December 20, 2006 18:32 WSPC/INSTRUCTION FILE redfusion3.pub

FUSION OVER A VECTOR SPACE

ANDREAS BAUDISCH, AMADOR MARTIN-PIZARRO, MARTIN ZIEGLER Institut für Mathematik Humboldt-Universität zu Berlin D-10099 Berlin, Germany Institut Camille Jordan Université Claude Bernard Lyon 1 69622 Villeurbanne cedex, France Mathematisches Institut Albert-Ludwigs-Universität Freiburg D-79104 Freiburg, Germany

Let T_{1} and T_{2} be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T_{1} ∪ T_{2} has a strongly minimal completion.

Keywords: Model Theory, Strongly minimal set, Fusion

Mathematics Subject Classification 2000: 03C45, 03C50

1. Introduction

In [1] E. Hrushovski answered negatively a question posed by G. Cherlin about the existence of maximal strongly minimal sets in a countable language by constructing the fusion of two strongly minimal theories:

Theorem . Let T_{1} and T_{2} be two countable strongly minimal theories, in disjoint languages, and with the DMP, the definable multiplicity property. Then T_{1} ∪ T_{2} has a strong minimal completion.

The above theorem was proved by extending Fraïssé’s amalgamation procedure to a given class in which Hrushovski’s “δ–function” will determine the pregeometry. In order to axiomatize the theory of the generic model, a set of representatives of rank 1 types or “codes” is chosen in a uniform way.

From now on, let F denote a fixed finite field and T_{0} the theory of infinite F – vector spaces in the language L_{0} = {0, +, −, λ}_{λ∈F}. In this article, we will prove the following:

Theorem 1.1. Let T_{1} and T_{2} be two countable strongly minimal extensions of T_{0} with the DMP, and assume that their languages L_{1} and L_{2} intersect in L_{0}. Then T_{1} ∪ T_{2} has a strongly minimal completion T^{μ}.

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This “fusion over a vector space” was proposed by Hrushovski in [1]. In the special case where both T_{1} and T_{2} are 1–based this fusion was already proved by A. Hasson and M. Hils [2]. These two articles also discuss fusions over more general T_{0}.

Our proof uses Hrushovski’s machinery. Schematically, it follows [3], which is a streamlined account of Hrushovski’s aforementioned paper.

In [4] and [5] it was explained how to apply Hrushovski’s method to construct “fields with black points” (see also [6]). In a similar way, the techniques exhibited here were used in [7] to construct “fields with red points” (fields with a predicate for an additive subgroup, of Morley rank 2), whose existence was conjectured in [8].

The theories T^{μ}, which depend on the choice of codes and of a certain function μ, have the following properties:

Theorem 1.2. Let M be a model of T^{μ}.

1. Let tr_{i} denote the transcendence degree in the sense of T_{i} and dim the F –linear dimension. Then for every finite subset A of M we have

dim(A) ≤ tr_{1}(A) + tr_{2}(A).

2. Let N be a model of T^{μ} which extends M. Then N ≺ M if N is an elementary extension of M in the sense of T_{1} and in the sense of T_{2}.

It follows^{a} from 1. that for every p there is a strongly minimal structure (K, +, ⊙, ⊗) such that (K, +, ⊙) and (K, +, ⊗) are algebraically closed fields of characteristic p and for every transcendental x the ⊙–powers

1_{⊙}, x, x ⊙ x, x ⊙ x ⊙ x, …

are algebraically independent in the sense of (K, +, ⊗), and vice versa.

2. Codes

Let us fix the following notation: T is a countable strongly minimal extension of T_{0} with the DMP, ℂ denotes the monster model of T, tr(a/A) the transcendence degree^{b} of the tuple a over A, MR(p) the Morley rank of the type p. Thus we have

tr(a/A) = MR(tp(a/A)).

We use

ϕ(x) ∼^{k} ψ(x)

or ϕ(x) ∼_{x}^{k} ψ(x) to express that the Morley rank of the symmetric difference of ϕ and ψ is smaller than k,

aWe will explain this at the end of the paper (p. 23). bThe maximal number of components of a which are algebraically independent over A.

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We denote by ⟨a⟩ We denote by the F –vector space of dimension dim(a) spanned by the components of the n–tuple a. Subspaces of ⟨a⟩ can be described in terms of subspaces U of F^{n} as

Ua = {∑_{i=1}^{n} u_{i}a_{i} | u ∈ U}.

We call a stationary type a group type (or coset type) if it is the generic type of a (coset of a) connected definable subgroup of (ℂ^{n}, +). These properties depend only on the parallel class. So we can call a formula of Morley degree 1 a group formula (or coset formula) if it belongs to a group type (or a coset type) of the same rank.

Given a group formula χ(x) of rank k, we denote by Inv(χ) the group of all H ∈ Gl_{n}(F) which map the generic realizations of χ to generic realizations, or, equivalently, for which H(χ) ∼^{k} χ. If χ is a coset formula, Inv(χ) is Inv(χ^{g}) where χ^{g} is the associated group formula^{c}.

A definable set X ⊂ ℂ^{n} of rank k is encoded by φ(x, y) if n = |x| and there is some tuple b such that X ∼^{k} φ(x, b).

A code c is a parameter free formula ϕ_{c}(x, y) where the variable x ranges over n_{c}–tuples of the home sort and y over a sort of T^{eq}, with the following properties.

C(i) All non–empty^{d} ϕ_{c}(x, b) have (constant) Morley rank k_{c} and Morley degree

1. C(ii) For every U ≤ F^{n_{c}} there is a number k_{c,U} such that for every realization a

of ϕ_{c}(x, b) we have:

tr(a/b, Ua) ≤ k_{c,U}.

Moreover, equality holds for generic a. (So we have k_{c} = k_{c,0}.) C(iii) dim(a) = n_{c} for all realizations a of ϕ_{c}(x, b). If a is generic, then

dim(a/ acl(b)) = n_{c} (this is equivalent to k_{c,U} = k_{c}−1 for all one–dimensional U).

C(iv) If ϕ_{c}(x, b) and ϕ_{c}(x, b′) are not empty and ϕ_{c}(x, b) ∼^{k_{c}} ϕ_{c}(x, b′), then b = b′. C(v) If some non–empty ϕ_{c}(x, b) is a coset formula, then all are. We call such a

code c a coset code. In this case, the group Inv(ϕ_{c}(x, b)) does not depend on b (whenever it is defined). Hence we denote it by Inv(c).

C(vi) For all b and m the set defined by ϕ_{c}(x + m, b) is encoded by ϕ_{c}. C(vii) There is a subgroup G_{c} of Gl_{n_{c}}(F) such that:

a) for all H ∈ G_{c} and all non–empty ϕ_{c}(x, b) there exists a (unique) b^{H} such that

ϕ_{c}(Hx, b) ≡ ϕ_{c}(x, b^{H}).

cThis is χ(x − m) for a generic realization m of χ(x). dCodes where all ϕ(x, b) are empty will not be considered.

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b) if H ∈ Gl_{n_{c}}(F) \ G_{c}, then no non–empty ϕ_{c}(Hx, b) is encoded by ϕ_{c}.

Two codes c and c′ are equivalent if for every b there is some b′ such that ϕ_{c}(x, b) ≡ ϕ_{c′}(x, b′) and vice versa. If c is a code and H ∈ Gl_{n_{c}}(F), then

ϕ_{c^{H}} (x, y) = ϕ_{c}(Hx, y)

is also a code. C(viia) states that c^{H} and c are equivalent if H lies in G_{c}.

Corollary 2.1. Let p ∈ S(b) be the generic type containing ϕ_{c}(x, b). Then b is the canonical base of p.

Proof. Immediate from C(iv).

A formula χ(x, d) is simple if it has Morley degree 1 and dim(a/ acl(d)) = |x| for all generic realizations a of χ(x, d). The second half of C(iii) states that all non–empty ϕ_{c}(x, b) are simple.

Lemma 2.2. Every simple formula χ(x, d) can be encoded by some code c.

I.e.

χ(x, d) ∼^{k_{c}} ϕ_{c}(x, b_{0})

for some parameter b_{0}. By C(iv) it follows that b_{0} is uniquely determined, thus b_{0} ∈ dcl^{eq}(d).

