(N(C)), as desired. Otherwise we consider B_{1} = C_{1} ∩ B. Then dim(B_{1}) + 1 = dim(B). Hence by assumption dim(N(B_{1})) + 1 < dim(N(B)). Choose Φ_{1} and Φ_{2} in N(B) linearly independent over ∧^{2}B_{1}. Then Φ_{1} and Φ_{2} are linearly independent over ∧^{2}C_{1}, as desired. □

Lemma 4.6 Let A be a finite subspace of a Δ-structure M. Then CSS(A) ⊆ pp(A).

Proof: Assume A ⊆ B ⊆ CSS(A) with dim(B) + 1 = dim(CSS(A)). Then δ(CSS(A)) < δ(B). This means dim(N(CSS(A)) > dim(N(B)) + 1, as desired. □

Lemma 4.5 and Lemma 4.6 imply

Corollary 4.7 If B ∈ pp(A) , then CSS(B) ∈ pp(A).

Lemma 4.8 Let A, B, C be finite subspaces of a Δ-structure M such that B ∈ pp(A) and C ∈ pp(A). Then B + C ∈ pp(A).

Proof: Assume A ⊆ D ⊆ B + C with dim(D) + 1 = dim(B + C). Then dim(D ∩ B) + 1 = dim(B) or dim(D ∩ C) + 1 = dim(C). Assume w.l.o.g. dim(D ∩ B) + 1 = dim(B). By assumption

(1) dim(N(D ∩ B)) + 1 < dim(N(B)).

We will show:

(2) dim(N(D)) − dim(N(D ∩ B)) ≤ dim(N(B + C)) − dim(N(B)).

Then (1) and (2) imply

dim(N(D)) + 1 = dim(N(D ∩ B)) + 1 + dim(N(D)) − dim(N(D ∩ B))
< dim(N(B)) + dim(N(B + C)) − dim(N(B))
= dim(N(B + C)

as desired. It remains to prove (2). Let Φ_{1}, …, Φ_{m} be a maximal set in N(D) linearly independent over N(D ∩ B). If we consider Φ_{1}, …, Φ_{m} as elements of N(B + C), then they are linearly independent over N(B). Otherwise a non-trivial linear combination of them would be in ∧^{2}B and therefore in ∧^{2}D ∩ ∧^{2}B = ∧^{2}(D ∩ B), a contradiction. □

5 The algebraic closure

H is used to denote a model of Δ and M for the corresponding Δ-structure. In this section we will show that past(A) ⊆ acl(A) for subspaces A of \overline{H}. Let A ⊆ B be finite subspaces of M with B ∈ pp(A). We will prove that there are finitely many Δ-structures isomorphic to B over A only. First we give an example that shows that B is not uniquely determined by its isomorphism type over A: Assume that there are geometrically independent elements a_{3}, c_{3}, a_{4}, c_{4}, a_{5}, c_{5} in M. Define a_{1} and a_{2} as solutions of the following problems:

a_{2} ∧ (a_{3} − c_{3}) + a_{4} ∧ a_{5} − c_{4} ∧ c_{5} = 0 a_{1} ∧ (a_{5} − c_{5}) + a_{3} ∧ a_{4} − c_{3} ∧ c_{4} = 0.

Then we introduce b_{1} and b_{2} as solutions of the following problems:

b_{2} ∧ a_{1} = a_{2} ∧ a_{3} + a_{4} ∧ a_{5} b_{1} ∧ a_{2} = a_{1} ∧ a_{5} + a_{3} ∧ a_{4}.

a_{1}, a_{2}, b_{1}, b_{2} are obtained by axion (Σ2). Let A be {b_{1}, b_{2}}, B = {b_{1}, b_{2}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}}, and C = {b_{1}, b_{2}, a_{1}, a_{2}, c_{3}, c_{4}, c_{5}}. Then N(B) is freely generated by b_{2} ∧ a_{1} − a_{2} ∧ a_{3} − a_{4} ∧ a_{5} and b_{1} ∧ a_{2} − a_{1} ∧ a_{5} − a_{3} ∧ a_{4}. N(C) is freely generated by b_{2} ∧ a_{1} − a_{2} ∧ c_{3} − c_{4} ∧ c_{5} and b_{1} ∧ a_{2} − a_{1} ∧ c_{5} − c_{3} ∧ c_{4}. Hence B and C are isomorphic over A as Δ-structures. By Lemma 4.4 B ∈ pp(A) and C ∈ pp(A). The next lemma is the key-lemma of this paper. We need the following definition.

Definition 5.1 Let M_{0} and B be subspaces of M. A Δ-basis Z of B respects M_{0} if Z = XY where X is a Δ-basis of B ∩ M_{0}.

Lemma 5.2 Let M and M_{0} be Δ-structures. Assume that M_{0} is a substructure of M. M_{0} = ⟨0⟩ is possible. Let XY be a Δ-basis of M such that the initial part X is a Δ-basis of M_{0}. Furthermore suppose that XY contains a Δ-basis of a finite subspace A. Let B_{1}, …, B_{r} be finite substructures of M that are linearly independent over A. B_{i} ≠ B_{j} modulo M_{0} for i ≠ j. Assume for each i A ⊆ B_{i}, dim(B_{i}) = m, A + (B_{i} ∩ M_{0}) ≠ B_{i}, and B_{i} ∈ pp(A+(B_{i}∩M_{0})). For each i let {z_{1}^{i}, …, z_{m}^{i}} be a Δ-basis of B_{i} induced by XY that extends XY ∩ A and respects M_{0}. We assume that g_{il}(z_{j}^{i}) = z_{j}^{l} induces an isomorphism of B_{i} onto B_{l} as Δ-structures with a unary predicate for the intersection with M_{0}. Then r ≤ m + 1.

Proof: We assume m + 2 = r and show a contradiction. Let B be B_{1} + … + B_{r}. B_{i} ≠ A + (B_{i} ∩ M_{0}) implies that there is some z_{j}^{i} ∉ A + M_{0}. We have z_{m}^{i} ∈ A since B_{i} ∈ pp(A + (B_{i} ∩ M_{0})). Choose k such that z_{j}^{i} ∈ A for j > k and z_{k}^{i} ∉ A. Then z_{k}^{i} ∉ M_{0}. Note z_{j}^{i} ∈ A if and only if z_{j}^{l} ∈ A and g_{il}(z_{j}^{i}) = z_{j}^{l} = z_{j}^{i} in this case. By Lemma 3.4 there is a Δ-basis Z of B induced by XY. Then each {z_{1}^{i}, …, z_{m}^{i}} is induced by Z. o_{{z_{1}^{i},...,z_{m}^{i}}} is a restriction of o_{Z} and o_{Z} is a restriction of o_{XY}. We write o(x) only. Note XY ∩ A is a Δ-basis for A. We can assume that Z contains XY ∩ A. Hence z_{k+1}^{1}, …, z_{m}^{1} are the last elements of Z. We get B = U_{k+1} ⊕ ⟨z_{k+1}^{1}, …, z_{m}^{1}⟩ where U_{k+1} ⊆ ⟨(XY)_{<z_{k+1}})⟩.

Let Φ_{j}^{1} be a generator of N(⟨z_{1}^{1}, …, z_{j}^{1}⟩) over N(⟨z_{1}^{1}, …, z_{j−1}^{1}⟩) if these two spaces are different. Otherwise let Φ_{j}^{1} be 0. Φ_{j}^{1} = 0 implies g_{1i}(Φ_{j}^{1}) = 0 and N(⟨z_{1}^{i}, …, z_{j}^{i}⟩) = N(⟨z_{1}^{i}, …, z_{j−1}^{i}⟩) for 1 ≤ i ≤ r. If Φ_{j}^{1} ≠ 0, then N(⟨z_{1}^{i}, …, z_{j}^{i}⟩) ≠ N(⟨z_{1}^{i}, …, z_{j−1}^{i}⟩) and g_{1i}(Φ_{j}^{1}) generates N(⟨z_{1}^{i}, …, z_{j}^{i}⟩) over N(⟨z_{1}^{i}, …, z_{j−1}^{i}⟩). We define Φ_{j}^{i} = g_{1i}(Φ_{j}^{1}).

With respect to the basis {z_{1}^{i}, …, z_{m}^{i}} and any h with 1 ≤ h < m we can write:

Φ_{j}^{i} = ( ∑_{h≤l≤j} z_{l}^{i} ∧ a_{l}^{ij}) + Φ_{j}^{i}(h)

where a_{l}^{ij} ∈ ⟨z_{1}^{i}, …, z_{l−1}^{i}⟩ and Φ_{j}^{i}(h) ∈ ∧^{2}⟨z_{1}^{i}, …, z_{h−1}^{i}⟩. Let h be k +1. Then

(3) For k < l ≤ j we have a_{l}^{i_{1}j} = a_{l}^{i_{2}j} ∈ A for all 1 ≤ i_{1}, i_{2} ≤ r.

