1 FUSION OVER A VECTOR SPACE ANDREAS BAUDISCH, AMADOR MARTIN-PIZARRO, MARTIN ZIEGLER Institut für Mathematik Humboldt-Universität zu Berlin ...

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FUSION OVER A VECTOR SPACE

ANDREAS BAUDISCH, AMADOR MARTIN-PIZARRO, MARTIN ZIEGLER Institut f¨ ur Mathematik Humboldt-Universit¨ at zu Berlin D-10099 Berlin, Germany Institut Camille Jordan Universit´ e Claude Bernard Lyon 1 69622 Villeurbanne cedex, France Mathematisches Institut Albert-Ludwigs-Universit¨ at Freiburg D-79104 Freiburg, Germany

Let T1 and T2 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 T1 ∪ T2 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 T1 and T2 be two countable strongly minimal theories, in disjoint languages, and with the DMP, the definable multiplicity property. Then T1 ∪ T2 has a strong minimal completion. The above theorem was proved by extending Fra¨ıss´e’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 T0 the theory of infinite F – vector spaces in the language L0 = {0, +, −, λ}λ∈F . In this article, we will prove the following: Theorem 1.1. Let T1 and T2 be two countable strongly minimal extensions of T0 with the DMP, and assume that their languages L1 and L2 intersect in L0 . Then T1 ∪ T2 has a strongly minimal completion T µ . 1

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This “fusion over a vector space” was proposed by Hrushovski in [1]. In the special case where both T1 and T2 are 1–based this fusion was already proved by A. Hasson and M. Hils [2]. These two articles also discuss fusions over more general T0 . 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 tri denote the transcendence degree in the sense of Ti and dim the F –linear dimension. Then for every finite subset A of M we have dim(A) ≤ tr1 (A) + tr2 (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 T1 and in the sense of T2 . It followsa 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 T0 with the DMP, C denotes the monster model of T , tr(a/A) the transcendence degreeb 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) ∼kx ψ(x) to express that the Morley rank of the symmetric difference of φ and ψ is smaller than k, a We

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

b The

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We denote by hai We denote by the F –vector space of dimension dim(a) spanned by the components of the n–tuple a. Subspaces of hai can be described in terms of subspaces U of F n as n nX o Ua = ui ai u ∈ U . i=1

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 (Cn , +). 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 ∈ Gln (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(χg ) where χg is the associated group formulac . A definable set X ⊂ Cn 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 φc (x, y) where the variable x ranges over nc –tuples of the home sort and y over a sort of T eq , with the following properties. C(i) All non–emptyd φc (x, b) have (constant) Morley rank kc and Morley degree 1. C(ii) For every U ≤ F nc there is a number kc,U such that for every realization a of φc (x, b) we have: tr(a/b, U a) ≤ kc,U . Moreover, equality holds for generic a. (So we have kc = kc,0 .) C(iii) dim(a) = nc for all realizations a of φc (x, b). If a is generic, then dim(a/ acl(b)) = nc (this is equivalent to kc,U = kc −1 for all one–dimensional U ). C(iv) If φc (x, b) and φc (x, b0 ) are not empty and φc (x, b) ∼kc φc (x, b0 ), then b = b0 . C(v) If some non–empty φc (x, b) is a coset formula, then all are. We call such a code c a coset code. In this case, the group Inv(φc (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 φc (x + m, b) is encoded by φc . C(vii) There is a subgroup Gc of Glnc (F ) such that: a) for all H ∈ Gc and all non–empty φc (x, b) there exists a (unique) bH such that φc (Hx, b) ≡ φc (x, bH ). c This

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

d Codes

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b) if H ∈ Glnc (F ) \ Gc , then no non–empty φc (Hx, b) is encoded by φc . Two codes c and c0 are equivalent if for every b there is some b0 such that φc (x, b) ≡ φc0 (x, b0 ) and vice versa. If c is a code and H ∈ Glnc (F ), then φcH (x, y) = φc (Hx, y) is also a code. C(viia) states that cH and c are equivalent if H lies in Gc . Corollary 2.1. Let p ∈ S(b) be the generic type containing φc (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 φc (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) ∼kc φc (x, b0 ) for some parameter b0 . By C(iv) it follows that b0 is uniquely determined, thus b0 ∈ dcleq (d). Proof. Set nc = |x|, kc = MR χ(x, d) and kc,U = tr(a/d, U a) for a generic realization a of χ(x, d). Let p be the global type of rank kc containing χ(x, d) and b0 its canonical base and choose some φ(x, b0 ) ∈ p of rank kc and degree 1. Hence, φ(x, b0 ) satisfies χ(x, d) ∼kc φc (x, b0 ) and has property C(iv) for all b and b0 realizing tp(b0 ). We can choose φ(x, b0 ) strong enough to ensure that C(iv) holds for all b and b0 . Consider now the set X of all b of same length and sort as b0 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, b0 ) is, and in this case Inv(φ(x, b)) = Inv(φ(x, b0 )). 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, U a) ≥ kc,U is equivalent to tr(U a/b) ≤ (kc − kc,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(b0 ) belong to X. So a finite part θ(y) of this type implies X. Then the formula φ0c (x, y) = φ(x, y) ∧ θ(y) has all properties, except possibly C(vi) and C(vii). Given any nc –tuple m and parameter b, the formula φ0c (x + m, b), if non–empty, has again rank kc and degree 1. If a is a generic realization, then a + m is a generic