Proof. Set n_{c} = |x|, k_{c} = MR χ(x, d) and k_{c,U} = tr(a/d, Ua) for a generic realization a of χ(x, d). Let p be the global type of rank k_{c} containing χ(x, d) and b_{0} its canonical base and choose some ϕ(x, b_{0}) ∈ p of rank k_{c} and degree 1. Hence, ϕ(x, b_{0}) satisfies χ(x, d) ∼^{k_{c}} ϕ_{c}(x, b_{0}) and has property C(iv) for all b and b′ realizing tp(b_{0}). We can choose ϕ(x, b_{0}) strong enough to ensure that C(iv) holds for all b and b′.

Consider now the set X of all b of same length and sort as b_{0} for which ϕ(x, y) satisfies C(i), C(ii), C(iii) and C(v). The latter means that ϕ(x, b) is a coset formula iff ϕ(x, b_{0}) is, and in this case Inv(ϕ(x, b)) = Inv(ϕ(x, b_{0})). Let us check that X is definable by a countable disjunction of formulae. This is clear for C(i) and C(iii). The second part in C(iii) is a special case of C(ii), and the latter follows from the fact that tr(a/b, Ua) ≥ k_{c,U} is equivalent to tr(Ua/b) ≤ (k_{c} − k_{c,U}) for generic a in ϕ(x, b). We refer to [7] for C(v), where it is shown that the set of all b such that ϕ(x, b) is a group (coset) formula is definable.

All b realizing tp(b_{0}) belong to X. So a finite part θ(y) of this type implies X. Then the formula

ϕ′_{c}(x, y) = ϕ(x, y) ∧ θ(y)

has all properties, except possibly C(vi) and C(vii).

Given any n_{c}–tuple m and parameter b, the formula ϕ′_{c}(x + m, b), if non–empty, has again rank k_{c} and degree 1. If a is a generic realization, then a + m is a generic

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realization of ϕ′_{c}(x, b) and a + m | ⌣_{b} m. Let u be some vector in F^{n_{c}} such that ∑_{i} u_{i}a_{i} ∈ acl(b, m). Then ∑_{i} u_{i}(a_{i} + m_{i}) ∈ acl(b, m). By independence ∑_{i} u_{i}(a_{i} + m_{i}) ∈ acl(b), which implies u = 0. Therefore dim(a/ acl(b, m)) = n_{c} and ϕ′_{c}(x+m, b) is simple. We note also that for every U

tr(Ua/m, b) = tr(U(a + m)/m, b) = tr(U(a + m)/b),

which implies tr(a/m, b, Ua) = k_{c,U}.

Whence, each ϕ′_{c}(x + m, b) can be encoded by some formula ϕ′(x, y) which has all properties of codes except possibly C(vi) and C(vii). Since these properties can be expressed by a countable disjunction we conclude that there is a finite sequence of formulae ϕ_{1}, …, ϕ_{r} with all properties except possibly C(vi) and C(vii) which encode all formulas ϕ′_{c}(x + m, b) with m and b varying. Moreover, we may assume that for all i

⊧ ∀y ∃v, w ϕ_{i}(x, y) ∼_{x}^{k_{c}} ϕ′_{c}(x + v, w),

which implies that either all or none of the ϕ_{i} code coset formulas and if so, they have all the same invariant group Inv(ϕ(x, b_{0})).

To prevent double-encoding, set

θ_{i}(y) = ⋀_{j<i} ∀z ϕ_{j}(x, z) ≁_{x}^{k_{c}} ϕ_{i}(x, y).

Fix a sequence of different constants^{e} w_{1}, …, w_{r} and define

ϕ′′_{c}(x, y, y′) = ⋁_{i=1}^{r} ϕ_{i}(x, y) ∧ θ_{i}(y) ∧ y′ =^{.} w_{i}.

ϕ′′_{c}(x, y) has all properties except possibly C(vii). To prove C(vi) fix m and b, w such that ϕ′′_{c}(x + m, b, w) is not empty. Then w equals some w_{j} and ϕ′′_{c}(x + m, b, w) is equivalent to ϕ_{j}(x + m, b). We know that ϕ_{j}(x, b) ∼ ϕ′_{c}(x + m′, b′) for some m′ and b′. It follows that: ϕ_{j}(x+m, b) ∼ ϕ′_{c}(x+(m+m′), b′). Since ϕ′_{c}(x+(m+m′), b′) can be encoded by one of the ϕ_{i}, property C(vi) holds.

Only property C(vii) remains to be obtained. Change the notation slightly and assume χ(x, d) ∼^{k_{c}} ϕ′′_{c}(x, b_{0}). Define G_{c} to be the set of all A ∈ Gl_{n_{c}}(F) such that there is some m and some realization b of p = tp(b_{0}) such that ϕ′′_{c}(Ax, b_{0}) ∼^{k_{c}} ϕ′′_{c}(x + m, b). To show that G_{c} is a group, consider another A′ ∈ G_{c}. Then there are m′ and b′ ⊧ p such that ϕ′′_{c}(A′x, b) ∼^{k_{c}} ϕ′′_{c}(x + m′, b′). This yields ϕ′′_{c}(AA′x, b_{0}) ∼^{k_{c}} ϕ′′_{c}(A′x + m, b) ≡ ϕ′′_{c}(A′(x + A^{′−1}m), b) ∼^{k_{c}} ϕ′′_{c}(x + (A^{′−1}m + m′), b′), and so AA′ ∈ G_{c}.

There is a ρ(y) ∈ p such that for no A ∈ Gl_{n_{c}}(F) \ G_{c} there are some b which satisfies ρ and some tuple m with ϕ′′_{c}(Ax, b_{0}) ∼^{k_{c}} ϕ′′_{c}(x + m, b), i.e.

⊧ ⋀_{A∈Gl_{nc} (F)\G_{c}} ¬ρ_{A}(b_{0}),

eIf T has no constants, use definable elements in a sort of T^{eq}.

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where

ρ_{A}(y) = ∃z, y′ ρ(y′) ∧ ϕ′′_{c}(Ax, y) ∼_{x}^{k_{c}} ϕ′′_{c}(x + z, y′).

Whence the formula

σ(y) = ⋀_{A∈G_{c}} ρ_{A}(y) ∧ ⋀_{A∈Gl_{nc} (F)\G_{c}} ¬ρ_{A}(y)

is satisfied by b_{0}. An easy calculation shows

⊧ ∀y (σ(y) → ( ⋀_{A∈G_{c}} σ^{A}(y) ∧ ⋀_{A∈Gl_{nc} (F)\G_{c}} ¬σ^{A}(y))),

where:

σ^{A}(y) = ∃y′ σ(y′) ∧ ϕ′′_{c}(Ax, y) ∼_{x}^{k_{c}} ϕ′′_{c}(x, y′).

Write now

ϕ′′′_{c}(x, y) = ϕ′′_{c}(x, y) ∧ σ(y).

It is clear that ϕ′′′_{c} still encodes χ(x, d) and has all properties except possibly C(vii). For C(vi) assume ϕ′′_{c}(x+m, b) ∼^{k_{c}} ϕ′′_{c}(x, b′). b′ satisfies ρ_{A} iff , ϕ′′_{c}(Ax, b′) ∼^{k_{c}} ϕ′′_{c}(x+ m′, b′′) for some m′ and some realization b′′ of ρ, or, equivalently, ϕ′′_{c}(Ax, b) ∼^{k_{c}} ϕ′′_{c}(x + (m′ − A^{−1}m), b′′). Therefore b satisfies ρ_{A} iff b′ satisfies ρ_{A}. This implies that b satisfies σ_{A} iff b′ satisfies σ_{A}. So C(vi) holds.

Now, C(vii) is satisfied by ϕ′′′_{c} and G_{c} only in the weaker form that ϕ′′′_{c}(Hx, b) is encoded by ϕ′′′_{c} iff H ∈ G_{c}. By C(iv) we can define for each A ∈ G_{c} a function b ↦ b^{A} such that

ϕ′′′_{c}(Ax, b) ∼^{k_{c}} ϕ′′′_{c}(x, b^{A})

and set:

ϕ_{c}(x, y) = ⋀_{A∈G_{c}} ϕ′′′_{c}(A^{−1}x, y^{A}).