By definition z_{l}^{i_{1}} = z_{l}^{i_{2}} ∈ A for k < l ≤ m. If (3) was not true, then we could chose j > k maximal such that there are i and l such that a_{l}^{ij} ∉ A. Choose l maximal in this situation. Note a_{l}^{ij} ∈ A implies a_{l}^{i_{1}j} = a_{l}^{ij} for all i_{1}. But a_{l}^{ij} ∉ A implies that a_{l}^{1j}, …, a_{l}^{rj} are linearly independent over A, since B_{1}, …, B_{r} are linearly independent over A. As mentioned above B = U_{k+1} ⊕ ⟨z_{k+1}^{1}, …, z_{m}^{1}⟩, z_{k+1}^{1}, …, z_{m}^{1} are the last elements of the Δ-basis Z, and U_{k+1} = {u ∈ B : o(u) < o(z_{k+1}^{1})}. Then Φ_{j}^{1}, …, Φ_{j}^{r} are linearly independent over ∧^{2}U_{k+1}, since z_{l}^{1} ∧ a_{l}^{1j}, …, z_{l}^{1} ∧ a_{l}^{rj} are linearly independent over ∧^{2}(U_{k+1} + ⟨z_{k+1}^{1}, …, z_{l−1}^{1}⟩). We get dim(N(B) − dim(N(U_{k+1})) ≥ r = m + 2. But this is impossible for the Δ-basis Z. End of the proof of (3) □

There is some m_{0} such that z_{1}^{i}, …, z_{m_{0}−1}^{i} ∈ M_{0} and z_{m_{0}}^{i}, …, z_{m}^{i} are linearly independent over M_{0}. As above o(x) = o_{Z}(x) = o_{XY} (x). We claim for every h with m_{0} ≤ h ≤ m:

(4)_{h} For every i (1 ≤ i ≤ r) o(z_{h}^{i}) < o(z_{h+1}^{i}) < … < o(z_{m}^{i}) is the end segment of o(B).

For i ≠ j (1 ≤ i, j ≤ r) and h ≤ l ≤ m we have o(z_{l}^{i} − z_{l}^{j}) < o(z_{h}^{i}).
If Φ_{j}^{i} = (∑_{h≤l≤j} z_{l}^{i} ∧ a_{l}^{ij}) + Φ_{j}^{i}(h)
where a_{l}^{ij} ∈ ⟨z_{1}^{i}, …, z_{l−1}^{i}⟩
and Φ_{j}^{i}(h) ∈ ∧^{2}⟨z_{1}^{i}, …, z_{h−1}^{i}⟩,
then a_{l}^{ij} ∈ A and o(a_{l}^{ij}) < o(z_{h}^{i}) if z_{l}^{1} ∉ A.

Note that (4)_{k+1} follows from (3). (4)_{h} implies o(z_{l}^{i}) = o(z_{l}^{j}) for h ≤ l ≤ m. Furthermore every Δ-basis Z of B induced by XY can be Δ-transformed into a Δ-basis of the form Z′{z_{h}^{i}, …, z_{m}^{i}}. Let U_{h} be the subspace {u ∈ B : o(u) < o(z_{h}^{1})}. In (4)_{h} we can write a_{l}^{ij} ∈ A ∩ U_{h} if z_{l}^{i} ∉ A and Φ_{j}^{i}(h) ∈ ∧^{2}U_{h}. First we show:

(5) For 1 < h (4)_{h} implies that for some i o(z_{h−1}^{i}) is the immediate predecessor of o(z_{h}^{i}). m_{0} = 1 and (4)_{1} is impossible.

Proof of (5): First of all m_{0} = 1 and (4)_{1} would imply a_{k}^{ij} = 0 for all i and j. Hence Φ_{j}^{1} ∈ N(⟨{z_{1}^{1}, …, z_{m}^{1}} \ {z_{k}^{1}}⟩) for 1 ≤ j ≤ m. This contradicts B_{1} ∈ pp(A). Note M_{0} ∩ B_{1} = ⟨0⟩ in this case.

Now we consider the case 1 < h. Let α be the immediate predecessor of o(z_{h}^{1}). Let U_{h−1} be the subspace of all u ∈ ⟨Z⟩ with o(u) < α. We assume that there is some h such that 1 < h, (4)_{h} is true, and o(z_{l}^{i}) < α for all i with 1 ≤ i ≤ r and all l with 1 ≤ l < h. We show a contradiction. As a consequence of this assumption we get in (4)_{h} that for l with h ≤ l ≤ m and z_{l}^{i} ∉ A a_{l}^{ij} ∈ U_{h} ∩ A ⊆ U_{h−1} and Φ_{j}^{i}(h) ∈ ∧^{2}U_{h−1}. By (4)_{h} U_{h} is generated by all z_{l}^{i} with 1 ≤ i ≤ r and 1 ≤ l < h and all z_{l}^{i} − z_{l}^{1} with 1 < i ≤ r and h ≤ l ≤ m. Hence by the assumption of the proof there is some l with h ≤ l ≤ m and some i with 1 < i ≤ r such that o(z_{l}^{i} −z_{l}^{1}) = α. Now choose l_{0} minimal such that h ≤ l_{0} ≤ m and z_{l_{0}}^{i} − z_{l_{0}}^{1} ∈ U_{h} \ U_{h−1}. If there is another z_{t}^{i} − z_{t}^{1} ∈ U_{h} \ U_{h−1} for the considered i, then z_{t}^{i} ∉ A. There is some q (0 < q < p) with q(z_{l_{0}}^{i} − z_{l_{0}}^{1}) − (z_{t}^{i} − z_{t}^{1}) ∈ U_{h−1}. For t fixed as above and all 1 ≤ s ≤ r we replace z_{t}^{s} by y_{t}^{s} = z_{t}^{s} − qz_{l_{0}}^{s}. Then y_{t}^{i} − y_{t}^{1} = (z_{t}^{i} − qz_{l_{0}}^{i}) − (z_{t}^{1} − qz_{l_{0}}^{1}) = (z_{t}^{i} − z_{t}^{1}) − q(z_{l_{0}}^{i} − z_{l_{0}}^{1}) ∈ U_{h−1}. Note that the assumptions of the lemma and (4)_{h} remain true after these Δtransformations. Hence we can assume w.l.o.g. z_{t}^{i} − z_{t}^{1} ∈ U_{h−1} for all t ≠ l_{0} with h ≤ t ≤ m. As mentioned above z_{l_{0}}^{i} ∉ A. Since Z = Z′{z_{h}^{1}, …, z_{m}^{1}} contains a basis for A and m_{0} ≤ h ≤ l_{0} we get z_{l_{0}}^{i} ∉ A + (B_{i} ∩ M_{0}). By assumption B_{i} ∈ pp(A + (B_{i} ∩ M_{0})). We get Φ_{j_{1}}^{i} and Φ_{j_{2}}^{i} that are linearly independent over ∧^{2}{z_{1}^{i}, …, z_{l_{0}−1}^{i}, z_{l_{0}+1}^{i}, …, z_{m}^{i}}. z_{l_{0}}^{i} ∉ A implies that z_{l_{0}}^{i} does not occur in any a_{l}^{ij}. Hence z_{l_{0}}^{i} ∧ a_{l_{0}}^{ij_{s}} is the only summand in Φ_{j_{s}}^{i} where z_{l_{0}}^{i} occurs. Hence a_{l_{0}}^{ij_{1}} ≠ 0, a_{l_{0}}^{ij_{2}} ≠ 0, and a_{l_{0}}^{ij_{1}} ≠ a_{l_{0}}^{ij_{2}} is necessary to realize the independence of Φ_{j_{1}}^{i} and Φ_{j_{2}}^{i} over ∧^{2}⟨z_{1}^{i}, …, z_{l_{0}−1}^{i}, z_{l_{0}+1}^{i}, …, z_{m}^{i}⟩. Since a_{l}^{ij} ∈ A we have a_{l}^{ij} = a_{l}^{1j}. By (4)_{h} we get

Φ_{j_{1}}^{i} − Φ_{j_{1}}^{1} = ( ∑_{h≤l≤m}(z_{l}^{i} − z_{l}^{1}) ∧ a_{l}^{ij_{1}}) + Φ_{j_{1}}^{i}(h) − Φ_{j_{1}}^{1}(h).

