Hrushovski’s Fusion^{∗}

A. Baudisch, A. Martin-Pizarro, M. Ziegler

March 4, 2007

Abstract

We present a detailed and simplified exposition of Hrushovki’s fusion of two strongly minimal theories.

1 Introduction

A definable set whose definable subsets (in some saturated model) are either finite or cofinite is called strongly minimal. Examples of strongly minimal structures are the trivial one, infinite dimensional vector spaces, or algebraically closed fields. It was observed that the geometrical behaviour of these archetypical examples could be generalized to the pregeometry of algebraic closure on strongly minimal sets, exhibiting a first example of regular types. This led B. Zilber to conjecture that all strongly minimal sets could be classified according to these three basic ones. This conjectured was refuted by E. Hrushovski [9], who adapted Fraïssé’s construction in a ingenious way in order to collapse to finite rank a candidate for the counterexample. The amalgamation procedure can be described in the following way: the goal is to construct a countable universal model starting from a given collection of finitely generated structures. In this model there is a unique type of rank ω. The decisive part (or collapse) is to modify this construction in order to algebraize types of finite rank. In order to do so, a collection of representatives (or codes) of these types needs to be chosen and one assigns a maximal length of an independence sequence of realisations to each code. The structure so obtained after amalgamation has now finite rank. Note that the prescribed maximal length must reflect any interaction between different codes, since some realizations of one code may yield realizations for another.

Using the same procedure, E. Hrushovski also merged two strongly minimal theories over a trivial geometry into a new one, their fusion. This answered negatively a question of G. Cherlin on the existence of a maximal strongly minimal theory. More precisely, consider two countable strongly minimal theories T_{1} and T_{2} with the definable multiplicity property (in short, DMP) whose respective languages L_{1} and L_{2} are disjoint. Recall that a ω-stable theory has the DMP if Morley rank and degree are definable on the parameters of any given definable set.

In this survey we give a detailed and slightly simplified exposition of the following theorem proved by E. Hrushovski.

∗blackfusion.tex,v 1.19, March 4, 2007. This is the english version of fusion2.tex,v 2.20.

Theorem ([8]). T_{1} ∪ T_{2} admits a strongly minimal completion T^{μ}. Its models satisfy the following: Let tr_{i} denote the transcendence degree in T_{i}. Given a finite subset A of M, then

|A| ≤ tr_{1}(A) + tr_{2}(A).

Our presentation grew out of a seminar held at the Humboldt-Universität Berlin in 2003. Several articles on this topic (among others [7] and [6]) have been published, however we believe that this survey will be beneficial for the mathematical community in order to become more acquainted with Hrushovski’s fusion method. The authors used the simplified approach in three subsequent articles: In [4] to reprove a theorem of Poizat and Baldwin–Holland ([10], [1]) about the existence of fields of Morley rank 2 with a distinguished subset in any characteristic, in [5] to construct fields of Morley rank 2 with a distinguished additive subgroup, and in [3] to prove the fusion over a vector space over a finite field, which had been proposed by E. Hrushovski.

The simplified technique was also helpful in [2], where a bad field was constructed: a field of Morley rank 2 with a distinguished multiplicative subgroup. This solved a long standing open problem.

Finally our techniques were used in [11] to prove the following generalization of Theorem 1: Let T_{1} and T_{2} be two countable complete theories in disjoint languages of finite Morley rank and of the same Morley degree. Assume that in T_{1} and T_{2} Morley rank and Morley degree are definable. Then T_{1} ∪ T_{2} has a “nice” complete extension of any rank which is a common multiple of the ranks of T_{1} and T_{2}.

2 Codes

All throughout the following sections (and until specified otherwise) T denotes a countable strongly minimal theory with DMP.

First, let us fix some notation: tr(a/B) is the transcendence degree of a over B^{1} and MR(p) denotes Morley rank of the type p. Note that

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

We write

ϕ(x) ∼^{k} ψ(x)

or ϕ(x) ∼_{x}^{k} ψ(x), if the symmetric difference of ϕ and ψ has smaller Morley rank than k.

A formula χ(x, b) is simple if it has Morley degree 1 and the components of a generic realization are pairwise different and not in acl(b). If a is an n–tuple and s is some subset of {1, …, n}, then a_{s} is {a_{i} | i ∈ s}.

A code c is a parameter-free formula

ϕ_{c}(x, y),

where |x| = n_{c} and y lies in some sort of T^{eq}, with the following properties.

1The maximal length of a B-independent subtuple of a.

(i) ϕ_{c}(x, b) is either empty^{2} or simple. Furthermore, ϕ_{c}(x, b) implies that the components of x are pairwise different.

(ii) All non-empty ϕ_{c}(x, b) have Morley rank k_{c} and Morley degree 1.