Since ϕ_{c}(x, b) ∼^{k_{c}} ϕ′′′_{c}(x, b) only C(viia) needs to be check: Given H ∈ G_{c},

ϕ_{c}(Hx, b) ≡ ⋀_{A∈G_{c}} ϕ′′′_{c}(A^{−1}Hx, b^{A}) ≡ ⋀_{A∈G_{c}} ϕ′′′_{c}(A^{−1}x, b^{HA}) ≡ ϕ_{c}(x, b^{H}).

Lemma 2.3. There is a set C of codes with the following properties:

C(viii) Every simple formula is encoded by a unique c ∈ C. C(ix) For all c ∈ C and all H ∈ Gl_{n_{c}}(F) the code c^{H} is equivalent to some code

in C.^{f}

fWe will construct C so that every c^{H} is equivalent to some c^{H′} which belongs to C. (We identify codes with equivalent formulas.)

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Proof. Work inside an ω–saturated model M of T and enumerate all simple formulas χ_{i}, i = 1, 2, … with parameters in M. We need only show that all χ_{i} can be encoded in C. We construct C as an increasing union of finite sets ∅ = C_{0} ⊂ C_{1} ⊂ · · ·. Assume that C_{i−1} is defined and closed under the action of Gl(F) in the sense of C(ix). If χ_{i} can be encoded in C_{i−1}, we set C_{i} = C_{i−1}. Otherwise choose some code c′ which encodes χ_{i}. Let ρ(b) express, that ϕ_{c′}(x, b) cannot be encoded in C_{i−1} and define

ϕ_{c}(x, y) = ϕ_{c′}(x, y) ∧ ρ(y).

Then ϕ_{c} still encodes χ_{i}. Moreover ϕ_{c} determines again a code: only C(vii) needs to be considered. So assume that ⊧ ρ(b) and let H be in G_{c′}. We need to show that ⊧ ρ(b^{H}). Otherwise ϕ_{c′}(Hx, b) can be encoded in C_{i−1}. Since C_{i−1} is closed under H^{−1}, also ϕ_{c′}(x, b) can be encoded in C_{i−1}, which is a contradiction.

Choose now a system of right representatives A_{1}, …, A_{r} of G_{c} in Gl_{n_{c}}(F) and set C_{i} = C_{i−1} ∪ {c^{A_{1}}, …, c^{A_{r}}}.

3. Difference sequences

As in the previous section, T denotes a countable strongly minimal extension of T_{0} with the DMP.

Let us recall the following lemma, which will be useful to distinguish whether or not a formula determines a coset of a group, according to the independence among generic realizations.

Lemma 3.1. Let ϕ(x) be a formula over B, of Morley degree 1, and e_{0} and e_{1} two generic B–independent realizations. If H ∈ Gl_{n}(F) and e_{0} | ⌣_{B} e_{0} − He_{1}, then ϕ(x) is a coset formula and H ∈ Inv(ϕ(x)).

Proof. It follows from

MR(He_{1}/B, He_{1} − e_{0}) = MR(e_{0}/B, He_{1} − e_{0}) = MR(e_{0}/B) ≥ MR(He_{1}/B)

that e_{0}, He_{1} and He_{1} − e_{0} are pairwise independent over B. By [9] e_{0}, He_{1} and He_{1} − e_{0} are generic elements of B–definable cosets of a B–definable group G. Whence ϕ(x) is a coset formula and HG = G.

We fix now for every code c a number m_{c} ≥ 0 such that for no ϕ_{c}(x, b) there is a Morley sequence (e_{i}) of length m_{c} and some b′ from the same sort as b with e_{i} |̸ ⌣_{b} b′ for all i.

Theorem 3.2. For every code c and any number μ > m_{c} there exists a parameter free formula Ψ_{c}(x_{0}, …, x_{μ}), whose realizations are called difference sequences (of length μ), with the following properties.

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P(i) If e′_{0}, …, e′_{μ}, f is a Morley sequence of ϕ_{c}(x, b), then e′_{0} − f, …, e′_{μ} − f is a

difference sequence.^{g} P(ii) For every difference sequence e_{0}, …, e_{μ} there is a unique b with ⊧ ϕ_{c}(e_{i}, b)

for all i (we call the base of the sequence). Furthermore, b is uniquely determined if ϕ_{c}(e_{i}, b) holds for at least m_{c} many i’s.^{h}

P(iii) If e_{0}, …, e_{μ} is a difference sequence then so is

e_{0} − e_{i}, …, e_{i−1} − e_{i}, −e_{i}, e_{i+1} − e_{i}, …, e_{μ} − e_{i}.

P(iv) Let e_{0}, …, e_{μ} be a difference sequence with base b. We distinguish two cases: Suppose c is not a coset code:

a) If e_{i} is generic in ϕ_{c}(x, b), then e_{i} |̸ ⌣_{b} e_{i} − He_{j} for all H ∈ Gl_{n_{c}}(F) and i ≠ j.

Suppose c is a coset code:

b) ϕ_{c}(x, b) is a group formula.

c) Ψ_{c}(e_{0}, …, e_{i−1}, e_{i} − e_{j}, e_{i+1}, …, e_{μ}) for all i ≠ j.^{i}

d) Ψ_{c}(e_{0}, …, e_{i−1}, He_{i}, e_{i+1}, …, e_{μ}) for all H ∈ Inv(c).^{i}

e) If e_{i} is a generic realization of ϕ_{c}(x, b), then e_{i} |̸ ⌣_{b} e_{i} − He_{j} for all i ≠ j and H ∈ Gl_{n_{c}}(F) \ Inv(c).

P(v) For all H ∈ G_{c}

Ψ_{c}(x_{0}, …, x_{μ}) ≡ Ψ_{c}(Hx_{0}, …, Hx_{μ}).

The derived sequences of of (e_{i}) consist of all difference sequences obtained from (e_{i}) by iteration of the transformations described in P(iii). Note that all permutations can be derived and have the same base (by P(ii)). We will later use a more refined notation: if in the derivation process only indices ≤ λ are involved, then we call the resulting derivation a λ–derivation.

Proof. Consider the following property DS(e_{0}, …, e_{μ}):

There is some b′ and a Morley sequence e′_{0}, …, e′_{μ}, f′ of ϕ_{c}(x, b′) such that e_{i} = e′_{i} − f′. This is clearly a partial type.

Claim: DS has all properties of Ψ_{c}.

Proof: Assume e_{i} = e′_{i} − f′ for a Morley sequence (e′_{i}), f′ of ϕ_{c}(x, b′). Then (e_{i}) is a Morley sequence of ϕ_{c}(x + f′, b′) over b′, f′. If ϕ_{c}(x + f′, b′) ∼ ϕ_{c}(x, b), then (e_{i})

gIn general b will not be the base of (e′_{i}) in the sense of P(ii). hIt follows that b ∈ dcl(e_{i_{1}}, …, e_{i_{mc}}) for all 0 ≤ i_{1} < ⋯ i_{m_{c}} ≤ μ. iBy P(ii) and μ > m_{c} this new sequence has also base b.

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is a Morley sequence of ϕ_{c}(x, b).^{j} P(ii) Suppose ⊧ ϕ_{c}(e_{i}, b′′) for m_{c}–many i’s. Then there exists such an i with e_{i} | ⌣_{b} b′′. Hence MR(ϕ_{c}(x, b) ∧ ϕ_{c}(x, b′′)) = k_{c} and therefore b = b′′. P(iii) Fix i ∈ {0, …, μ} and note that e′_{0}, …, e′_{i−1}, f′, e′_{i+1}, …, e′_{μ}, e′_{i} is again a Morley sequence for ϕ_{c}(x, b′). Hence, the sequence

\begin{matrix}e′_{0} − e′_{i}, …, e′_{i−1} − e′_{i}, f′ − e′_{i}, e′_{i+1} − e′_{i}, …, e′_{μ} − e′_{i} = \\ e_{0} − e_{i}, …, e_{i−1} − e_{i}, −e_{i}, e_{i+1} − e_{i}, …, e_{μ} − e_{i} \end{matrix}

also satisfies DS. P(iva) If c is not a coset code, then ϕ_{c}(x, b) is not a coset formula and the claim follows from Lemma 3.1. P(ivb) If c is a coset code, then ϕ_{c}(x, b′) is a coset formula. Since f′ is a generic realization, ϕ_{c}(x, b) ∼ ϕ_{c}(x + f′, b′) is a group formula. P(ivc) Extend the Morley sequence e_{0}, …, e_{μ} of ϕ_{c}(x, b) by f. If ϕ_{c}(x, b) is a group formula, and i ≠ j, then

e_{0} + f, …, e_{i−1} + f, e_{i} − e_{j} + f, e_{i+1} + f, …, e_{μ} + f, f

is again a Morley sequence of ϕ_{c}(x, b). It follows that

e_{0}, …, e_{i−1}, e_{i} − e_{j}, e_{i+1}, …, e_{μ}

realizes DS. P(ivd) Choose f as above. If H ∈ Inv(c), then

e_{0} + f, …, e_{i−1} + f, He_{i} + f, e_{i+1} + f, …, e_{μ} + f, f

is also a Morley sequence of ϕ_{c}(x, b). It follows that

e_{0}, …, e_{i−1}, He_{i}, e_{i+1}, …, e_{μ}

realizes DS. P(ive) Immediate from Lemma 3.1. P(v) If ϕ_{c}(Hx, b′) ≡ ϕ_{c}(x, b′′), then He′_{0}, …, He′_{μ}, Hf is a Morley sequence of ϕ_{c}(x, b′′) and (He_{i}) = (He′_{i} − Hf) satisfies DS.

jSince b is canonical.