By the assumption above z_{l}^{i} − z_{l}^{1} ∈ U_{h−1} for l ≠ l_{0}. As mentioned above we have a_{l}^{ij} ∈ U_{h} ∩ A ⊆ U_{h−1} for z_{l}^{i} ≠ z_{l}^{1} ∉ A and Φ_{j_{1}}^{i}(h), Φ_{j_{1}}^{1}(h) ∈ ∧^{2}U_{h−1}. Hence

Φ_{j_{1}}^{i} − Φ_{j_{1}}^{1} = (z_{l_{0}}^{i} − z_{l_{0}}^{1}) ∧ a_{l_{0}}^{ij_{1}} + Φ

where Φ ∈ ∧^{2}U_{h−1}. Analogously

Φ_{j_{2}}^{i} − Φ_{j_{2}}^{1} = (z_{l_{0}}^{i} − z_{l_{0}}^{1}) ∧ a_{l_{0}}^{ij_{2}} + Ψ

where Ψ ∈ ∧^{2}U_{h−1}. Since 0 ≠ a_{l_{0}}^{ij_{1}} ≠ a_{l_{0}}^{ij_{2}} we have a contradiction to Lemma 3.7. End of the proof of (5) □

By induction from h + 1 to h we will show (4)_{h} for m_{0} ≤ h ≤ k + 1. Then (4)_{m_{0}} gives the desired contradiction. If m_{0} = 1 this follows from (5). Otherwise by (5) o_{Z}(z_{m_{0}−1}^{1}) is the immediate predecessor of o_{Z}(z_{m_{0}}^{1}). (4)_{m_{0}} implies o(z_{l}^{i} − z_{l}^{1}) ≤ o(z_{m_{0}−1}^{1}) for m_{0} ≤ l ≤ m. Hence z_{l}^{i} − z_{l}^{1} ∈ M_{0} and B_{i} = B_{1} modulo M_{0}, a contradiction. Let us assume (4)_{h+1} for m_{0} < h+1 ≤ k +1. We show (4)_{h}.

Proof of (4)_{h}: By the induction hypothesis (4)_{h+1} there is a Δ-basis W′{w, z_{h+1}^{1}, …, z_{m}^{1}} of B induced by XY that contains the Δ-basis XY ∩A of A. U_{h+1} = ⟨W′{w}⟩ and U_{h} is ⟨W′⟩. By induction z_{l}^{i} − z_{l}^{1} ∈ U_{h+1} for h + 1 ≤ l ≤ m. A basis induced by XY is also induced by every Δ-transformation of XY. Hence we can assume w.l.o.g. that W′{w, z_{h+1}^{1}, …, z_{m}^{1}} is containded in XY. Since {z_{1}^{i}, …, z_{m}^{i}} is induced by XY z_{l}^{i} ∈ U_{h+1} for 1 ≤ l ≤ h and 1 ≤ i ≤ r. Hence U_{h+1} is generated by all these z_{l}^{i} with 1 ≤ i ≤ h and 1 ≤ l ≤ r and all z_{h+1}^{i} − z_{h+1}^{1}, …, z_{m}^{i} − z_{m}^{1} for all 1 < i ≤ r. By (5) and (4)_{h+1} we get:

There is some z_{h}^{i} ∈ U_{h+1} \ U_{h}. If z_{h}^{i} ∈ A for some i, then z_{h}^{i} ∈ A for all i and they are equal. For the first assertion of (4)_{h} we have to show z_{h}^{i} ∈ U_{h+1} \ U_{h} for all 1 ≤ i ≤ r. This and the next assertions (6), (7), (8), and (9) are trivial if z_{h}^{i} ∈ A. Hence we assume z_{h}^{i} ∉ A. Since B_{i} ∈ pp(A + (B_{i} ∩ M_{0})) we can choose j_{1}, j_{2} such that Φ_{j_{1}}^{i} and Φ_{j_{2}}^{i} are linearly independent over ∧^{2}⟨z_{1}^{i}, …, z_{h−1}^{i}, z_{h+1}^{i}, …, z_{m}^{i}⟩. z_{h}^{i} ∉ A implies that z_{h}^{i} does not occur in a_{l}^{ij} for h < l. Hence z_{h}^{i} ∧ a_{h}^{ij_{s}} is the only summand in Φ_{j_{s}}^{i} where z_{h}^{i} occurs. We have

(6) If z_{h}^{i} ∉ A, then for j_{1} and j_{2} as chosen above 0 ≠ a_{h}^{ij_{1}} ≠ a_{h}^{ij_{2}} ≠ 0 for all i.

Furthermore we show:

(7) If z_{h}^{i} − z_{h}^{t} ∉ U_{h} for some t ≠ i, then we can assume after some Δtransformations that z_{l}^{i} − z_{l}^{t} ∈ U_{h} for h < l ≤ m.

Assume that this is not true. We have z_{h}^{i} − z_{h}^{t} ∈ U_{h+1} \ U_{h}. Then there is some l with h < l ≤ m and z_{l}^{i} − z_{l}^{t} = q(z_{h}^{i} − z_{h}^{t}) mod U_{h} for some 0 < q < p. We can replace every z_{l}^{s} by z_{l}^{s} − qz_{h}^{s} where 1 ≤ s ≤ r and l is fixed above. This is a Δ-transformation. (4)_{h+1} remains true. If we iterate this argument, then we can assume (7). Now we consider

Φ_{j_{1}}^{i} − Φ_{j_{1}}^{t} = ( ∑_{h<l≤m}(z_{l}^{i} − z_{l}^{t}) ∧ a_{l}^{ij_{1}}) + z_{h}^{i} ∧ a_{h}^{ij_{1}} − z_{h}^{t} ∧ a_{h}^{tj_{1}} + Φ_{j_{1}}^{i}(h) − Φ_{j_{1}}^{t}(h)

where for h < l ≤ m a_{l}^{ij_{1}} = a_{l}^{tj_{1}} ∈ A∩U_{h+1} ⊆ U_{h} by (4)_{h+1} and a_{h}^{ij_{1}}, a_{h}^{tj_{1}} ∈ U_{h}, and Φ_{j_{1}}^{i}(h), Φ_{j_{1}}^{t}(h) ∈ ∧^{2}U_{h}. If z_{h}^{t} ∈ U_{h}, then by (7)

Φ_{j_{1}}^{i} − Φ_{j_{1}}^{t} = z_{h}^{i} ∧ a_{h}^{ij_{1}} + Φ

where Φ ∈ ∧^{2}U_{h}. Analogously

Φ_{j_{2}}^{i} − Φ_{j_{2}}^{t} = z_{h}^{i} ∧ a_{h}^{ij_{2}} + Ψ

where Ψ ∈ ∧^{2}U_{h}. By (6) we get a contradiction to Lemma 3.7. We have shown

(8) z_{h}^{i} ∈ U_{h+1} \ U_{h} for 1 ≤ i ≤ r.

Let us consider Φ_{j_{1}}^{i} −Φ_{j_{2}}^{t} again. Now we assume that a_{h}^{ij} = a_{h}^{tj} for j ∈ {j_{1}, j_{2}}. Then by (7)

Φ_{j_{1}}^{i} − Φ_{j_{1}}^{t} = (z_{h}^{i} − z_{h}^{t}) ∧ a_{h}^{ij_{1}} + Φ′

and

Φ_{j_{2}}^{i} − Φ_{j_{2}}^{t} = (z_{h}^{i} − z_{h}^{t}) ∧ a_{h}^{ij_{2}} + Ψ′

where Φ′, Ψ′ ∈ ∧^{2}U_{h}. If z_{h}^{i} − z_{h}^{t} ∉ U_{h}, then we get a contradiction by (6). We have shown

(9) If a_{h}^{1j} = a_{h}^{ij} for 1 < i ≤ r and all j, then z_{h}^{i} − z_{h}^{1} ∈ U_{h}.

(8) and (9) are trivial if z_{h}^{i} ∈ A. Now we have to consider this case too. Next we claim

(10) a_{h}^{ij} ∈ A for 1 ≤ i ≤ r and h ≤ j ≤ m.

If this was not true we could fix j such that a_{h}^{ij} ∉ A for all i. Then a_{h}^{1j}, …, a_{h}^{rj} are linearly independent over A. Note that by definition a_{h}^{ij} ∈ U_{h}. Hence z_{h}^{1} ∧ a_{h}^{1j}, …, z_{h}^{1} ∧ a_{h}^{rj} are linearly independent over ∧^{2}U_{h}. We consider

Φ_{j}^{i} = ( ∑_{h<l≤m} z_{l}^{i} ∧ a_{l}^{ij}) + z_{h}^{i} ∧ a_{h}^{ij} + Φ_{j}^{i}(h)

where Φ_{j}^{i}(h) ∈ ∧^{2}U_{h}, and a_{l}^{ij} ∈ ⟨z_{1}^{i}, …, z_{l−1}^{i}⟩. By (4)_{h+1} for h < l ≤ m we know z_{l}^{i} − z_{l}^{1} ∈ U_{h+1}, a_{l}^{ij} = a_{l}^{1j} ∈ A, and a_{l}^{1j} ∈ A ∩ U_{h+1} if z_{l}^{1} ∉ A. Furthermore by (8) z_{h}^{i} ∈ U_{h+1} \ U_{h}. Hence z_{h}^{i} = s_{i}z_{h}^{1} + v_{i} with 0 < s_{i} < p and v_{i} ∈ U_{h}. Let us rewrite

Φ_{j}^{i} = ( ∑_{h<l≤m} z_{l}^{1} ∧a_{l}^{1j})+( ∑_{{h<l≤m \atop z^{1}_{l}∉A}}((z_{l}^{i} −z_{l}^{1})∧a_{l}^{1j})+s_{i}(z_{h}^{1} ∧a_{h}^{ij})+(v_{i} ∧a_{h}^{ij})+Φ_{j}^{i}(h).