(iii) For each subset s of {1, …, n_{c}} there exists an integer k_{c,s} such that for every realization a of ϕ_{c}(x, b)

tr(a/ba_{s}) ≤ k_{c,s}.

Equality holds for generic a.

(iv) If both ϕ_{c}(x, b) and ϕ_{c}(x, b′) are non-empty and ϕ_{c}(x, b) ∼^{k_{c}} ϕ_{c}(x, b′), then b = b′.

For ϕ_{c}(x, b) to have Morley rank k_{c} in (ii) is equivalent to k_{c,∅} = k_{c}. The simplicity of ϕ_{c}(x, b) in (i) is equivalent to Morley degree 1 and k_{c,{i}} = k_{c} − 1 for all i.

Corollary 2.1. Let p ∈ S(b) be the unique type of Morley rank k_{c} containing ϕ_{c}(x, b). Then b is the canonical basis of p.

Proof. This follows easily from (iv).

Lemma 2.2. Let χ(x, d) be a simple formula. Then there is some code c and some b_{0} ∈ dcl^{eq}(d) such that χ(x, d) ∼^{k_{c}} ϕ_{c}(x, b_{0}).

We say that c encodes χ(x, d).

Proof. Set k_{c} = MR(χ(x, d)) and n_{c} = |x|. Let p be the global type of rank k_{c} containing χ(x, d), with canonical basis b_{0}. We find a formula ϕ(x, b_{0}) in p of rank k and degree 1. Choose a generic realization a_{0} of ϕ(x, b_{0}). For each s ⊂ {1, …, n_{c}} set k_{c,s} = MR(a_{0}/b_{0}a_{0}_{s}). By strengthening ϕ(x, b_{0}) appropriately we may assume that ϕ_{c}(a, b_{0}) implies that the components of a are pairwise different, and that tr(a/b_{0}a_{s}) ≤ k_{c,s} and tr(a_{s}/b_{0}) ≤ (k_{c} − k_{c,s}) for all realizations a of ϕ(x, b_{0}).

Consider now the following property E(b, b′):

• ϕ(x, b) implies that the components of x are pairwise disjoint.

• ϕ(x, b) has Morley rank k_{c} and degree 1.

• tr(a/ba_{s}) ≤ k_{c,s} and tr(a_{s}/b) ≤ (k_{s} − k_{c,s}) for all realizations a of ϕ(x, b).

• ϕ(x, b) ∼^{k_{c}} ϕ(x, b′) implies that b = b′.

E holds for all b, b′ realizing the type of b_{0}. Moreover, E is equivalent to an infinite disjunction of formulae ϵ(y, y′). Therefore, there is some θ(y) ∈ tp(b_{0}) such that ⊧ θ(y) ∧ θ(y′) → E(y, y′). Set

ϕ_{c}(x, y) = ϕ(x, y) ∧ θ(y).

Let a be a generic realization of ϕ_{c}(x, b). Then tr(a/ba_{s}) = k_{c,s} follows from tr(a/ba_{s}) ≤ k_{c,s}, tr(a_{s}/b) ≤ (k_{c} −k_{c,s}) and tr(a/b) = k_{c}. By simplicity of χ(x, d) we have k_{c,{i}} < k_{c} for all i, which in turn implies that all non–empty ϕ_{c}(x, b) are simple.

2We assume that ϕ_{c}(x, b) is non-empty for some b.

Let c be a code, ϕ_{c}(x, b) non-empty and p ∈ S(b) the type of rank k_{c} determined by ϕ_{c}(x, b). Hence, b lies in the definable closure of a sufficiently large segment of a Morley sequence of p. Let m_{c} be some upper bound for the length of such a segment.

Lemma 2.3. For every code c and every integer μ ≥ m_{c} − 1 there exists some formula Ψ_{c}(x_{0}, …, x_{μ}, y) without parameters satisfying the following:

(v) Given a Morley sequence e_{0}, …, e_{μ} of ϕ_{c}(x, b), then ⊧ Ψ_{c}(e_{0}, …, e_{μ}, b).

(vi) For all e_{0}, …, e_{μ}, b realizing Ψ_{c} the e_{i}’s are pairwise disjoint realizations of ϕ_{c}(x, b).

(vii) Let e_{0}, …, e_{μ}, b realize Ψ_{c}. Then b lies in the definable closure of any m_{c} many e_{i}’s.

We say that “x_{0}, …, x_{μ} is a pseudo Morley sequence of c over y”.

Proof. The statement “(e_{i}) is a Morley sequence of ϕ_{c}(x, b)” can be described by a partial type M(e_{0}, …, b). Likewise, properties (vi) and (vii) can be described by an infinite disjunction D(e_{0}, …, b). Since non-empty ϕ_{c}(x, b) are simple, it follows that ⊧ M → D. Hence we may choose a sufficiently strong formula Ψ_{c} in M with the desired properties.