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This proves the claim.

We will take for Ψ_{c} a finite part of DS. Property P(i) will hold automatically. The Properties P(ii), P(iva), P(ivb), P(ive) can be described by countable disjunctions, which follow from DS. Therefore these properties follow from a sufficiently strong part of DS, which we call Ψ′_{c}.

Assume c to be a non–coset code. Write

V_{i}(x_{0}, …, x_{μ}) = (x_{0} − x_{i}, …, x_{i−1} − x_{i}, −x_{i}, x_{i+1} − x_{i}, …, x_{μ} − x_{i})

and

V_{H}(x_{0}, …, x_{μ}) = (Hx_{0}, …, Hx_{μ}).

Let V be the finite group generated by V_{0}, …, V_{μ} and V_{H} for H ∈ G_{c}. The formula

Ψ(x̄) = ⋀_{V ∈V} Ψ′_{c}(V (x̄))

has now properties P(iii) and P(v), and it still belongs to DS, since DS satisfies P(iii) and P(v).

If c is a coset code, consider the group generated by {V_{H}}_{H∈G_{c}} and the operations described in P(ivc) and P(ivd), and define. Ψ_{c} analogously. It satisfy then P(ivc) and P(ivd) and P(v), and therefore^{k} also P(iii).

We choose an appropriate Ψ_{c} (depending on μ) for every code c in such a way that

Ψ_{c^{H}} (x_{0}, …) = Ψ_{c}(Hx_{0}, …).

For two codes c and c′ to be equivalent we also impose that

Ψ_{c} ≡ Ψ_{c′}.

Corollary 3.3. Lemma 2.3 remains true if Ψ_{c} is also taken into account.

Proof. This follows from P(v) and the proof of Lemma 2.3.

4. The δ–function

Consider now two strongly minimal theories^{l} T_{1} and T_{2} which intersect in T_{0}, the theory of infinite F –vector spaces.

By considering their morleyization, we may assume that :

kNote that −1 ∈ Inv(c). lIn this section neither countability nor the DMP will be required.

(I) December 20, 2006 18:32 WSPC/INSTRUCTION FILE redfusion3.pub

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QE-Assumption . Both theories T_{i} have quantifier elimination. Their languages L_{i} are relational, except for the function symbols in L_{0}.

We may also assume that codes ϕ_{c} and formulas Ψ_{c} for T_{1} and T_{2} are quantifier free, as well as T_{i}–types tp_{i}(a/B). This assumption will be dropped only in section 9.

Let K be the class of all models A of T_{1}^{∀} ∪ T_{2}^{∀}. So, A is an F –vector space, which occurs at the same time as a subspace of ℂ_{1} and as a subspace of ℂ_{2}, where ℂ_{i} the monster model of T_{i}.

For finite A ∈ K, define

δ(A) = tr_{1}(A) + tr_{2}(A) − dim A.

We have that:

δ(0) = 0  (4.1)
δ(⟨a⟩) ≤ 1  (4.2)
δ(A + B) + δ(A ∩ B) ≤ δ(A) + δ(B)  (4.3)

Moreover, if dim(A/B) is finite^{m}, then we also set

δ(A/B) = tr_{1}(A/B) + tr_{2}(A/B) − dim A/B.

In case B is finite, we have that δ(A/B) = δ(A + B) − δ(B).

We say that B is strong in A, if B ⊂ A and δ(A′/B) ≥ 0 for all finite A′ ⊂ A and denote this by

B ≤ A.

A proper strong extension B ≤ A is minimal, if there is no A′ properly contained between B and A such that B ≤ A′ ≤ A.^{n}

Let B ⊂ A and a be in A. We call a algebraic over B, if a is algebraic over B either in the sense of T_{1} or of T_{2}. We call A transcendental over B, if no a ∈ A \ B is algebraic over B.

Lemma 4.1. B ≤ A is minimal iff δ(A/A′) < 0 for all A′ which lie properly between B and A.

Proof. One direction is clear, since A′ ≤ A implies δ(A/A′) ≥ 0. Conversely, if δ(A/A′) ≥ 0 for some A′, we may assume that δ(A/A′) is maximal. Then A′ ≤ A and A is not minimal over B.

mWe do not assume B ⊂ A. nNote that B is strong in all A′ ⊂ A.

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Lemma 4.2. Let B ≤ A be a minimal extension. One of the three following holds:

δ(A/B) = 0 and A = ⟨B, a⟩ for some element a ∈ A \ B algebraic over B ( algebraic minimal extension)

(II) δ(A/B) = 0, with A transcendental over B. ( prealgebraic minimal extension) (III) δ(A/B) = 1 and A = ⟨B, a⟩, for some element a transcendental over B

( transcendental minimal extension)

Note that in the prealgebraic case dim A/B ≥ 2.

Proof. Minimality implies that there is no C, properly contained between B and A with δ(C/B) = 0. We distinguish two cases.

δ(A/B) = 0. If there is an a ∈ A\B which is algebraic over B, then δ(⟨B, a⟩/B) = 0. Therefore ⟨B, a⟩ = A.

δ(A/B) > 0. For each a ∈ A\B it follows that δ(⟨B, a⟩/B) ≠ 0. Hence δ(⟨B, a⟩/B) = 1 and therefore ⟨B, a⟩ ≤ A. By minimality ⟨B, a⟩ = A.

We define the class K^{0} ⊂ K as

K^{0} = {M ∈ K | 0 ≤ M}.

It is easy to see that K^{0} can be axiomatized by a set of universal L_{1} ∪ L_{2}–sentences. The following results are also easy.

Lemma 4.3. Fix M in K^{0} and define

d(A) = min_{A⊂A′⊂M} δ(A′)

for all finite subspaces A of M. Then d is (on finite subspaces) the dimension function of a pregeometry i.e., d satisfies (4.1), (4.2), (4.3) and

d(A) ≥ 0  (4.4)
A ⊂ B ⇒ d(A) ≤ d(B).  (4.5) □

Lemma 4.4. Let M be in K^{0} and A a finite subspace. Let A′ be an extension of A, minimal with δ(A′) = d(A). Then A′ is the smallest strong subspace of M which contains A. We denote it by cl(A). □

We call cl(A) the closure of A.

For arbitrary subsets X of M we will use the notation δ(X) = δ⟨X⟩ and d(X) = d⟨X⟩.

Note that δ(A) ≤ dim(A).

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5. Prealgebraic codes

From now on, T_{1} and T_{2} are two countable strongly minimal extensions of T_{0} with the DMP. We assume the QE-Assumption of section 4, as in the next three sections 6, 7 and 8.

Choose for each T_{i} a set C_{i} of codes as in Corollary 3.3. A prealgebraic code c = (c_{1}, c_{2}) consists of two codes c_{1} ∈ C_{1} and c_{2} ∈ C_{2} with the following properties:

• n_{c} := n_{c_{1}} = n_{c_{2}} = k_{c_{1}} + k_{c_{2}}

• For all proper, non–zero subspaces U of F^{n_{c}}

k_{c_{1},U} + k_{c_{2},U} + dim U < n_{c}.  (5.1)

Set m_{c} = max(m_{c_{1}}, m_{c_{2}}). Note that simplicity of the ϕ_{c_{i}}(x, b) implies that n_{c} ≥ 2. Note also that for every H ∈ Gl_{n_{c}}(F)

c^{H} = (c_{1}^{H}, c_{2}^{H})

is a prealgebraic code.