Since z_{l}^{i} − z_{l}^{1} ∈ U_{h+1} for h < l ≤ m we can write z_{l}^{i} − z_{l}^{1} = r_{l}^{i}z_{h}^{1} + w_{l}^{i} with 0 ≤ r_{l}^{i} < p for all h < l ≤ m and w_{l}^{i} ∈ U_{h}. By the linear independence of B_{1}, …, B_{r} over A the following is a basis of B:

Z^{∗} = {z_{1}^{1}, …, z_{m}^{1}} ∪ {z_{l}^{i} ∉ A : 1 < i ≤ r, 1 ≤ l < h}
∪ {w_{l}^{i} : 1 < i ≤ r, h < l ≤ m, z_{l}^{1} ∉ A}
∪ {v_{i} : 1 < i ≤ r}.

We do not speak about a Δ-basis. Note that {z_{1}^{1}, …, z_{m}^{1}} contains a basis for A. Z^{∗} \{z_{h}^{1}, … z_{m}^{1}} is a basis for U_{h}. Furthermore a_{t}^{1j} ∈ {z_{l}^{1} : 1 ≤ l ≤ h}∩A for h < t ≤ m and z_{t}^{1} ∉ A. By definition a_{h}^{ij} ∈ {z_{l}^{i} : 1 ≤ l < h}.

We use the representation of Φ_{j}^{i} above and the basis Z^{∗} described above to show that the linear indepedence of a_{h}^{2j}, …, a_{h}^{rj} over A + B_{1} implies the linear independence of Φ_{j}^{2}, …, Φ_{j}^{r} over ∧^{2}U_{h}. This is a contradiction to (Δ3) and (Δ4) since r − 1 = m + 1 > m − h + 1 = dim(B) − dim(U_{h}).

First we assume z_{h}^{1} ∉ A. Then U_{h+1} ∩ A ⊆ U_{h}. We have

Φ_{j}^{i} = s_{i}(z_{h}^{1} ∧ a_{h}^{ij}) + Φ_{i}

where Φ_{i} ∈ ∧^{2}B_{1} + ∧^{2}U_{h}.

Now we assume that z_{h}^{1} ∈ A. Then for h < l ≤ m we have a_{l}^{1j} = t_{l}^{j}z_{h}^{1} + b_{l}^{j} where b_{l} ∈ A ∩ U_{h}. We get

Φ_{j}^{i} = s_{i}(z_{h}^{1} ∧ a_{h}^{ij}) − ∑_{{h<l≤m \atop z^{1}_{l}∉A}} t_{l}^{j}(z_{h}^{1} ∧ w_{l}^{i}) + Ψ_{i}

where Ψ_{i} ∈ ∧^{2}B_{1} + ∧^{2}U_{h}. Note a_{h}^{ij} ∈ ⟨z_{1}^{i}, …, z_{h−1}^{i}⟩ ⊆ Z^{∗} and w_{l}^{i} ∈ Z^{∗}. This completes the proof of (9). □

By (9) and (10) we have

(11) z_{h}^{i} − z_{h}^{1} ∈ U_{h} for 1 ≤ i ≤ r.

The next step is to show:

(12) a_{l}^{ij} ∈ A ∩ U_{h} for h ≤ l ≤ m with z_{l}^{i} ∉ A.

Note a_{h}^{ij} ∈ U_{h} by definition and a_{h}^{ij} ∈ A by (10). By induction z_{l}^{i} ∉ A implies a_{l}^{ij} ∈ A ∩ U_{h+1} for h < l ≤ m. If z_{h}^{1} ∉ A it follows a_{l}^{ij} ∈ A ∩ U_{h+1} ⊆ U_{h} as desired. Therefore we consider the case z_{h}^{1} ∈ A. For z_{l}^{i} ∉ A we can write a_{l}^{1j} = a_{l}^{ij} = t_{l}^{j}z_{h}^{1} + b_{l}^{j} where b_{l}^{j} ∈ A ∩ U_{h} and 0 ≤ t_{l}^{j} < p. Let I_{j} be

{l : h < l ≤ m, t_{l}^{j} ≠ 0, z_{l}^{1} ∉ A}.

If the assertion of (12) was not true, then there would be some j_{0} with I_{j_{0}} ≠ ∅. Under this assumption we show that Φ_{j_{0}}^{2} − Φ_{j_{0}}^{1}, …, Φ_{j_{0}}^{r} − Φ_{j_{0}}^{1} are linearly independent over ∧^{2}U_{h}. This gives the desired contradiction since dim(B)−dim(U_{h}) = m−h+1 ≤ m < r−1 and there are r−1 many relations in N(B) linearly independent over ∧^{2}U_{h}. To prove the linear independence consider

Φ_{j_{0}}^{i} − Φ_{j_{0}}^{1} = ( ∑_{h≤l≤m}(z_{l}^{i} − z_{l}^{1}) ∧ a_{l}^{ij_{0}}) + Φ_{j_{0}}^{i}(h) − Φ_{j_{0}}^{1}(h).

Note Φ_{j_{0}}^{i}(h)−Φ_{j_{0}}^{1}(h) ∈ ∧^{2}U_{h}. For z_{l}^{i} ∈ A we have z_{l}^{i} −z_{l}^{1} = 0, e.g. z_{h}^{i} −z_{h}^{1} = 0 since we have assumed that z_{h}^{1} ∈ A. As above assume that z_{l}^{i} −z_{l}^{1} = r_{l}^{i}z_{h}^{1} +w_{l}^{i} where 0 ≤ r_{l}^{i} < p and w_{l}^{i} ∈ U_{h} ∩ Z^{∗}. Then W^{∗} = {w_{l}^{i} : 1 < i ≤ r, h < l ≤ m, z_{l}^{1} ∉ A} ⊆ U_{h} ∩ Z^{∗} is a set linearly independent over B_{1}. For l ∉ I_{j_{0}} we have (z_{l}^{i} − z_{l}^{1}) ∧ a_{l}^{ij_{0}} = r_{l}^{i}(z_{h}^{1} ∧ b_{l}^{j_{0}}) + w_{l}^{i} ∧ b_{l}^{j_{0}} ∈ ⟨z_{h}^{1}⟩ ∧ (A ∩ U_{h}) + ∧^{2}U_{h} and for l ∈ I_{j_{0}} we have (z_{l}^{i} − z_{l}^{1}) ∧ a_{l}^{ij_{0}} = r_{l}^{i}(z_{h}^{1} ∧ b_{l}^{j_{0}}) + w_{l}^{i} ∧ t_{l}^{i}z_{h}^{1} + w_{l}^{i} ∧ b_{l}^{j_{0}}. Hence

Φ_{j_{0}}^{i} − Φ_{j_{0}}^{1} = ( ∑_{{h<l≤m \atop l∈I_{j_{0}}}} w_{l}^{i} ∧ t_{l}^{i}z_{h}^{1}) + Ψ_{j_{0}}^{i}

where Ψ_{j_{0}}^{i} ∈ ⟨z_{h}^{1}⟩∧(A∩U_{h})+∧^{2}U_{h} ⊆ ∧^{2}B_{1} +∧^{2}U_{h}. The linear independence of W^{∗} over B_{1} and W^{∗} ⊆ U_{h} implies the desired linear independence of Φ_{j_{0}}^{2} − Φ_{j_{0}}^{1}, …, Φ_{j_{0}}^{r} − Φ_{j_{0}}^{r} over ∧^{2}U_{h}. □

To finish the proof of (4)_{h} we show

(13) z_{l}^{i} − z_{l}^{1} ∈ U_{h} for h ≤ l ≤ m and 1 < i ≤ r.

Let us fix some i. By (11) z_{h}^{i} − z_{h}^{1} ∈ U_{h}. By induction z_{l}^{i} − z_{l}^{1} ∈ U_{h+1} for h < l ≤ m. We assume that there is some l_{0} with h < l_{0} ≤ m and z_{l_{0}}^{i} − z_{l_{0}}^{1} ∈ U_{h+1} \ U_{h}. Choose l_{0} minimal. As above for (7) we find some suitable Δ-transformations such that w.l.o.g. z_{l}^{i} − z_{l}^{1} ∈ U_{h} for l ≠ l_{0} with h ≤ l ≤ m. Since B_{i} ∈ pp(A + (B_{i} ∩ M_{0})) there are j_{1} and j_{2} such that Φ_{j_{1}}^{i} and Φ_{j_{2}}^{i} are linearly independent over ∧^{2}⟨z_{1}^{i}, …, z_{l_{0}−1}^{i}, z_{l_{0}+1}^{i}, …, z_{m}^{i}⟩. As above it follows 0 ≠ a_{l_{0}}^{ij_{1}} ≠ a_{l_{0}}^{ij_{2}} ≠ 0. By our assumptions and by (12) we have

Φ_{j_{1}}^{i} − Φ_{j_{1}}^{1} = (z_{l_{0}}^{i} − z_{l_{0}}^{1}) ∧ a_{l_{0}}^{ij_{1}} + Φ
Φ_{j_{2}}^{i} − Φ_{j_{2}}^{1} = (z_{l_{0}}^{i} − z_{l_{0}}^{1}) ∧ a_{l_{0}}^{ij_{2}} + Ψ

where Φ, Ψ ∈ ∧^{2}U_{h}. We get the desired contradiction to Lemma 3.7. □

Corollary 5.3 Let M be a Δ-structure. Let X be a Δ-basis of M that extends a Δ-basis of a finite substructure A. Let B_{1}, …, B_{r} be finite substructures of M that contain A and are linearly independent over A. Assume that B_{i} ∈ pp(A) and that there are Δ-bases {z_{1}^{i}, …, z_{m}^{i}} of B_{i} induced by X for each i (1 ≤ i ≤ r). Furthermore suppose that these Δ-bases contain the Δ-basis of A contained in X. We assume that g_{it}(z_{l}^{i}) = z_{l}^{t} induces an isomorphism of B_{i} onto B_{t} as Δ-structures. Then r ≤ m + 1.