Choose now for every code (and every μ)^{3} a formula Ψ_{c} as above.

Let c be a code and σ some permutation of {1, …, n_{c}}. Then c^{σ} defined by

ϕ_{c^{σ}}(x^{σ}, y) = ϕ_{c}(x, y)

is also a code. Similarly,

Ψ_{c^{σ}}(x̄^{σ}, y) = Ψ_{c}(x̄, y)

defines a pseudo Morley sequence of c^{σ}.

We consider two codes c and c′ to be equivalent if n_{c} = n_{c′}, m_{c} = m_{c′}, and

• For every b there is some b′ such that ϕ_{c}(x, b) ≡ ϕ_{c′}(x, b′) and Ψ_{c}(x̄, b) ≡ Ψ_{c′}(x̄, b′) in T.

• Similarly permuting c and c′.

Theorem 2.4. There is a collection of codes C such that:

(viii) Every simple formula can be encoded by exactly one c ∈ C.

(ix) For every c ∈ C and every permutation σ, we have that c^{σ} is equivalent to a code in C.^{4}

In [8] it was stated that one could find such a set C closed under permutations, which is stronger than (ix). This is not true.

3In the proof of 2.4 this choice may be modified.

4In fact, we find a collection C such that every c^{σ} is equivalent to some permutation of c which lies in C.

Proof. Work inside some countable ω–saturated model M of T and list all simple formulae χ_{i}, i = 1, 2, …, with parameters in M. We need only show that every χ_{i} may be encoded by some c ∈ C. We build up C as the union of an increasing sequence ∅ = C_{0} ⊂ C_{1} ⊂ ⋯ of finite sets. Suppose by induction on i that C_{i−1} has already been constructed and satisfies (ix). If χ_{i} may be encoded by some element in C_{i−1}, then set C_{i} = C_{i−1}. Otherwise, choose some code c and some b_{0} with ϕ_{c}(x, b_{0}) ∼^{k_{c}} χ_{i}. Replace ϕ_{c} by

ϕ_{c}(x, y) ∧ “ϕ_{c}(x, y) cannot be encoded by any element of C_{i−1}.”

We obtain a new code which still encodes χ_{i}. Therefore, we may assume that no permutation of c encodes a formula which may be also encoded by some element of C_{i−1}. Let G be now the group of all permutations σ ∈ Sym(n_{c}) with

ϕ_{c}(x, b_{0}) ∼^{k_{c}} ϕ_{c^{σ}}(x, b′_{0}))

for some realization b′_{0} of p, the type of b_{0}. It follows that b′_{0} is uniquely determined and hence given by a ∅-definable function of b_{0}. Write b′_{0} = b_{0}^{σ}.

After strengthening ϕ_{c}(x, y) with an appropriate subset^{5} of p, we may assume that for all b with non-empty ϕ_{c}(x, b) and all σ there is a b^{σ} with ϕ_{c}(x, b) ∼^{k_{c}} ϕ_{c^{σ}}(x, b^{σ}) iff σ ∈ G.

It is easy to see that

ϕ_{d}(x, y) = ⋀_{σ∈G} ϕ_{c^{σ}}(x, y^{σ})

defines a code which still encodes χ_{i}. Likewise,

Ψ_{d}(x̄, y) = ⋀_{σ∈G} Ψ_{c^{σ}}(x̄, y^{σ})

defines pseudo Morley sequences of d.

Moreover, ϕ_{d}(x, y) ≡ ϕ_{d^{σ}}(x, y^{σ}) and Ψ_{d}(x̄, y) ≡ Ψ_{d^{σ}}(x̄, y^{σ}) for all σ ∈ G. Hence d and d^{σ} are equivalent. Finally, choose representatives ρ_{1}, …, ρ_{r} of the right cosets of G in Sym(n_{c}) and set C_{i} = C_{i−1} ∪ {d^{ρ_{1}}, …, d^{ρ_{r}}}.

3 The δ–function

From now on, T_{1} and T_{2} are two strongly minimal theories^{6}, formulated in two disjoint languages L_{1} and L_{2}.

We assume the following

QE-Hypothesis. T_{1} and T_{2} have quantifier elimination, and L_{1} and L_{2} are pure relational languages.

ρ(y) = ⋀_{σ∈G} ρ′(b^{σ}) ∧ ⋀_{σ∉G} ¬ρ′(b^{σ})

5 Choose ρ′(y) ∈ tp(b_{0}) such that ⊧ ¬ρ′(b_{0}^{σ}) for all σ ∉ G. The aforementioned subset of p is

6Countability and DMP of T_{i} will not be used in this section.

by considering Morleyizations of T_{1} and T_{2}. We may also assume that ϕ_{c} and Ψ_{c} are quantifier-free for the T_{1}–Codes and the T_{2}–Codes. Types tp_{i}(a/B) in each theory T_{i} are always quantifier-free types. These assumptions will be dropped in section 7.