Notation Unless otherwise stated, independence (a | ⌣_{b} c) means independent both in the sense of T_{1} and T_{2}. If c is a prealgebraic code, a (generic) realization of ϕ_{c}(x, b) is a (generic) realization of both ϕ_{c_{1}}(x, b_{1}) and ϕ_{c_{2}}(x, b_{2}). A Morley sequence of ϕ_{c}(x, b) is a Morley sequence for both ϕ_{c_{1}}(x, b_{1}) and ϕ_{c_{2}}(x, b_{2}). Similarly, for a set X of real elements, one defines X–generic realization of ϕ_{c}(x, b) and Morley sequence of ϕ_{c}(x, b) over X. A difference sequence for c with basis b = (b_{1}, b_{2}) is a difference sequence for c_{i} with basis b_{i} for each i = 1, 2.

We say c is a coset code if c_{1} and c_{2} are. We define then Inv(c) = Inv(c_{1})∩Inv(c_{2}). T_{1}^{eq} and T_{2}^{eq} have only the home sort in common. So b ∈ dcl^{eq}(A) (resp. acl^{eq}(A))

means that b is a pair consisting of an element in dcl^{eq}_{1}(A) (resp. acl^{eq}_{1}(A)) and an element in dcl^{eq}_{2}(A) (resp. acl^{eq}_{2}(A)). If M is a model of T_{1} ∪ T_{2}, then M^{eq} consists of imaginary elements in the sense of T_{1} and in the sense of T_{2}.

Lemma 5.1. Let B ≤ A be a prealgebraic minimal extension and a = (a_{1}, …, a_{n}) a basis for A over B. Then there is a prealgebraic code c and b ∈ acl^{eq}(B) such that a is a generic realization of ϕ_{c}(x, b).

Proof. Fix i ∈ {1, 2}. Choose d_{i} ∈ acl^{eq}_{i}(B) such that tp_{i}(a/Bd_{i}) is stationary. Since A/B is transcendental, we have dim(a/ acl_{i}(B)) = n. So we can find an L_{i}–formula χ_{i}(x) ∈ tp_{i}(a/Bd_{i}) of Morley rank k_{i} = MR_{i}(a/Bd_{i}). Since A/B is transcendental, χ(x) is simple. By 2.3 there is a T_{i}–code c_{i} ∈ C_{i} and b_{i} ∈ dcl^{eq}_{i}(Bd_{i}) with χ_{i}(x) ∼^{k_{i}} ϕ_{c_{i}}(x, b_{i}).

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Set c = (c_{1}, c_{2}) and b = (b_{1}, b_{2}). It follows from

k_{1} + k_{2} − n = tr_{1}(a/B) + tr_{2}(a/B) − dim(A/B) = δ(A/B) = 0

that n_{c} = k_{c_{1}} + k_{c_{2}}. Inequality (5.1) follows from Lemma 4.1:

k_{c_{1},U} + k_{c_{2},U} − (n − dim U) = tr_{1}(a/b, Ua) + tr_{2}(a/b, Ua) − dim(F^{n}/U)
=δ(A/B + Ua) < 0.

Lemma 5.2. Let B ∈ K, b ∈ acl^{eq}(B), c be a prealgebraic code, and a a B–generic realization of ϕ_{c}(x, b). Then ⟨B, a⟩ is a prealgebraic minimal extension of B.

Note that the isomorphism type of a over B is uniquely determined.

Proof. The proof follows from the above considerations. Note that subspaces of A containing B are of the form B + Ua for some subspace U of F^{n_{c}}.

Lemma 5.3. Let B ⊂ A be in K, c a prealgebraic code, b in acl^{eq}(B) and a ∈ A a realization of ϕ_{c}(x, b) in A not completely contained in B. Then

1. δ(a/B) ≤ 0.

2. If δ(a/B) = 0, then a is a B–generic realization of ϕ_{c}(x, b).

Proof. Let Ua = ⟨a⟩ ∩ B. Let Ua = ⟨a⟩ ∩ B. Since a is not contained in B, it follows that U is a proper subspace of F^{n_{c}}. Therefore

δ(a/B) = tr_{1}(a/B) + tr_{2}(a/B) − (n − dim U) ≤ k_{c_{1},U} + k_{c_{2},U} + dim U − n.

If U ≠ 0 the right hand side is negative. If U = 0, we have

δ(a/B) = tr_{1}(a/B) + tr_{2}(a/B) − n ≤ k_{c_{1}} + k_{c_{2}} − n = 0.

So δ(a/B) = 0 implies tr_{i}(a/B) = k_{c_{i}}.

Lemma 5.4. Let M ≤ N be a strong extension of elements in K. Given a prealgebraic code c, and natural numbers ε and r, there is some λ = λ(ε, r, c) ≥ 0 such that for every difference sequence e_{0}, …, e_{μ} in N, with basis b, and λ ≤ μ, either

• the basis of some λ–derived sequence of e_{0}, …, e_{μ} lies in dcl^{eq}(M),

or

• for every subset A of M′ with dim A ≤ ε the sequence e_{0}, …, e_{μ} contains a Morley sequence of ϕ_{c}(x, b) over M, A of length r.

Proof. By adding e_{0}, …, e_{m_{c}−1} to A, we may assume that b ∈ dcl^{eq}(M ∪ A). If at least (m_{c}+1) many of the e_{i} lie in the same class of N^{n_{c}}/M^{n_{c}}, we subtract one of these elements from the others and obtain a derived sequence with m_{c} many elements in M, which then has a base in dcl^{eq}(M). Therefore, we may assume that each class of N^{n_{c}}/M^{n_{c}} contains at most m_{c} many e_{i}’s.

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Fix an A of dimension ε and set

d = dim(e_{0}, …, e_{μ}/⟨M, A⟩).

Then dim(e_{0}, …, e_{μ}/M) ≤ d + ε. Thus by our assumption

μ + 1 ≤ m_{c} |F |^{(d+ε)n_{c}}.

Consider the following sets of indices.

X_{1} = {i ≤ μ | e_{i} generic over M, A, e_{0}, …, e_{i−1}}
X_{2} = {i ≤ μ | i ∉ X_{1} ∧ dim(e_{i}/M, A, e_{0}, …, e_{i−1}) > 0}

It is clear that

d ≤ (|X_{1}| + |X_{2}|) n_{c}.

With the notation δ(i) = δ(e_{i}/M, A, e_{0}, …, e_{i−1}), Lemma 5.3 implies that δ(i) < 0 if x ∈ X_{2}, and δ(i) = 0 otherwise. Since M ≤ N we have

0 ≤ δ(A, e_{0}, …, e_{μ}/M) = δ(A/M) + ∑_{i=1}^{μ} δ(i) ≤ ε − |X_{2}|.

If we put the three inequalities together, we obtain

μ + 1 ≤ m_{c} |F |^{(|X_{1}|n_{c}+ε n_{c}+ε)n_{c}}.

If μ is large enough, |X_{1}| ≥ r and (e_{i})_{i∈X_{1}} is our Morley sequence.

6. The class K^{μ}

Choose now a function μ^{∗} which assigns to every prealgebraic code c a natural number μ^{∗}(c). We assume that

M(i) for every m and n there are only finitely many c with μ^{∗}(c) = m and n_{c} = n.

The existence of such a function is ensured by the countability of C. Then we choose a function μ from prealgebraic codes to natural numbers such that

M(ii) μ(c) ≥ λ(n_{c}, 1, c) + 1 M(iii) μ(c) ≥ λ(0, λ(0, m_{c} + 1, c) + 1, c) M(iv) μ(c) ≥ λ(0, μ^{∗}(c) + 1, c) M(v) μ(c) = μ(d), if c is equivalent to some d^{H}.^{o}

From now on, all difference sequences of c will have fixed length μ(c) + 1. Condition M(v) ensures that, if c is equivalent to d^{H}, and (e_{i}) is a difference sequence for d, then (He_{i}) is a difference sequence for c.

oNote that every d^{H} can be equivalent to only one prealgebraic c.