Theorem 5.4 There is a function f(m) from the naturals into the naturals such that for every Δ-structure M and all finite subspaces A of M the number of subspaces B of M with A ⊆ B, B ∈ pp(A), and dim(B) = m is less or equal to f(m).

Proof: First we show that there is a function h(m) such that for every Δstructure M, every finite subspace A of M, and every B_{1}, …, B_{r} linearly independent over A with A ⊆ B_{i}, dim(B_{i}) = m, and B_{i} ∈ pp(A) we have r ≤ h(m). W.l.o.g. we can assume that M is countable. By Lemma 3.5 there is a Δ-basis X of M. By Lemma 3.4 we can assume that X extends a Δ-basis Z of A. Assume tha B_{1}, …, B_{r} are given as above. For each B_{i} we fix a Δ-basis z_{1}^{i}, …, z_{m}^{i} induced by X and containing Z. We say B_{i} ∼ B_{t} if g_{it}(z_{l}^{i}) = z_{l}^{t} induces an isomorphism of B_{i} onto B_{t} over A as Δ-structures. There is a finite bound h_{∼} for the number of possible ∼-classes. By Corollary 5.3 we have

r ≤ (m + 1)h_{∼} = h(m).

Now we show by induction on i (1 ≤ i < m) that there is a function f(m, i) such that for every Δ-structure M for every subspace A of M with m − i ≤ dim(A) < m the number of subspaces B with A ⊆ B, dim(B) = m, and B ∈ pp(A) is bounded by f(m, i).

Choose any M and any A with dim(A) = m − 1. Assume that r is chosen maximal such that there are B_{1}, …, B_{r} that are linearly independent over A, have size m, contain A, and are parts of the past of A. As shown above r ≤ h(m). If B is any other subspace of M with dim(B) = m, A ⊆ B, and B ∈ pp(A), then B ⊆ B_{1} + … + B_{r} since r was chosen maximal. It follows that f(m, 1) = 2^{(p^{h(m)m})} is a bound with the desired properties.

Now assume that f(m, i) is given. Let A ⊆ M be subspace of size m − (i + 1). Again we consider a maximal set {B_{1}, …, B_{r}} linearly independent over A with dim(B_{i}) = m, A ⊆ B_{i}, and B_{i} ∈ pp(A). We know r ≤ h(m). If B is any subspace of M with dim(B) = m, A ⊆ B, and B ∈ pp(A), then there is some a ∈ (B \ A) ∩ (B_{1} + … + B_{r}). Define A_{1} = A + ⟨a⟩. Then dim(A_{1}) = m − i, A_{1} ⊆ B, and B ∈ pp(A_{1}). By induction the number of these B is bounded by f(m, i). Furthermore A_{1} ⊆ B_{1} + … + B_{r}. Hence

f(m, i + 1) = f(m, i)2^{(p^{h(m)m})} + f(m, i)

has the desired properties. We can define f(m) = f(m, m − 1). □

Corollary 5.5 Let H be a model of Δ. Let A ⊆ B be finite subspaces of \overline{H}. If B ∈ pp(A), then B ⊆ acl(A). Hence past(A) ⊆ acl(A).

Corollary 5.6 If A is a finite subspace of a Δ-structure M, then past(A) is countable.

6 Extending a Δ-basis

Lemma 6.1 Let M be a Δ-structure and M_{0} be a substructure with past(M_{0}) ⊆ M_{0}. Let A be a finite subspace of M. Then there is a Δ-basis X = X_{0}X_{1} of A such that the initial part X_{0} is a Δ-basis of A ∩ M_{0}.

Proof: We use induction on dim(A) = m. The case m = 1 is clear. Assume that the assertion is proved for all A with dim(A) = m. Now we consider A with dim(A) = m + 1. If A ⊆ M_{0}, then the assertion follows by Lemma 3.4. We assume A ⊈ M_{0}. Let A_{0} be A ∩ M_{0}. Then there exists some A_{1} with A_{0} ⊆ A_{1} ⊆ A such that dim(A_{1}) = dim(A) − 1 and dim(N(A_{1})) + 1 ≥ dim(N(A)). Otherwise we would get A ⊆ past(A_{0}) ⊆ past(M_{0}) ⊆ M_{0}, a contradiction to our assumption A ⊈ M_{0}. By induction hypothesis there is a Δ-basis X_{0}Y of A_{1} such that X_{0} is a Δ-basis of A_{1} ∩ M_{0} = A ∩ M_{0}. By the choice of A_{1} for every a ∈ A \ A_{1} X_{0}Y {a} is the desired Δ-basis of A. □

Note that we have defined X respects M_{0} if we are in the situation described in the Lemma 6.1. If H is a model of Δ, H_{0} is an elementary submodel, and M and M_{0} are the Δ-structures that correspond to H and H_{0} respectively, then we have past(M_{0}) ⊆ M_{0} by Corollary 5.5.

Lemma 6.2 Let M be a countable Δ-structure and M_{0} be a substructure with past(M_{0}) ⊆ M_{0}. Then there exists a Δ-basis of M that respects M_{0}.

Proof: We use a similar argument as in Lemma 3.5. Let M be ∪_{1≤i<ω}N_{i} such that N_{i} ⊆ N_{i+1} and dim(N_{i}) = i. We consider the following tree. The set P_{i} of points of level i is the set of the Δ-bases of N_{i} that respect M_{0}. By Lemma 6.1 P_{i} ≠ ∅. A Δ-basis of N_{i} is connected with a Δ-basis of N_{i+1} if it is a subset. Every point of the tree has finite valency and by Lemma 6.1 there are points at every level. As in Lemma 3.4 it is easily seen that from every point at level i there is a path to level 1. By König’s Lemma there is an infinite path. The union of the Δ-bases of this path is the desired Δ-basis of M that respects M_{0}. □

Corollary 6.3 Let M_{0} be a substructure of a countable Δ-structure M such that past(M_{0}) ⊆ M_{0}. Then every Δ-basis X of M_{0} can be extended to a Δ-basis of M.

Proof: By Lemma 6.2 there is a Δ-basis ZY of M that respects M_{0} where Z is a Δ-basis of M_{0}. If we replace Z by X, then we get the desired Δ-basis of M. □

Lemma 6.4 Let B ⊆ C be finite subspaces of a Δ-structure M. Assume that C ⊆ cl_{1}(B). If B ⊆ C_{1} ⊆ C, dim(C_{1}) + 1 = dim(C), and dim(N(C_{1})) + 1 ≥ dim(N(C)), then C_{1} ⊆ cl_{1}(B).

Proof: Since C ⊆ cl_{1}(B) there are c_{1}, …, c_{n} such that C = B + ⟨c_{1}, …, c_{n}⟩ and B+⟨c_{1}, …, c_{i+1}⟩ is an algebraic one-extension of B+⟨c_{1}, …, c_{i}⟩. Assume that Φ_{i+1} generates N(B + ⟨c_{1}, …, c_{i+1}⟩) over N(B + ⟨c_{1}, …, c_{i}⟩). Choose i_{0} minimal such that c_{i_{0}} ∉ C_{1}. By dim(C_{1}) + 1 = dim(C) after some Δtransformations we can assume that c_{i_{0}+1}, …, c_{n} ∈ C_{1}. By the assumption Φ_{i_{0}} generates N(C) over N(C_{1}). Hence we can choose all Φ_{i} with i ≠ i_{0} in N(C_{1}) as desired. □

Lemma 6.5 Let A ⊆ B ⊆ C ⊆ D be a finite subspaces of a Δ-structure M. If B ∈ pp(A), C ⊆ cl_{1}(B), and D ∈ pp(C), then there is a B_{1} ∈ pp(A) such that D = B_{1} + C.