Let K be the class of all models of T_{1}^{∀} ∪ T_{2}^{∀}. We also allow ∅ to be in K. If ℂ_{i} is some monster model of T_{i}, we may see elements of K simultaneously as subsets of ℂ_{1} and ℂ_{2}.

Given a finite A ∈ K, define

δ(A) = tr_{1}(A) + tr_{2}(A) − |A|.

The following hold

δ(∅) = 0 (1) δ({a}) ≤ 1 (2) δ(A ∪ B) + δ(A ∩ B) ≤ δ(A) + δ(B) (3)

If A \ B is finite, we set

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

For B finite, it follows 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. Denote this by

B ≤ A.

An element a ∈ A is algebraic over B ⊂ A if a is algebraic over B either in the sense of T_{1} or in the sense of T_{2}. A is transcendental over B if no a ∈ A \ B is algebraic over B.

B ≨ A is minimal if B ≤ A′ ≤ A for no A′ properly contained between B and A.

Lemma 3.1. A proper strong extension B ≤ A is minimal if and only if δ(A/A′) < 0 for all A′ properly contained between B and A.

Proof. One direction is clear, since A′ ≤ A implies that δ(A/A′) ≥ 0. On the other hand, if δ(A/A′) ≥ 0 for some A′, we may choose A′ such that δ(A/A′) is maximal. Hence A′ ≤ A, which implies that A is not minimal over B.

Note that A \ B is finite for minimal extensions.

Lemma 3.2. Let B ≤ A be a minimal extension of elements in K. Then one of the following hold:

(I) 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 and A is transcendental over B. (prealgebraic minimal extension)

(III) δ(A/B) = 1 and A = B ∪ {a} for some a in A transcendental over B. (transcendental minimal extension)

Note that |A \ B| ≥ 2 in the prealgebraic case.

Proof. If A \ B contains some algebraic element a, then δ(a/B) = 0. Hence B ∪ {a} = A.

Otherwise, two cases apply: if δ(A/B) = 0, then the extension is prealgebraic minimal. Otherwise, δ(A′/B) ≥ 1 for each B ⊊ A′ ⊂ A. Given any a ∈ A \ B, we have that B ∪ {a} ≤ A and hence B ∪ {a} = A.

Define K^{0} ⊂ K as the subclass

K^{0} = {M ∈ K | ∅ ≤ M}.

It is easy to see that K^{0} may be described by a collection of universal L_{1} ∪ L_{2}– sentences. The following lemmas follow easily from (1), (2) and (3).

Lemma 3.3. Let M in K^{0} and A a finite subset of M. Set

d(A) = min_{A⊂A′⊂M} δ(A′).

Then d is the dimension function of a pregeometry, i.e. d satisfies (1), (2), (3) and

d(A) ≥ 0 (4) A ⊂ B ⇒ d(A) ≤ d(B) (5)

Lemma 3.4. Let M ∈ K^{0} and A a finite subset of M. Take a minimal superset A′ of A with δ(A′) = d(A). Then A′ is the smallest strong subset cl(A) of M containing A, called the closure of A.

4 Prealgebraic codes

From now on, T_{1} and T_{2} are two countable strongly minimal theories with DMP as in Theorem 1. The QE-Hypothesis from section 3 holds all throughout this and the next sections (5,6).

Fix for each T_{i} a collection C_{i} of codes as in Theorem 2.4. A prealgebraic code c = (c_{1}, c_{2}) consists of a code c_{1} ∈ C_{1} and a code c_{2} ∈ C_{2} with the following properties:

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

• For each proper non-empty subset s of {1, …, n_{c}},

k_{c_{1},s} + k_{c_{2},s} − (n_{c} − |s|) < 0

Set m_{c} = max(m_{c_{1}}, m_{c_{2}}). Note that simplicity of ϕ_{c_{i}}(x, b) implies that

n_{c} ≥ 2.

For each permutation σ, the code

c^{σ} = (c_{1}^{σ}, c_{2}^{σ})

is also prealgebraic.

Some explanatory remarks: T_{1}^{eq} and T_{2}^{eq} share only their home sort. An element b ∈ dcl^{eq}(B) is a pair b = (b_{1}, b_{2}) with b_{i} ∈ dcl^{eq}_{i}(B) for i = 1, 2. Likewise for acl^{eq}(B). A generic realization of ϕ_{c}(x, b) (over B) is a generic realization of ϕ_{c_{i}}(x, b_{i}) (over B) in T_{i} for i = 1, 2. A Morley sequence of ϕ_{c}(x, b) is a Morley sequence both of ϕ_{c_{1}}(x, b_{1}) and ϕ_{c_{2}}(x, b_{2}). A pseudo Morley sequence of c over b is a realization of both Ψ_{c_{1}}(x̄, b_{1}) and Ψ_{c_{2}}(x̄, b_{2}). We say that M is independent from A over B if M is independent from A over B both in T_{1} and T_{2}.