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The class K^{μ} consists of all elements A of K^{0} which do not contain a difference sequence for any prealgebraic code.

Lemma 6.1. Let B ≤ M ∈ K^{μ} and A/B prealgebraic minimal. Then there are only finitely many B–isomorphic copies of A strong in M.

Proof. Let a be a basis of A/B. Choose d ∈ acl^{eq}(B) such that the types tp_{i}(a/Bd_{i}) are stationary. It suffices to show that for all such d the partial type tp_{1}(a/Bd_{1}) ∪ tp_{2}(a/Bd_{2}) has only finitely many realizations in M. For this we choose a prealgebraic code c and b ∈ acl^{eq}(B) with ⊧ ϕ_{c}(a, b) by 5.1. We now show that ϕ_{c}(x, b) has only finitely many realizations in M. If not, there is an infinite sequence e_{0}, … of realizations such that e_{i} is not contained in ⟨B, e_{0}, …, e_{i−1}⟩ (since the latter set is finite). Strongness of B in M yields that e_{0} is a B–generic realization by 5.3. From δ(e_{0}/B) = 0 we conclude that ⟨B, e_{0}⟩ ≤ M. If we proceed in this way, we see that e_{0}, … is a Morley sequence of ϕ_{c}(x, b) over B. Now P(i) yields that e_{1} − e_{0}, …, e_{μ(c)+1} − e_{0} is a difference sequence of c. Contradiction.

Corollary 6.2. Let B ≤ M ∈ K^{μ} and B ⊂ A finite with δ(A/B) = 0. Then there are only finitely many B ≤ A′ ⊂ M, which are isomorphic to A over B.

Note that automatically A′ ≤ M.

Proof. Decompose the extension A/B into a sequence of minimal extensions.

Corollary 6.3. Let X be a finite subset of M ∈ K^{μ}. Then the d–closure of X:

cl_{d}(X) = {x ∈ M| d(Xx) = d(X)}

is at most countable.

Proof. Note that cl_{d}(X) is the union of all A′ ⊂ M with cl(X) ⊂ A′ and δ(A′/ cl(X)) = 0.

Lemma 6.4. Let M ∈ K^{μ}, M ≤ M′ a minimal extension and (e_{i}) a difference sequence for a prealgebraic code c with base b ∈ acl^{eq}(M). Then c has a difference sequence (e′_{i}) with the same base b such that M contains e′_{0}, …, e′_{μ(c)−1}. In particular, e′_{μ(c)} is an M–generic realization of ϕ_{c}(b), which generates M′ over M as a vector space. Also b must be in dcl^{eq}(M).

Proof. Let e_{i} be any element which does not lie in M. By strongness of M in M′ and Lemma 5.3, it follows that e_{i} is an M–generic realization of ϕ_{c}(x, b). We have δ(⟨M, e_{i}⟩/M) = 0 and whence ⟨M, e_{i}⟩ ≤ M′. By minimality ⟨M, e_{i}⟩ = M′.

After permutation we may assume that e_{0}, …, e_{ν−1} are in M and e_{ν}, …, e_{μ(c)} are not. Since M ∈ K^{μ}, it follows that ν ≤ μ(c). As above, for i ≥ ν, e_{i} is an M–generic realization of ϕ_{c}(x, b) which generates M′/M, so e_{i} − H_{i}e_{μ(c)} ∈ M for some H_{i} ∈ Gl_{n_{c}}(F). Therefore e_{i} | ⌣_{b} e_{i} − H_{i}e_{μ(c)}.

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If c is a not coset code, it follows from P(iva) that i = μ(c). So we have ν = μ(c). Suppose that c is a coset code. If ν ≤ i < μ(c), then H_{i} ∈ Inv(c) by P(ive). By

P(ivc) and P(ivd) the difference sequence

e_{0}, …, e_{ν−1}, e_{ν} − H_{ν}e_{μ(c)}, …, e_{μ(c)−1} − H_{μ(c)−1}e_{μ(c)}, e_{μ(c)}

is as stated in the claim. Note that the above sequence has same base b.

7. Amalgamation

Theorem 7.1. K^{μ} (and therefore also the class of all finite elements of K^{μ}) has the amalgamation property with respect to strong embeddings.

Proof. Consider B ≤ M and B ≤ A in K^{μ}. We want to find a strong extension M′ ∈ K^{μ} of M and a B ≤ A′ ≤ M′ isomorphic to A over B. We may assume that A/B and M/B are minimal. We will show that either some “free amalgam” M′ of M and A is in K^{μ} or that M and A are isomorphic over B.

Case 1: A/B is algebraic. Then A = ⟨B, a⟩ for an element a which is (e.g.) algebraic over B in the sense of T_{1} and transcendental over B in the sense of T_{2}. There are two (non exclusive) subcases.

Subcase 1.1: tp_{1}(a/B) is realized in M. Choose some realization a′ in M. Hence, a′/B is transcendental in the sense of T_{2} and a′ ↦ a defines an isomorphism between M = ⟨B, a′⟩ and A over B.

Subcase 1.2: There is some a′ ∉ M, which realizes tp_{1}(a/B) (in the sense of T_{1}). Define the structure M′ = ⟨M, a⟩ by setting a to have the same T_{1}–type over M as a′ and being transcendental over M in the sense of T_{2} i.e. M′ is a free amalgam of A and M over B in the sense that M are A are independent over B and linearly independent^{p} over B. It is easy to see that, in free amalgams, M ≤ M′ and A ≤ M′. By Lemma 7.2 below, M′ belongs to K^{μ}.

Case 2: A/B is transcendental. We may assume that M ∩ A = B. Since A/B is transcendental, we find M′ = M +A in K, such that M and A are independent over B. So M′ is a free amalgam of M and A, and M′ is a minimal extension of M and of A. If M′ ∈ K^{μ}, we are done. Otherwise, 7.3 shows that, by symmetry, we may assume that M′ contains a difference sequence (e_{i}) of a prealgebraic code c with base b ∈ acl^{eq}(M). Also by Lemma 7.2 , dim(M′/M) > 1 and A/B is prealgebraic. By minimality and Lemma 6.4, we may also assume that e_{0}, …, e_{μ(c)−1} are in M and e_{μ(c)} is an M–generic realization of ϕ_{c}(x, b), which generates M′ over M. Write e_{μ(c)} = m+a for m ∈ M and a ∈ A. Therefore δ(a/B) = δ(a/M) = δ(e_{μ(c)}/M) = 0.

pI.e. dim(A/B) = dim(A/M).

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Whence a generates A over B. We apply now Lemma 5.4 and M(ii) to the extension (M′/A) and m and obtain two subcases:

Subcase 2.1: There is a (μ(c) − 1)–derived difference sequence (e′_{i}) with basis b′ ∈ dcl^{eq}(A). Since e′_{i} ∈ M for i ≤ μ(c) − 1, the base b′ is in dcl^{eq}(M) ∩ dcl^{eq}(A) ⊂ acl^{eq}(B). Hence e′_{μ(c)} is an M–generic realization of ϕ_{c}(x, b′) which generates M′ over M. Again there are two cases.

Subsubcase 2.1.1: e′_{μ(c)} ∈ A. Since A ∈ K^{μ}, there is an e′_{i} ∈ M not in A. By minimality e′_{i} generates M over B and e′_{μ(c)} ↦ e′_{i} defines a B-isomorphism between A and M.

Subsubcase 2.1.2: e′_{μ(c)} ∉ A. Then e′_{μ(c)} is an A–generic realization of ϕ_{c}(x, b′). Write e′_{μ(c)} = m′ + a′ for m′ ∈ M and a′ ∈ A. Since e′_{μ(c)}, m′ and a′ are pairwise independent over b′, then, for i = 1, 2, ϕ_{c_{i}}(x, b′_{i}) is a coset formula by [9] and whence a group formula by C(v) and P(ivb). It follows that −m′ and a′ are generics of the same Bb′_{i}–definable coset of a Bb′_{i}–definable connected group. Thus they have the same type over B. As above m′ generates M over B and a′ generates A over B. So the map a′ ↦ −m′ defines an isomorphism between A and M over B.