Proof: We show the assertion by induction on dim(D) − dim(A) = m. If D ∈ pp(A), then we are done. Otherwise there exists D_{1} such that A ⊆ D_{1} ⊆ D, dim(D_{1}) + 1 = dim(D), and dim(N(D_{1})) + 1 ≥ dim(N(D)). Then B ⊆ D_{1} since otherwise dim(B∩D_{1})+1 = dim(B) and dim(N(B∩D_{1}))+1 ≥ dim(N(B)), a contradiction to B ∈ pp(A). Note A ⊆ B ∩ D_{1}. Furthermore C ⊈ D_{1}. C ⊆ D_{1} would imply D ∈ pp(D_{1}), a contradiction to the defining condition of D_{1}. Hence D = D_{1} +C. Let C_{1} be D_{1} ∩C. Then dim(C_{1})+1 = dim(C) and dim(N(C_{1})) + 1 ≥ dim(N(C)). Furthermore B ⊆ D_{1} ∩ C = C_{1}. By Lemma 6.4 C ⊆ cl_{1}(B) implies C_{1} ⊆ cl_{1}(B). Let c ∈ C \ D_{1}. Then C_{1} ⊕ ⟨c⟩ = C is an algebraical one-extension of C_{1}. We have D = D_{1} ⊕ ⟨c⟩ and c ∈ cl_{1}(D_{1}). We claim D_{1} ∈ pp(C_{1}). Otherwise there is some D_{2} with C_{1} ⊆ D_{2} ⊆ D_{1}, dim(D_{2})+1 = dim(D_{1}), and dim(N(D_{2}))+1 ≥ dim(N(D_{1})). Since C_{1} ⊆ D_{1} and c ∈ cl_{1}(C_{1}) we have dim(N(D_{2} ⊕⟨c⟩)) = dim(N(D_{2}))+1. But then dim(N(D_{2} ⊕ ⟨c⟩)) + 1 = dim(N(D_{2})) + 2 ≥ dim(N(D_{1})) + 1 ≥ dim(N(D)). This contradicts D ∈ pp(C). Now we have B ∈ pp(A), B ⊆ C_{1}, C_{1} ⊆ cl_{1}(B) and D_{1} ∈ pp(C_{1}). Since dim(D_{1}) < dim(D) and A ⊆ D_{1} ⊆ D we obtain by induction D_{1} = B_{1} +C_{1} where B_{1} ∈ pp(A). Hence D = B_{1} +C, as desired. □

6 EXTENDING A Δ-BASIS 28 Corollary 6.6 Let A be a finite subspace in a Δ-structure M. Then past(cl_{1}(past(A))) = cl_{1}(past(A)).

7 Completeness

In this section we show that Δ is a complete theory. Let H be a model of Δ, M be the corresponding Δ-structure, and \overline{a} be a tuple of elements linearly independent over Z(H). Then we also use \overline{a} to denote the corresponding tuple of images in M that are linearly independent. The elementary type of \overline{a} in H is determined by the isomorphism type of past(\overline{a}) in the corresponding Δ-structure M. The next lemma will be used for ω-saturated H to find finite subspaces of M with no proper past. Note that we assume A ⊆ B if we write B ∈ pp(A). If B ∈ pp(A) and A ⊆ A_{1}, then B + A_{1} ∈ pp(A_{1}). If B ∈ pp(A), B_{0} ⊆ B, and N(B) = N(B_{0}), then B_{0} ∈ pp(A ∩ B_{0}). If X is a Δ-basis of M that contains a Δ-basis Y of a finite subspace A, y is the maximal element of Y, and B ∈ pp(A), then B ⊆ ⟨X_{≤y}⟩. Furthermore for an infinite set A and a finite set C we have C ⊆ past(A) if there is a finite subset A_{1} of A and a finite set B ∈ pp(A_{1}) such that C ⊆ B.

Lemma 7.1 Let H be a model of Δ, M be the corresponding Δ-structure, and A be a finite subspace of M. Assume natural numbers m and n are given. Then there are elements b_{1}, …, b_{n} in M linearly independent over A such that for every B ∈ pp(A ⊕ ⟨b_{1}, …, b_{n}⟩) with dim(B) ≤ m there is some B_{A} with A ⊆ B_{A}, B = B_{A} ⊕ ⟨b_{1}, …, b_{n}⟩, and N(B) = N(B_{A}). Hence B_{A} ∈ pp(A).

Proof: Let h = 3^{m+2}n+1. Assume w.l.o.g. dim(A) ≥ 4 and M is countable. By Lemma 3.5 there is a Δ-basis X of past(A). There is some a_{3} ∈ A such that o_{X}(a_{3}) is the last element of the order given by X. W.l.o.g. a_{3} is the last element of X. If N(past(A)) ≠ N(past(⟨X_{<a_{3}}⟩)), then assume that a_{3} ∧ d − Ψ with d ∈ ⟨X_{<a_{3}}⟩ and Ψ ∈ ∧^{2}⟨X_{<a_{3}}⟩ is a generator of N(past(A)) over N(past(⟨X_{<a_{3}}⟩)). Choose a_{1}, a_{2} in past(A) such that a_{1}, a_{2}, a_{3} are linearly indpendent and d ∉ ⟨a_{1}, a_{2}, a_{3}⟩. Then the problem

x ∧ a_{1} = a_{3} ∧ a_{2}

has no solution in past(A). Let a_{4} be a solution in M. It exists by axiom (Σ2). By Lemma 6.5

past(A ⊕ ⟨a_{4}⟩) = past(A) ⊕ ⟨a_{4}⟩.

X{a_{4}} is a Δ-basis of this space and the problem

x ∧ a_{2} = a_{4} ∧ a_{3}

has no solution in it. Let a_{5} be a solution. Then by Lemma 6.5

past(A ⊕ ⟨a_{4}, a_{5}⟩) = past(A) ⊕ ⟨a_{4}, a_{5}⟩

and X{a_{4}, a_{5}} is a Δ-basis of this subspace. We can continue in this way and obtain a set Y = {a_{4}, …, a_{h}} where a_{i+2} is a solution of

x ∧ a_{i−1} = a_{i+1} ∧ a_{i}.

Then by Lemma 6.5

past(A ⊕ ⟨Y ⟩) = past(A) ⊕ ⟨Y ⟩

and XY is a Δ-basis of this space. For 1 ≤ i ≤ 3^{m+1}n let a_{i}^{m+1} be a_{3(i−1)+4}. We define A_{m+1} = {a_{1}^{m+1}, …, a_{3^{m+1}n}^{m+1}}. A_{m+1} is a subset of Y. Note that the problems

x ∧ a_{3(l−1)+1}^{m+1} = a_{3(l−1)+2}^{m+1} ∧ a_{3l)}^{m+1}

for 1 ≤ l ≤ 3_{n}^{m} have no solutions in past(A) ⊕ ⟨Y ⟩.

Using axiom (Σ2) we define inductively A_{j} = {a_{1}^{j}, …, a_{3^{j−1}n}^{j}} where a_{l}^{j} is a solution of the following problem

x ∧ a_{3(l−1)+1}^{j+1} = a_{3(l−1)+2}^{j+1} ∧ a_{3l}^{j+1}.

and show that this problem has no solution in past(A)⊕⟨Y A_{m} … A_{j+1}{a_{1}^{j}, …, a_{l−1}^{j}}. Define b_{i} = a_{i}^{1} for 1 ≤ i ≤ n. Then Z = XY A_{m} … A_{1} is a Δ-basis of ⟨Z⟩. By Corollary 6.5 we have past(⟨Z⟩) ⊆ ⟨Z⟩. Hence by Lemma 6.3 there is a Δ-basis W of M such that Z is an initial segment of W.

Now assume B ∈ pp(A ⊕ ⟨b_{1}, …, b_{n}⟩) with dim(B) = m. As mentioned above past(⟨Z⟩) ⊆ ⟨Z⟩. Hence B ⊆ ⟨Z⟩. Let B_{A} be B ∩ ⟨XY A_{m} … A_{2}⟩. Then B = B_{A} ⊕ ⟨b_{1}, …, b_{n}⟩. It is sufficient to prove N(B) = N(B_{A}). Hence we assume that N(B) ≠ N(B_{A}) and show a contradiction. Choose i minimal such that N(B) = N(⟨Z_{≤b_{i}}⟩ ∩ B). Assume that Φ_{1} generates N(B) ∩ ⟨Z_{≤b_{i}}⟩ over N(B∩⟨Z_{<b_{i}}⟩). Then Φ_{1} generates N(⟨Z_{≤b_{i}}⟩) over N(⟨Z_{<b_{i}}⟩). By Lemma 3.7

Φ_{1} = b_{i} ∧ a_{3(i−1)+1}^{2} − Ψ_{1}

where Ψ_{1} ∈ ∧^{2}⟨Z_{<b_{i}}⟩. Then a_{3(i−1)+1}^{2} ∈ B and Ψ_{1} ∈ ∧^{2}B. Since B ∈ pp(A ⊕ ⟨b_{1}, …, b_{n}⟩) there is some Φ_{2} ∈ N(B) linearly independent from Φ_{1} over ∧^{2}(B ∩⟨Z \{a_{3(i−1)+1}^{2}}⟩). If we write Φ_{2} as a linear combination of basic commutators with respect to Z, then for summands of the form a_{3(i−1)+1}^{2} ∧ d we get d ∈ ⟨a_{3^{2}(i−1)+1}^{3}, b_{i}⟩. Hence the linear independence of Φ_{1} and Φ_{2} over ∧^{2}(B ∩ ⟨Z \ {a_{(i−1)+1}^{2}}⟩) implies a_{3^{2}(i−1)+1}^{3} ∈ B and

Φ_{2} = a_{3(i−1)+1}^{2} ∧ a_{3^{2}(i−1)+1}^{3} − Ψ_{2}

where Ψ_{2} ∈ ∧^{2}(⟨Z \ {a_{3^{2}(i−1)+1}^{3}}⟩ ∩ B). Since B ∈ pp(A ⊕ ⟨b_{1}, …, b_{n}⟩) there is some Φ_{3} ∈ N(B) such that Φ_{2} and Φ_{3} are linearly independent over ∧^{2}(B ∩ ⟨Z \ {a_{3^{2}(i−1)+1}^{3}⟩). We continue in this way and get a_{3^{j−1}(i−1)+1}^{j} ∈ B for 1 ≤ j ≤ m + 1. This contradicts dim(B) = m. □

Theorem 7.2 1. Δ is complete.