Lemma 4.1. Let B ≤ B ∪{a_{1}, …, a_{n}} be a prealgebraic minimal extension and a = (a_{1}, …, a_{n}). Then there is some prealgebraic code c and b ∈ acl^{eq}(B) such that a is a generic realization of ϕ_{c}(a, b).

Proof. Fix i ∈ {1, 2} and choose d_{i} ∈ acl^{eq}_{i}(B) such that tp_{i}(a/Bd_{i}) is stationary, and χ_{i}(x, d_{i}) ∈ tp_{i}(a/Bd_{i}) with Morley rank MR_{i}(a/Bd_{i}) and degree 1. Since A/B is transcendental, the formula χ_{i}(x, d_{i}) is simple. Choose some T_{i}–code c_{i} ∈ C_{i} and some b_{i} ∈ dcl^{eq}_{i}(d_{i}) with χ_{i}(x, d_{i}) ∼^{k_{c_{i}}} ϕ_{c_{i}}(x, b_{i}). It follows from δ(A/B) = 0 that k_{c_{1}} + k_{c_{2}} = n. Moreover, k_{c_{1},s} + k_{c_{2},s} − (n − |s|) < 0 holds by 3.1.

The following lemma is proved similarly.

Lemma 4.2. Let B ∈ K, c a prealgebraic code and b ∈ acl^{eq}(B). Take a generic realization a = (a_{1}, …, a_{n_{c}}) of ϕ_{c}(x, b) over B. Then B ∪ {a_{1}, …, a_{n_{c}}} is a prealgebraic minimal extension of B.

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

Lemma 4.3. Let B ⊂ A in K, c a prealgebraic Code, b in acl^{eq}(B) and a ∈ A a realization of ϕ_{c}(x, b) which does not lie completely in B. Then

1. δ(a/B) ≤ 0.

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

Proof. Let s = {i | a_{i} ∈ B}. Since a is not completely contained in B, then s is a proper subset of {1, …, n_{c}}. Therefore

δ(a/B) = tr_{1}(a/B) + tr_{2}(a/B) − (n − |s|) ≤ k_{c_{1},s} + k_{c_{2},s} − (n − |s|)

If s ≠ ∅, then the right-hand side is negative. If s = ∅, we have that

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 that tr_{i}(a/B) = k_{c_{i}}.

Lemma 4.4. Let M ≤ N be a strong extension of structures in K and e_{0}, …, e_{μ} a pseudo Morley sequence of c in N over b. Then one of the following hold:

• b ∈ dcl^{eq}(M)

• At least μ − n_{c}m_{c} + 1 many e_{i}’s lie in N \ M.

Proof. Permute the e_{i}’s so that e_{0}, …, e_{r_{0}−1} are in M and e_{r_{1}}, …, e_{μ(c)} lie in N \ M. Hence 0 ≤ r_{0} ≤ r_{1} ≤ μ(c) + 1. Possibly the e_{i}’s do not form a pseudo Morley sequence of c after permutation, however they are still disjoint realizations of ϕ_{c}(x, b). Assume b ∉ dcl^{eq}(M). Then (vii) implies r_{0} < m_{c}. We need only to show that r_{1} ≤ m_{c}n_{c}. Suppose that m_{c} ≤ r_{1}.

Define δ(i) = δ(e_{i}/Me_{0} ⋯ e_{i−1}). For i < r_{1} the following upper bound holds^{7} δ(i) ≤ (n_{c}−1). If m_{c} ≤ i < r_{1}, then δ(i) < 0 since b ∈ dcl^{eq}(Me_{0} ⋯ e_{i−1}) by 4.3. Therefore

0 ≤ δ(e_{0} ⋯ e_{r_{1}−1}/M) = ∑_{i<r_{1}} δ(i) = ∑_{i<m_{c}} δ(i) + ∑_{m_{c}≤i<r_{1}} δ(i)
≤ m_{c}(n_{c} − 1) − (r_{1} − m_{c}).

The above inequality proves the claim.

5 The class K^{μ}

Let μ^{∗} be a function that assigns to each prealgebraic code c some natural number μ^{∗}(c). We suppose that

• μ^{∗}(c) ≥ m_{c} − 1

• For all triples l, m, n with m > 0 there are only finitely many c’s with μ^{∗}(c) = l, m_{c} = m and n_{c} = n for each m > 0 and n. (Such μ^{∗} exist since there are only countably many codes.)