Subcase 2.2: e_{0}, …, e_{μ(c)−1} contains a B, m–generic realization of ϕ_{c}(x, b), say e_{0}. For i = 1, 2, e_{0} and e_{μ(c)} have the same T_{i}–type over B, m, b_{i}. Whence e_{0} − m and a have the same T_{i}–type over B, m, b_{i}, a forteriori over B. Whence a ↦ e_{0} − m defines a B–isomorphism between A and M.

Lemma 7.2. Let M ∈ K^{μ}, M ≤ M′ and dim(M′/M) = 1. Then, M′ ∈ K^{μ}.

Proof. Assume M′ ∉ K^{μ} and (e_{i}) is a difference sequence in M′ for a prealgebraic code c with base b witnessing this fact. Since dim(M′/M) = 1 and n_{c} ≥ 2, no e_{i} is an M–generic realization. By the choice of μ(c) and Lemma 5.4 we may assume that b ∈ dcl^{eq}(M). By Lemma 5.3 we conclude that all e_{i} lie in M. Contradiction.

Lemma 7.3. Let M′ be a free amalgam of M and A over B and (e_{i}) a difference sequence in M′. Then there is a derived sequence with base in acl^{eq}(M) or a derived sequence with base in acl^{eq}(A).

Actually we find the base in dcl^{eq}(M), dcl^{eq}(A) or acl^{eq}(B).

Proof. Let b be the base of s = (e_{i}). If no derivation has a base in dcl^{eq}(M), Lemma 5.4 and M(iii) yield a subsequence s′ of length λ(0, m_{c} + 1, c) + 1 which is a Morley sequence of ϕ_{c}(x, b) over M. Again by 5.4, applied to M′/A, if there is no derivation with base in dcl^{eq}(A), there is a subsequence s′′ of s′ of length

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m_{c} + 1, say e_{0}, …, e_{m_{c}}, which is also a Morley sequence of ϕ_{c}(x, b) over A. Set E = {e_{0}, …, e_{m_{c}−1}}. Hence, b ∈ dcl^{eq}(E) and

e_{m_{c}} | ⌣_{b} M, E , e_{m_{c}} | ⌣_{b} A, E .

Write every e ∈ E as the sum of an element of M and an element of A. Define E_{M} to be the set of all elements in M which occur as summands, and likewise E_{A}, and set E′ = E_{M} ∪ E_{A}. Then also b ∈ dcl^{eq}(E′) and, since E′ and E are interdefinable over M and as well as over A, we have

e_{m_{c}} | ⌣_{b} M, E′ , e_{m_{c}} | ⌣_{b} A, E′ ,

which implies

e_{m_{c}} | ⌣_{B,E′} M , e_{m_{c}} | ⌣_{B,E′} A .

Furthermore

M | ⌣_{B,E′} A .

Write e_{m_{c}} = m + a for m ∈ M and a ∈ A. Then e_{m_{c}}, m, and a are pairwise independent over B, E′. Fix i = 1, 2. Then ϕ_{c_{i}}(x, b_{i}) is a group formula for a definable group G_{i} and b_{i} is the canonical parameter of G_{i}. Moreover, a is a generic element of an acl^{eq}_{i}(B, E′)–definable coset of G_{i} and b_{i} is definable from the canonical base of p = tp_{i}(a/ acl^{eq}_{i}(B, E′)). Note that a | ⌣_{B,E_{A}} E′. So the canonical base of p is in acl^{eq}_{i}(A), hence b ∈ acl^{eq}(A). By symmetry b ∈ acl^{eq}(M), and since M and A are independent over B, this yields b ∈ acl^{eq}(B).

We call M ∈ K^{μ} rich, if for all finite B ≤ M and all finite B ≤ A ∈ K^{μ} there is an B ≤ A′ ≤ M, which is B–isomorphic to A. We will show in the next section (8.3) that rich structures are models of T_{1} ∪ T_{2}.

Corollary 7.4. There is a unique countable rich structure K^{μ}. All rich structures are (L_{1} ∪ L_{2})_{∞,ω}–equivalent. □

8. The theory T^{μ}

Lemma 8.1. Let M ∈ K^{μ}, b ∈ dcl^{eq}(M), c a prealgebraic code and M′ a prealgebraic minimal extension of M, generated by an M–generic realization a of ϕ_{c}(x, b) as in 5.2. If M′ does not belong to K^{μ}, one of the following is true.

(a) M′ contains a difference sequence (e_{i}) for c whose elements but one lie in M.

(b) M′ contains a difference sequence for a prealgebraic code c′ with base b′ which contains a Morley sequence of ϕ_{c′}(x, b′) over M of length μ^{∗}(c′) + 1.

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Proof. If M′ ∉ K^{μ} there is a difference sequence (e′_{i}) in M′ for a prealgebraic code c′ with base b′. If case (b) does not occur, by M(iv) and Lemma 5.4 we may assume that b′ ∈ dcl^{eq}(M) and furthermore that (e′_{i}) is as in Lemma 6.4. So n_{c′} = n_{c} = dim(M′/M) and we have He′_{μ(c′)} + m = a for some H ∈ Gl_{n_{c}}(F) and m ∈ M. By C(vi) there is a d ∈ dcl^{eq}(M) with ϕ_{c_{i}}(x + m, b_{i}) ∼^{k_{c_{i}}} ϕ_{c_{i}}(x, d_{i}) (i = 1, 2). Then He′_{μ(c′)} is an M–generic realization of ϕ_{c}(x, d), i.e. e′_{μ(c′)} is an M–generic realization of ϕ_{c^{H}} (x, d). By C(ix) there is a prealgebraic code c′′ which is equivalent to c^{H}. We have ϕ_{c^{H}} (x, d) ≡ ϕ_{c′′}(x, b′′) for some b′′ ∈ dcl^{eq}(M). By C(viii) and C(iv) we conclude c′′ = c′ and b′′ = b′.

Finally note that (e′_{i}) is a difference sequence for c^{H}. So (e_{i}) = (He′_{i}) is the desired difference sequence for c as in (a).

Corollary 8.2.

1. Let c be a prealgebraic code. That a structure M ∈ K contains no difference sequence for c can by expressed by a single sentence α_{c}.

2. Let c be a prealgebraic code, M ∈ K^{μ} a model of T_{1} ∪ T_{2}. That no extension of M in K^{μ} is generated by a generic realization of some ϕ_{c}(x, b) with b ∈ dcl^{eq}(M) can be expressed by an sentence β_{c}.

3. Let M ∈ K^{μ} be a model of T_{1}∪T_{2}. That M has no prealgebraic minimal extension in K^{μ} can be expressed by a set of sentences.

Proof. 1. Let α_{c} = ¬∃x_{0}, …, x_{μ(c)} (Ψ_{c_{1}}(x_{0}, …, x_{μ(c)}) ∧ Ψ_{c_{2}}(x_{0}, …, x_{μ(c)})).

2. Fix i = 1, 2 and let M be a submodel of ℂ_{i}. Let m ∈ M, ϕ(x, m) an L_{i}–formula of Morley rank k and degree 1, and a ∈ ℂ_{i} be an M–generic realization of ϕ(x, m). There is a uniform way to translate a quantifier free property ψ(a, m) of a, m into a quantifier free property ψ^{∗}(m) of m: Set

ψ^{∗}(y) = MR_{x}(ϕ(x, y) ∧ ψ(x, y)) =^{.} k

This shows that, if M ∈ K and a is an M–generic realization of ϕ_{c}(x, b), then any L_{1} ∪ L_{2}–sentence α about ⟨M, a⟩ can be translated into an L_{1} ∪ L_{2}–sentence α^{c}(b) about M.

Now there is only a finite set C_{c} of codes c′ which can occur in (b) of 8.1 since (μ^{∗}(c′) + 1)n_{c′} ≤ dim(M′/M) = n_{c}. So set

β_{c} = ∀y_{c} α_{c}^{c}(y_{c}) ∧ ⋀_{c′∈C_{c}} ∀y_{c′} α_{c′}^{c}(y_{c′}).

The variables y_{c}, y_{c′} are understood to range over appropriate sorts of M^{eq}.

3. This follows from 2. and Lemma 5.1.

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We now introduce the theory T^{μ} described by the following axioms, which by the above are elementarily expressible.

Axioms of T^{μ} M is model of T^{μ} iff

(i) M ∈ K^{μ} (ii) M is a model of T_{1} ∪ T_{2} (iii) No prealgebraic minimal extension of M belongs to K^{μ}.