2. Let G and H be two models of Δ. Let (a_{1}, …, a_{n}) be a tuple of elements in G linearly independent over Z(G). Let (b_{1}, …, b_{n}) be a tuple of elements in H linearly independent over Z(H). Let M and N be the corresponding Δ-structures respectively. Assume there is an isomorphism f_{n} of ⟨a_{1}, …, a_{n}⟩ onto ⟨b_{1}, …, b_{n}⟩ defined by f_{n}(a_{i}) = b_{i}. Let g_{n} be the induced S-isomorphism of the corresponding finite substructures of M and N. We also use a_{i} and b_{i} to denote their images in M and N respectively. If g_{n} can be extended to an isomorphism of past(⟨a_{1}, …, a_{n}⟩) onto past(⟨b_{1}, …, b_{n}⟩), then (a_{1}, …, a_{n}) in G and (b_{1}, …, b_{n}) in H have the same type.

Proof: 2) implies 1). To show 2) we can assume w.l.o.g. that G and H are ω_{1}-saturated. We describe a winnning strategy for player II in the Ehrenfeucht - Fraissé - Game. If the considered sequences are empty (n = 0), then 1) follows. We assume that (m − n) rounds are played and the following situation is provided. Note that for a ∈ G and b ∈ H we use a and b also to denote the corresponding elements in M and N respectively.

(EF)_{m}) There are extensions (a_{1}, …, a_{m}) of (a_{1}, …, a_{n}) in G and (b_{1}, …, b_{m}) of (b_{1}, …, b_{n}) in H of elements linearly independent over Z(G) and Z(H) respectively and an isomorphism f_{m} of ⟨a_{1}, …, a_{m}⟩ onto ⟨b_{1}, …, b_{m}⟩ defined by f_{m}(a_{i}) = b_{i}. We assume that f_{m} gives the desired isomophism for the chosen elements. Let g_{m} be the induced isomorphism of the corresponding Δ-structures. There is an isomorphism h_{m} of past(⟨a_{1}, …, a_{m}⟩) in M onto past(⟨b_{1}, …, b_{m}⟩) in N that extends g_{m}.

Now we assume w.l.o.g. that player I has chosen an element c ∈ G. If n = m = 0 and c ∈ Z(G), then player II fixes some element c_{1} ∈ G \ Z(G) and assumes that player I has chosen this element first. Working with c is then already the second round for him. Hence w.l.o.g. if n = m = 0, then first a_{1} ∈ G \ Z(G) is chosen. If 0 < m and c is linearly dependent from a_{1}, …, a_{m} modulo Z(G), then we can assume w.l.o.g. that c ∈ Z(G) \ (⟨a_{1}, …, a_{m}⟩)′. By axiom (Σ2) there is some d such that c = [a_{1}, d]. Then d is linearly independent from a_{1}, …, a_{m} modulo Z(G). Hence w.l.o.g. player I has chosen an element a_{m+1} linearly independent from a_{1}, …, a_{m} modulo Z(G). Now we see by Lemma 2.2 that it is sufficient to play the game in the corresponding Δ-structures only. We have to find b_{m+1} in N linearly independent from b_{1}, …, b_{m} such that we can extend h_{m} to an isomorphism h_{m+1} from past(⟨a_{1}, …, a_{m+1}⟩) onto past(⟨b_{1}, …, b_{m+1}⟩) with h_{m+1}(a_{m+1}) = b_{m+1}.

Let A_{i} be the subspace ⟨a_{1}, …, a_{i}⟩ of M. For every t let C_{t} the subspace of M generated by all B with dim(B) ≤ t and B_{t} ∈ pp(A_{m+1}). By Theorem 5.4 C_{t} is finite and past(A_{m+1}) is countable. Hence there is a countable elementary subgroup G_{0} of G such that past(A_{m+1}) ⊆ M_{0} where M_{0} is the substructure of M corresponding to G_{0}. By Lemma 6.2 there is a Δ-basis XY of M_{0} where the initial segment X is a Δ-basis for past(A_{m}). Let us fix t for the next considerations. XY induces a Δ-basis X_{t}Y_{t} of C_{t} such that the initial part X_{t} is a Δ-basis of C_{t} ∩ past(A_{m}). Using some Δ-transformations we can assume w.l.o.g. that X_{t} ⊆ X and Y_{t} ⊆ Y. The same argument as for Lemma 3.4 works.

(14) There is some isomorphism h_{m+1}^{t} of C_{t} into N extending the restriction of h_{m} to C_{t} ∩ past(A_{m}) such that for every B with dim(B) ≤ t and B ∈ pp(h_{m+1}^{t}(A_{m+1})) we have B ⊆ h_{m+1}^{t}(C_{t}).

Proof: Since XY_{t} is a Δ-basis of past(A_{m}) + C_{t} we can find a permutation ZW of Y_{t} such that N(⟨XZ⟩) = N(⟨X⟩) and W ⊆ cl_{1}(⟨XZ⟩).

If Z ≠ ∅, then by Lemma 7.1 and the ω_{1}-saturation of H there is a set Z^{N} with |Z| = |Z^{N}| and past(h_{m}(A_{m}) ⊕ ⟨Z^{N}⟩) = past(h_{m}(A_{m})) ⊕ ⟨Z^{N}⟩. For Z = ∅ let Z^{N} be ∅. Using a one-one correspondence between Z and Z^{N} we can extend h_{m} to an isomorphism u of cl_{1}(⟨XZ⟩) onto cl_{1}(⟨h_{m}(X)Z^{N}⟩). Axiom (Σ2) is used. Then u is defined on ⟨XZW ⟩. Let B ∈ pp(u(A_{m+1})) with dim(B) ≤ t. We show B ⊆ u(C_{t}).

By construction u(A_{m+1}) ∈ cl_{1}(past(h_{m}(A_{m}) ⊕ ⟨Z^{N}⟩)). By Corollary 6.6 there is some B_{1} such that B_{1} ∈ pp(h_{m}(A_{m}) ⊕ ⟨Z^{N}⟩) and B ⊆ cl_{1}(B_{1}). By the properties of Z^{N} there are E_{2} ∈ pp(h_{m}(A_{m})) and E_{3} ⊆ ⟨Z^{N}⟩ such that B_{1} = E_{2} ⊕ E_{3}. Then u^{−1}(E_{2}) ∈ pp(A_{m}) and u^{−1}(E_{3}) ⊆ ⟨Z⟩. Hence F_{1} = u^{−1}(B_{1}) ⊆ past(A_{m}) ⊕ ⟨Z⟩. Since B ⊆ cl_{1}(B_{1}) there is a chain B_{1} ⊆ B_{2} ⊆ … ⊆ B_{k} such that B_{i+1} is an algebraic one-extension of B_{i} and B ⊆ B_{k}. Using axiom (Σ2) we find a chain F_{1} ⊆ F_{2} ⊆ … ⊆ F_{k} in M such that F_{i+1} is an algebraic one-extension of F_{i} and u(F_{i}) = B_{i}. By the isomorphism u between B_{k} and F_{k} we obtain u^{−1}(B) ∈ pp(A_{m+1}). Hence u^{−1}(B) ⊆ C_{t} and therefore B ⊆ u(⟨X_{t}ZW ⟩) = u(C_{t}). The restriction of u on C_{t} is the desired isomorphism h_{m+1}^{t}. This proves (14). □

By ω_{1}-saturation and (14) we get an element b_{m+1} such that for every t there is a subspace D_{t} in N and an isomorphism h_{m+1}^{t} of C_{t} onto D_{t} that is an extension of h_{m}, sends a_{m+1} to b_{m+1}, and D_{t} is generated by all subspaces B with dim(B) ≤ t and B ∈ pp(⟨b_{1}, … b_{m+1}⟩). Now we consider the following tree. At level t we have isomorphisms of C_{t} onto D_{t} extending h_{m} and sending a_{m+1} to b_{m+1}. An isomorphism of level t+1 is connected with an isomorphism of level t if the first one extends the second one. Since the C_{t} are finite we can apply König’s Lemma and get the desired isomorphism h_{m+1}. □

Corollary 7.3 Let H_{0} and H_{1} be subgroups with (Σ2)^{w} of a model G of Δ. Assume past(\overline{H_{i}}) ⊆ \overline{H_{i}}. Assume that there is an isomorphism f of H_{0} onto H_{1}. Then H_{0} and H_{1} have the same type.