• μ^{∗}(c) = μ^{∗}(d) if c is equivalent to some permutation of d^{8}.

Define

μ(c) = m_{c}n_{c} + μ^{∗}(c).

Note that μ(c) ≥ m_{c}.

From now on, a pseudo Morley sequence denotes a pseudo Morley sequence of length μ(c) + 1 for a prealgebraic code c. Given such a pseudo Morley sequence (e_{i}), so is every (e_{i}^{σ}) for every permutation σ by (ix).

The class K^{μ} consists of the elements M ∈ K^{0} which do not contain any pseudo Morley sequence.

Lemma 5.1. Let B be a finite strong subset of M ∈ K^{μ} and A/B a prealgebraic minimal extension. Then there are only finitely many B–isomorphic copies of A in M.

7Note that δ(A/B) ≤ |A/B| in general.

8Note that each permutation is equivalent to at most one prealgebraic code.

Proof. Let A = B ∪ {a} for some tuple a and choose d ∈ acl^{eq}(B) with tp_{i}(a/Bd_{i}) stationary. We need only show that tp_{1}(a/Bd_{1}) ∪ tp_{2}(a/Bd_{2}) has only finitely many realizations in M. Choose a prealgebraic code c by 4.1 and b ∈ acl^{eq}(B) with ⊧ ϕ_{c}(a, b). We show that ϕ_{c}(x, b) cannot be infinitely often realized in M. Otherwise, we obtain at least (μ(c)+1) many such realizations e_{i} with e_{i} ∉ B ∪ {e_{0}, …, e_{i−1}}. It follows from 4.3 that the e_{i}’s form a Morley sequence of ϕ_{c}(x, b) over B and hence a pseudo Morley sequence of c over b by (v), which contradicts that M is in K^{μ}.

Corollary 5.2. Let B ≤ M ∈ K^{μ}, B ⊂ A finite with δ(A/B) = 0. Then there are only finitely many A′ such that: B ≤ A′ ⊂ M and A′ is B-isomorphic to A.

Note that A′ ≤ M automatically.

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

Corollary 5.3. Let B be a finite subset of M ∈ K^{μ}. Then the d–closure of B:

cl_{d}(B) = {x ∈ M| d(Bx) = d(B)}

is countable.

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

Lemma 5.4. If M ∈ K^{μ}, M ≤ N and |N \ M| = 1, then N is also in K^{μ}.

Proof. Let (e_{i}) be a pseudo Morley sequence of c over b in N. There is at most one e_{i} not in M. Now b ∈ dcl^{eq}(M) since μ(c) ≥ m_{c}. It follows that every e_{i} is either in M or in N \ M by 4.3. The latter cannot hold, since n_{c} ≥ 2. Hence (e_{i}) is completely contained in M. Contradiction.

Theorem 5.5. K^{μ} (and hence the class of all finite structures in K^{μ}) has the amalgamation property with respect to strong embeddings.

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

Case 1: A/B is an algebraic minimal extension. Suppose that A = B ∪ {a} for some a algebraic over B in T_{1} and transcendental over B in T_{2}. Two possible (non-exclusive) cases may arise.

Subcase 1.1: tp_{1}(a/B) is realized in M by some a′. Then a′/B is transcendental in T_{2}. Hence B ∪ {a′} is B–isomorphic to A and strong in M. By minimality, M = B ∪ {a′}.

Subcase 1.2: There is some a′ ∉ M realizing tp_{1}(a/B). Define M′ = M ∪ {a} by letting a have the type of a′ over M in the sense of T_{1} and be transcendental over M in the sense of T_{2}. Then M′ is a free amalgam of M and A over B, i.e. M ∩ A = B and M is independent from A over B. It is easy to see that M ≤ M′ and A ≤ M′ for such amalgams. Now, M′ ∈ K^{μ} by 5.4.

Case 2: A/B is transcendental. Then there is a free amalgam M′ of M and A as above. Suppose that M′ is not in K^{μ}. Then M′ contains a pseudo Morley sequence (e_{i}) of c over b. Apply Lemma 4.4 to the extension M′/M to obtain one of the following cases.

Subcase 2.1: b ∈ dcl^{eq}(M). Since M is in K^{μ}, not all members of the pseudo Morley sequence lie in M. Let e_{i} ∉ M. By 4.3 e_{i} is a generic realization of ϕ_{c}(x, b) over M. Independence of M and e_{i} over B yields that b in acl^{eq}(B) by 2.1. Since A ∈ K^{μ}, there is some e_{j} not completely contained in A. Again, e_{j} is a generic realization of ϕ_{c}(x, b) over B. It follows that M = B ∪ {e_{j}} and A = B ∪ {e_{i}} are isomorphic over B.