Theorem 8.3. Rich structures are exactly the ω–saturated models of T^{μ}.

Proof. Let M be an ω–saturated model of T^{μ}. In order to show that M is rich, we consider a finite strong subspace B of M and a minimal extension A ∈ K^{μ} of B. We want to find a copy B ≤ A′ ≤ M of A/B.

case (I): A/B is algebraic. Since M is a model of T_{1} ∪ T_{2}, it has no proper algebraic extension in K. So A′ exists by 7.1.

case (II): A/B is prealgebraic. Since M has no prealgebraic minimal extension, 7.1 forces to obtain a copy of A in M.

case (III): A/B is transcendental. Since A/B is generated by a transcendental element we have to find an a′ ∈ M which is transcendental over B such that ⟨B, a′⟩ ≤ M. Since this equivalent to realize a partial type, and since M is ω– saturated, it suffices to find a′ in an elementary extension M′ of M. Choose M’ uncountable. By 6.3 cl_{d}(B) ≤ M′ is countable. For every a′ ∈ M′ \ cl_{d}(B), we have δ(a′/B) = 1 and ⟨B, a′⟩ ≤ M′.

Assume now that M is rich. We show first that M is a model of T^{μ}.

Axiom (ii): By Lemma 7.2 there are elements in K^{μ} of arbitrary finite dimension. So M is infinite and we need only show that M is algebraically closed in the sense of T_{1} and of T_{2}.

Let a be an element in acl_{1}(M) and transcendental over M in the sense of T_{2}. Therefore, a is 1–algebraic over a finite subset B of M. We may assume that B ≤ M. Since (by Lemma 7.2) B ≤ ⟨B, a⟩ ∈ K^{μ}, there is a copy of a over B in M. This implies that M acl_{1}–closed. Likewise M is algebraically closed in the sense of T_{2}.

Axiom (iii): Let M′ be a prealgebraic minimal extension generated by an M–generic realization a of ϕ_{c}(x, b). Assume M′ ∈ K^{μ}. Choose a finite subspace C_{0} ≤ M with b ∈ dcl^{eq}(C_{0}). Then C_{0} ≤ ⟨C_{0}, a⟩. Since M is rich, M contains a copy e_{0} of a over C_{0} with C_{1} = ⟨C_{0}, e_{0}⟩ ≤ M. Continuing this way we obtain an infinite Morley sequence

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e_{0}, e_{1}, … of ϕ_{c}(x, b). By P(i), e_{1} − e_{0}, …, e_{μ(c)+1} − e_{0} is a difference sequence for c.

Choose an ω–saturated M′ ≡ M. By the above we know that M′ is rich. Since M′ ≡_{∞,ω} M, this implies that M is ω–saturated.

9. Proof of the Theorem

In this section quantifier elimination for T_{1} and T_{2} will no longer be required. Hence, replace in the class K embeddings by elementary maps in the sense of T_{1} and in the sense of T_{2}, which we call bi-elementary maps.

Corollary 9.1. T^{μ} is complete. Two tuples a and a′ in two models M and M′ have the same type iff there is bi-elementary bijection

f : cl(a) → cl(a′)

which maps a to a′.

Proof. K^{μ} is a model of T^{μ}. So is T^{μ} consistent. Let M be any model of T^{μ}. By theorem 8.3 there is a rich M′ ≡ M. So M′ ≡_{∞,ω} K^{μ}, which proves completeness.

To prove the second statement choose ω–saturated elementary extensions M ≺ N and M′ ≺ N′. It is easy to see^{q} that M ≤ N and M′ ≤ N′, so “cl” does not increase.

Since M′ and N′ are rich, f is even ∞, ω–elementary.

For the converse suppose that a and a′ have the same type. There is a bi-elementary map f : cl(a) → M′ which maps a onto a′. We write A′ for f(cl(a)). Then d(a) = δ(cl(a)) = δ(A′). It follows d(a′) ≤ d(a) and d(a′) = d(a) by symmetry. A′ has, like cl(a), no proper subset A′′ which contains a′ and with δ(A′′) = d(a′). This implies A′ = cl(a′).

Theorem 9.2. T^{μ} is strongly–minimal and d is the dimension function of the natural pregeometry on models of T^{μ}, i.e.

MR(a/B) = d(a/B).

Proof. Let a be a single element. Types tp(a/B) with d(a/B) = 0 are algebraic by Corollary 6.2. It follows from 9.1, that there is only one type with d(a/B) = 1.^{r}

qIf M ≰ N, there is a tuple a ∈ N with δ(a/M) < 0. We find a finite B ≤ M with δ(a/B) < 0. This is witnessed by the truth of an L_{1} ∪ L_{2}–formula ϕ(a, b̄). However, ϕ(x, b̄) is not satisfiable in M, whence M ⊀ N. rThis is the type of elements a which are transcendental over cl(B) and for which ⟨cl(B), a⟩ is strong in the considered model.

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This implies strong minimality. The rest of the claim follows from the fact that d describes the algebraic closure.

This completes the proof of 1.1.

Proof. [Proof of Theorem 1.2, 2.] Let M be an elementary submodel of N in the sense of T_{1} and T_{2}. By Corollary 9.1 we need only show that M is strong in N. Suppose not and pick a smallest extension M ⊂ H ⊂ N with negative δ(H/M). We may decompose H/M into a sequence M ≤ K ⊂ H, where δ(K/M) = 0 and H = ⟨K, a⟩ for some element a with δ(a/K) = −1. Since M is a model of Axiom (iii), we have M = K. a is algebraic over M in the sense of T_{1} (and T_{2}), whence by Axiom (ii) we have a ∈ M. Contradiction.

Corollary 9.3. If T_{1} and T_{2} are model-complete, then T^{μ} is also model-complete.

We now prove the last remark of the introduction. Let T_{1} and T_{2} be both the theory of algebraically closed fields of characteristic p formulated in L_{1} = {+, ⊙} and L_{2} = {+, ⊗}. Let T^{μ} be a fusion over

T_{0}, the theory of 𝔽_{p}–vector spaces. Let x be transcendental (in the sense of T^{μ}), x_{i} the i–th power in the sense of T_{1} and X = {x_{i} | i ∈ ℕ}. Let S be any subset of X. Then dim(S) = |S| and tr_{1}(S) ≤ 1. It follows from Theorem 1.2, 1. that tr_{2}(S) ≥ |S| − 1. We claim that tr_{2}(S) = |S|, which is clear for S = {x_{0}}. Assume the contrary. Then, for some n > 0, we have tr_{2}(x_{1} …, x_{n}/x_{0}) < n. But x_{n+1} is also transcendental, therefore it has the same type as x. So tr_{2}(x_{n+1}, …, x_{(n+1)n}/x_{0}) < n. It follows

tr_{2}(x_{1}, …, x_{n}, x_{n+1}, …, x_{(n+1)n}/x_{0}) < 2n − 1,

which is impossible.

Remark 9.4. E. Hrushovski stated in [1] that the DMP survives the fusion. M. Hils explained a proof of this fact to us, which shows also that T^{μ} has the DMP.

References

[1] Ehud Hrushovski. Strongly minimal expansions of algebraically closed fields. Israel J. Math., 79:129–151, 1992.

[2] Assaf Hasson and Martin Hils. Fusion over sublanguages. J. Symbolic Logic, 2005. to appear.

[3] A. Baudisch, A. Martin-Pizarro, and M. Ziegler. Hrushovski’s Fusion. In F. Haug, B. Löwe, and T. Schatz, editors, Festschrift für Ulrich Felgner zum 65. Geburtstag, volume 4 of Studies in Logic, pages 15–31. College Publications, London, 2007.

[4] Bruno Poizat. Le carré de l’egalité. J. Symbolic Logic, 64(3):1338–1355, 1999.

[5] J. Baldwin and K. Holland. Constructing ω-stable structures: rank 2 fields. J. Symbolic Logic, 65(1):371–391, 2000.

[6] A. Baudisch, A. Martin-Pizarro, and M. Ziegler. On fields and colors. Algebra i Logika,

   45(2), 2006. (http://arxiv.org/math.LO/0605412).

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[7] A. Baudisch, A. Martin-Pizarro, and M. Ziegler. Red fields. J. Symbolic Logic, 2005. to appear.

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[9] Martin Ziegler. A note on generic types. (http://arxiv.org/math.LO/0608433), 2006.