Proof: If K is a finite subspace of \overline{H_{0}}, then f(past(K) = past(f(K)). □

8 Stability

We need a modified version of Theorem 5.4.

Theorem 8.1 There is a function g(m) from the naturals into the naturals such that for every pair H_{0} ⪯ H of models of Δ where H_{0} is an elementary substructure of H the following holds: Let M_{0} and M respectively be the corresponding Δ-structures. If A is a finite subspace of M, then g(m) is a bound for the number of subspaces B modulo M_{0} where dim(B) = m, A ⊆ B, and B ∈ pp(A + (B ∩ M_{0})).

Proof: It is sufficient to consider countable models H only.

First we show that there is a function h(m) such that for all H, H_{0}, M, M_{0} as above, for all finite subspaces A of M and all B_{1}, …, B_{t} that are linearly independent over A with dim(B_{i}) = m, A ⊆ B_{i}, B_{i} ≠ A+(M_{0}∩B_{i}), B_{i} ≠ B_{j} modulo M_{0} for i ≠ j, and B_{i} ∈ pp(A + (B_{i} ∩ M_{0})) we have t ≤ h(m). Since we can restrict us to countable M by Lemma 6.2 we can assume that we have a Δ-basis XY of M where the initial part X is a Δ-basis of M_{0}. W.l.o.g. we can suppose that XY contains a basis Z for A. Let {z_{1}^{i}, …, z_{m}^{i}} be a Δ-basis of B_{i} that contains Z and is induced by XY. Then there is some m_{i} such that z_{1}^{i}, …, z_{m_{i−1}}^{i} ∈ M_{0} and z_{m_{i}}^{i}, …, z_{m}^{i} are linearly independent over M_{0}. We say B_{i} ∼ B_{j} if g(z_{l}^{i}) = z_{l}^{j} induces an isomorphism of B_{i} onto B_{j} as Δ-structures with an additional predicate for for the intersection with M_{0}. Let h_{∼} be a bound for the number of ∼-classes. By Lemma 5.2 there are at most m + 1 elements in each ∼-class. Hence h(m) = (m + 1)h_{∼}.

Next we show by induction on i that there is a function g(m, i) such that for all H, H_{0}, M, M_{0} as above and every finite subspace A of M with (m − i) ≤ dim(A) < m the number of subspaces B of M modulo M_{0} with dim(B) = m, A ⊆ B, and B ∈ pp(A + (M_{0} ∩ B)) is less or equal to g(m, i).

First assume dim(A) = m − 1. We choose a maximal set {B_{1}, …, B_{t}} of subspaces B_{i} with A ⊆ B_{i}, B_{i} ≠ A + (M_{0} ∩ B_{i}), and B_{i} ∈ pp(A + (M_{0} ∩ B_{i})). that are linearly independent over A and different modulo M_{0}. As shown above t ≤ h(m). Let B be a subspace with A ⊆ B, B ∈ pp(A + (M_{0} ∩ B). Then either B = A modulo M_{0} or by the maximality of t we have B ⊆ B_{1}+…+B_{t} modulo M_{0}. Hence g(m, 1) = 2^{(p^{h(m)m})} has the desired properties.

Now assume that g(m, i) is given and we want to construct g(m, i + 1). Let A be a subspace of M of size m − (i + 1). Again we consider a maximal set {B_{1}, …, B_{t}} linearly independent over A with the following properties: B_{i} ≠ B_{j} modulo M_{0} for i ≠ j, dim(B_{i}) = m, A ⊆ B_{i}, B_{i} ≠ A + (M_{0} ∩ B_{i}), and B_{i} ∈ pp(A + (M_{0} ∩ B_{i})). As shown above t ≤ h(m). Let B be a subspace of M with dim(B) = m, A ⊆ B, and B ∈ pp(A + (M_{0} ∩ B)). Then B = A modulo M_{0} or there is some a ∈ (B ∩ (B_{1} + … + B_{t})) \ A. If we define A_{1} = A + ⟨a⟩, then B ∈ pp(A_{1} + (M_{0} ∩ B). The number of possible A_{1} in B_{1} + … + B_{t} is bounded by 2^{(}p^{h(m)m}). Hence by induction

g(m, i + 1) = g(m, i)2^{(p^{h(m)m})} + g(m, i)

has the desired properties. We define g(m) = g(m, m − 1). □

Lemma 8.2 Let H be a model of Δ considered as an elementary submodel of the monster model C of Δ. Let M ⊆ M_{C} be the corresponding Δ-structures. Let \overline{a} = (a_{1}, …, a_{n}) be a tuple of elements in M_{C} linearly independent over M.

1. There is a finite subset A of M such that N(M + ⟨\overline{a}⟩) = N(M) + N(⟨A ∪ \{\overline{a}\}⟩). Then furthermore dim(N(⟨A ∪ \{\overline{a}\}⟩)) − dim(N(⟨A⟩)) < n.

2. There is a countable elementary submodel H\overline{a} of H such that for the corresponding Δ-substructure M\overline{a} of M the following holds: past(M\overline{a} + ⟨\overline{a}⟩) ∩ M = M\overline{a}, past(M + ⟨\overline{a}⟩) = past(M\overline{a} + ⟨\overline{a}⟩) + M, and N(past(M + ⟨\overline{a}⟩)) = N(past(M\overline{a} + ⟨\overline{a}⟩)) + N(M).

Proof: ad 1) If \overline{a} ⊆ M, then A = ⟨\overline{a}⟩. Otherwise let Φ_{1}, …, Φ_{t} be elements of N(M+⟨\overline{a}⟩) that are linearly independent over N(M). Then t < n. Otherwise we use \overline{a} also to denote the corresponding tuple in C/Z(C) = \overline{C}. Then ⟨\overline{a}⟩ ⊆ cl(\overline{H}) ⊆ acl(\overline{H}) = \overline{H} by Lemma 4.1, a contradiction. Choose t maximal. Then Φ_{1}, …, Φ_{t} generate N(M + ⟨\overline{a}⟩) over N(M). Let A be any finite subspace of M such that each Φ_{j} is defined over A. ad 2) By Theorem 8.1 we can find a countable sum B of finite subspaces B_{i} of M_{C} such that B_{i} ∈ pp(⟨\overline{a}⟩ + (M ∩ B_{i})) and past(M + ⟨\overline{a}⟩) = M + B. By 1) there is a countable subspace C of M such that N(M + B) = N(C + B) + N(M). Let H_{0} be a countable elementary submodel of H and let M_{0} be the corresponding Δ-substructure of M such that B ∩ M and C are subspaces of M_{0}. Then past(M_{0} + ⟨\overline{a}⟩) + M = past(M + ⟨\overline{a}⟩) and N(past(M + ⟨\overline{a}⟩)) = N(M)+N(past(M_{0}+⟨\overline{a}⟩)). Now we choose an elementary chain of countable elementary submodels H_{i} of H and the corresponding Δ-substructures M_{i} of M such that M_{i} + (past(M_{i} + ⟨\overline{a}⟩) ∩ M) ⊆ M_{i+1}. Then H\overline{a} = ∪_{0≤i<ω}H_{i} has the desired properties. □

Corollary 8.3 Δ is a stable theory.

Proof: We count the 1-types tp(a/H) over a model H of Δ. As above we consider H as an elementary submodel of some monster model C. Let M ⊆ M_{C} be the corresponding Δ-structures. Let us fix some g ∈ G \ Z(G). If a − c ∈ Z(C) where c ∈ H, then by axiom (Σ2) there is some d ∉ Z(C) such that (a − c) = [g, d]. Hence tp(a/H) is algebraic over tp(d/H).It is sufficient to count types that contain the formulas x − c ∉ Z for all c ∈ H.

Let us consider tp(a/H) where a is not in Z(C) modulo H. We choose a countable elementary submodel H_{a} of H and the corresponding M_{a} according to Lemma 8.2 2). Now let b be any element in C such that tp(a/H_{a}) = tp(b/H_{a}), past(M_{a}+⟨b⟩)∩M = M_{a}, past(M +⟨b⟩) = past(M_{a}+⟨b⟩)+M, and N(past(M+⟨b⟩)) = N(past(M_{a}+⟨b⟩))+N(M). tp(a/H_{a}) = tp(b/H_{a}) implies the existence of some isomorphism f of past(M_{a} + ⟨a⟩) onto past(M_{a} + ⟨b⟩) that is the identity on M_{a} and sends a to b since past ⊆ acl. By the properties of M_{a} with respect to a and b we see that we can extend f to an isomorphism g past(M + ⟨a⟩) onto past(M + ⟨b⟩) that is the identity on M. By Theorem 7.2 we have tp(a/H) = tp(b/H).

We have shown that |S_{1}(H)| is the sum of all |S_{1}(H_{0})| where H_{0} varies over all countable elementary submodels of H. It follows stability. □

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