Subcase 2.2: More than μ^{∗}(c) many e_{i}’s lie in M′ \ M. Since μ^{∗}(c) + 1 ≥ m_{c}, we have that b ∈ dcl^{eq}(A). Proceed now as in subcase 2.1.

A structure M ∈ K^{μ} is rich if for every finite B ≤ M and every finite B ≤ A ∈ K^{μ} there is some B-isomorphic copy of A in M. We will show in the next section that rich structures are models of T_{1} ∪ T_{2}.

Corollary 5.6. There is a unique (up to isomorphism) countable rich structure K^{μ}. Any two rich structures are (L_{1} ∪ L_{2})_{∞,ω}–equivalent.

6 The theory T^{μ}

Lemma 6.1. Let M ∈ K^{μ}, b ∈ acl^{eq}(M), a ⊧ ϕ_{c}(x, b) generic over M and M′ the prealgebraic minimal extension M ∪ {a_{1}, ⋯ a_{n_{c}}}. If M′ is not in K^{μ}, then one of the following hold.

(a) M′ contains a pseudo Morley sequence of c over b, all whose elements but possibly one are contained in M

(b) M′ contains a pseudo Morley sequence for some code c′ with more than μ^{∗}(c′) many elements in M′ \ M. Moreover, m_{c′} > 0.

Proof. Let (e′_{i}) be a pseudo Morley sequence of c′ over b′ in M′. If (b) does not hold, it follows that b′ ∈ dcl^{eq}(M) by 4.4. There must be some e′_{i} not completely contained in M, which is a M–generic realization of ϕ_{c′}(x, b′) by 4.3. Minimality of M′/M yields that e′_{i} is some permutation of a. After permutation of the pseudo Morley sequences, we may assume that e′_{i} = a. Hence ϕ_{c′}(x, b′) ∼^{k_{c}} ϕ_{c}(x, b), so c = c′ and b = b′.

Corollary 6.2.

1. Let c be a prealgebraic code. The statement “M contains no pseudo Morley sequence for c” can be expressed by a universal L_{1} ∪ L_{2}–sentence.

2. Let c be as above, M ∈ K^{μ} a model of T_{1} ∪ T_{2}. The statement “For no b ∈ dcl^{eq}(M) and generic realization a of ϕ_{c}(x, b) is M ∪ {a_{1}, …, a_{n_{c}}} in K^{μ}” can be expressed by an inductive L_{1} ∪ L_{2}–sentence.

Proof. 1. Let Ψ^{i}(x̄) be quantifier-free and T_{i}–equivalent to ∃y Ψ_{c_{i}}(x̄, y). Hence, the desired sentence is

¬∃x̄ (Ψ^{1}(x̄) ∧ Ψ^{2}(x̄))

2. Let i ∈ {1, 2} and M be some elementary substructure of ℂ_{i}. Take m ∈ M and ϕ(x, m) be some L_{i}–formula of rank k and degree 1. Pick some M–generic realization a ∈ ℂ_{i} of ϕ(x, m). Then every quantifier-free property ψ(a, m) of a, m is equivalent to some quantifier-free property ψ^{∗}(m) of m: Set

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

The above shows that for all M ∈ K and for all M–generic realization a of ϕ_{c}(x, b) every L_{1} ∪ L_{2}–sentence over M ∪ {a_{1} …, a_{n_{c}}} can be transformed into one L_{1} ∪ L_{2}–sentence over M, b.

The claim follows now by 6.1, since only finitely many codes c′ need to be considered in case (b), namely those with m_{c′} > 0 and

(μ^{∗}(c′) + 1)n_{c′} ≤ |M′ \ M| = n_{c}.

Models M of the L_{1} ∪ L_{2}–theory T^{μ} will be described by the following properties. Lemmas 4.1 and 4.2 and the above show that the axioms can be first-order described.

Axioms of T^{μ}.

(a) M ∈ K^{μ}

(b) T_{1} ∪ T_{2}

(c) No prealgebraic minimal extension of M lies in K^{μ}.

It is easy to see that M is a model of (b) if and only if M is infinite and has no algebraic minimal extensions. Hence, M is a model of (b) and (c) if and only if M is infinite and has no minimal (or proper) extensions M′ ∈ K^{μ} with δ(M′/M) = 0.

Theorem 6.3. An L_{1} ∪ L_{2} structure is rich if and only if it is an ω–saturated model of T^{μ}.

Proof. Let M ⊧ T^{μ} be ω–saturated. To show that M is rich, we need only consider a finite strong subset B of M and a minimal strong extension A of B in K^{μ}. We aim to show that M contains a B-isomorphic copy of A.

Case (I/II): A/B is algebraic or prealgebraic. We can amalgam M and A in K^{μ}, however M has no proper algebraic or prealgebraic extensions in K^{μ}. Therefore there must be a B–copy of A in M.

Case (III): A = B ∪{a} is transcendental. We want some a′ ∈ M transcendental over B with B ∪ {a′} ≤ M. By saturation of M (note that a′ satisfies a given partial type over B), it suffices to find a′ in some elementary extension M′ of M. If M′ is uncountable, there is some a′ ∈ M′ \ cl_{d}(B) by 5.3. Equivalently, B ∪ {a′} ≤ M′.

Let M be now a rich structure.

Axiom (b): Let a be some element in acl_{1}(M) transcendental over M in T_{2}. There is some finite subset B of M witnessing 1–algebraicity of a. We may assume that B ≤ M. By lemma 5.4, B ≤ B ∪ {a} ∈ K^{μ}, so there exists some copy of a over B in M. It follows that M is acl_{1}–closed. Since M is infinite^{9} it is a model of T_{1}. Likewise for T_{2}.

Axiom (c): Let a be a generic realization of ϕ_{c}(x, b) over M such that M ∪ {a} is in K^{μ}. Choose some finite strong subset C of M with b ∈ dcl^{eq}(C). Then C ≤ C ∪ {a}, so there is a copy a′ of a in M over C with C′ = C ∪ {a′} ≤ M by richness of M. Iterate to obtain a sufficiently large Morley sequence a′, a′′, … of ϕ_{c}(x, b) in M. This contradicts that M ∈ K^{μ}.

Choose now some ω–saturated M′ ≡ M. The first part of the proof yields that M′ is rich. So M′ ≡_{∞,ω} M. So M is also ω–saturated.

7 Proof of the Main Theorem

In this section we drop the QE-Hypothesis of section 3. Hence in our class K we replace isomorphic embeddings by bi-elementary maps, i.e. maps which are both T_{1} and T_{2} elementary.

Corollary 7.1. T^{μ} is complete. Two tuples a and a′ in models M and M′ have the same type if and only if there is some bi-elementary bijection

f : cl(a) → cl(a′)

with f(a) = a′.

Proof. The structure K^{μ} is a model of T^{μ}, so T^{μ} is consistent. Let M be any model of T^{μ}. By 6.3 there is some rich M′ ≡ M. Since M′ ≡_{∞,ω} K^{μ}, we have that T^{μ} is complete.

Let M ≺ N and M′ ≺ N′ be two ω–saturated elementary extensions. It is easy to see^{10} that M ≤ N and M′ ≤ N′, i.e. closure does not change. An isomorphism f : cl(a) → cl(a′) belongs to some back-and-forth system of partial isomorphisms between finite strong subsets of M′ and N′. Hence f is elementary.

For the other direction, suppose that a and a′ have the same type. Then there is some isomorphic embedding f : cl(a) → M′ mapping a to a′. Write A′ = f(cl(a)). Then d(a) = δ(cl(a)) = δ(A′). Therefore d(a′) ≤ d(a) and by symmetry d(a′) = d(a). Note that A′ has no proper subset A′′ containing a′ with δ(A′′) = d(a′) since cl(a) does not. Hence A′ = cl(a′).

9This follows also from 5.4.

10 If M ≰ N, there is some a ∈ N with δ(a/M) < 0. Find some finite B ≤ M with δ(a/B) < 0. a realizes some L_{1} ∪ L_{2}–formula witnessing this fact. However ϕ(x, b) is not realized in M, so M ⊀ N.

Theorem 7.2. T^{μ} is strongly minimal and d is the dimension function of the natural pregeometry on models of T^{μ}. In particular

MR(ā/B) = d(ā/B)

.

Proof. All types tp(a/B) with d(a/B) = 0 are algebraic by 5.2. It follows from 7.1 that there is only one type with d(a/B) = 1^{11}. Therefore T is strongly minimal. The rest follows easily since d describes the algebraic closure.

The above proves Theorem 1.

8 Remarks

It is easy to prove that T^{μ} has the following properties:

• T^{μ} has the DMP.

• For each i = 1, 2 every L_{i}–formula ϕ(x, b) preserves its Morley rank and degree from T_{i} in T^{μ}.

• ([7]) Let M be a model of T^{μ} which is an elementary substructure of N according to both T_{1} and T_{2}. Then M is an elementary substructure of N.

We prove the last property. Let M and N be models of T^{μ} with M ↾ L_{i} ≺ N ↾ L_{i} for i = 1, 2. We show that M ≤ N. It follows then from 7.1 that N ≺ M.

Recall that M has no extension M′ in N with δ(M′/M) = 0. If M is not strong, there is some a ∈ N with δ(a/M) = −1 which is algebraic over M in both T_{1} and T_{2}. Contradiction.

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