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realization of φ0c (x, b) and a + m ^ | b m. Let u be some vector in F nc such that P P P i ui ai ∈ acl(b, m). Then i ui (ai + mi ) ∈ acl(b, m). By independence i ui (ai + mi ) ∈ acl(b), which implies u = 0. Therefore dim(a/ acl(b, m)) = nc and φ0c (x+m, b) is simple. We note also that for every U tr(U a/m, b) = tr(U (a + m)/m, b) = tr(U (a + m)/b), which implies tr(a/m, b, U a) = kc,U . Whence, each φ0c (x + m, b) can be encoded by some formula φ0 (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 φ1 , . . . , φr with all properties except possibly C(vi) and C(vii) which encode all formulas φ0c (x + m, b) with m and b varying. Moreover, we may assume that for all i |= ∀y ∃v, w φi (x, y) ∼kxc φ0c (x + v, w), which implies that either all or none of the φi code coset formulas and if so, they have all the same invariant group Inv(φ(x, b0 )). To prevent double-encoding, set ^ θi (y) = ∀z φj (x, z) 6∼kxc φi (x, y). j

Fix a sequence of different constantse w1 , . . . , wr and define r _ . 00 0 φc (x, y, y ) = φi (x, y) ∧ θi (y) ∧ y 0 = wi . i=1

φ00c (x, y)

has all properties except possibly C(vii). To prove C(vi) fix m and b, w such that φ00c (x + m, b, w) is not empty. Then w equals some wj and φ00c (x + m, b, w) is equivalent to φj (x + m, b). We know that φj (x, b) ∼ φ0c (x + m0 , b0 ) for some m0 and b0 . It follows that: φj (x + m, b) ∼ φ0c (x + (m + m0 ), b0 ). Since φ0c (x + (m + m0 ), b0 ) can be encoded by one of the φi , property C(vi) holds. Only property C(vii) remains to be obtained. Change the notation slightly and assume χ(x, d) ∼kc φ00c (x, b0 ). Define Gc to be the set of all A ∈ Glnc (F ) such that there is some m and some realization b of p = tp(b0 ) such that φ00c (Ax, b0 ) ∼kc φ00c (x + m, b). To show that Gc is a group, consider another A0 ∈ Gc . Then there are m0 and b0 |= p such that φ00c (A0 x, b) ∼kc φ00c (x + m0 , b0 ). This yields φ00c (AA0 x, b0 ) ∼kc φ00c (A0 x + m, b) ≡ φ00c (A0 (x + A0−1 m), b) ∼kc φ00c (x + (A0−1 m + m0 ), b0 ), and so AA0 ∈ Gc . There is a ρ(y) ∈ p such that for no A ∈ Glnc (F ) \ Gc there are some b which satisfies ρ and some tuple m with φ00c (Ax, b0 ) ∼kc φ00c (x + m, b), i.e. ^ |= ¬ρA (b0 ), A∈Glnc (F )\Gc e If

T has no constants, use definable elements in a sort of T eq .

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where ρA (y) = ∃z, y 0 ρ(y 0 ) ∧ φ00c (Ax, y) ∼kxc φ00c (x + z, y 0 ). Whence the formula σ(y) =

^

^

ρA (y) ∧

A∈Gc

¬ρA (y)

A∈Glnc (F )\Gc

is satisfied by b0 . An easy calculation shows ^ |= ∀y σ(y) → σ A (y) ∧ A∈Gc

¬σ A (y) ,

^ A∈Glnc (F )\Gc

where: σ A (y) = ∃y 0 σ(y 0 ) ∧ φ00c (Ax, y) ∼kxc φ00c (x, y 0 ). Write now 00 φ000 c (x, y) = φc (x, y) ∧ σ(y).

It is clear that φ000 c still encodes χ(x, d) and has all properties except possibly C(vii). For C(vi) assume φ00c (x+m, b) ∼kc φ00c (x, b0 ). b0 satisfies ρA iff , φ00c (Ax, b0 ) ∼kc φ00c (x+ m0 , b00 ) for some m0 and some realization b00 of ρ, or, equivalently, φ00c (Ax, b) ∼kc φ00c (x + (m0 − A−1 m), b00 ). Therefore b satisfies ρA iff b0 satisfies ρA . This implies that b satisfies σA iff b0 satisfies σA . So C(vi) holds. 000 Now, C(vii) is satisfied by φ000 c and Gc only in the weaker form that φc (Hx, b) 000 is encoded by φc iff H ∈ Gc . By C(iv) we can define for each A ∈ Gc a function b 7→ bA such that kc 000 φ000 φc (x, bA ) c (Ax, b) ∼

and set: φc (x, y) =

^

−1 φ000 x, y A ). c (A

A∈Gc

Since φc (x, b) ∼kc φ000 c (x, b) only C(viia) needs to be check: Given H ∈ Gc , ^ ^ −1 −1 φc (Hx, b) ≡ φ000 Hx, bA ) ≡ φ000 x, bHA ) ≡ φc (x, bH ). c (A c (A A∈Gc

A∈Gc

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 ∈ Glnc (F ) the code cH is equivalent to some code in C.f f We

0

will construct C so that every cH is equivalent to some cH 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 χi , i = 1, 2, . . . with parameters in M . We need only show that all χi can be encoded in C. We construct C as an increasing union of finite sets ∅ = C0 ⊂ C1 ⊂ · · · . Assume that Ci−1 is defined and closed under the action of Gl(F ) in the sense of C(ix). If χi can be encoded in Ci−1 , we set Ci = Ci−1 . Otherwise choose some code c0 which encodes χi . Let ρ(b) express, that φc0 (x, b) cannot be encoded in Ci−1 and define φc (x, y) = φc0 (x, y) ∧ ρ(y). Then φc still encodes χi . Moreover φc determines again a code: only C(vii) needs to be considered. So assume that |= ρ(b) and let H be in Gc0 . We need to show that |= ρ(bH ). Otherwise φc0 (Hx, b) can be encoded in Ci−1 . Since Ci−1 is closed under H −1 , also φc0 (x, b) can be encoded in Ci−1 , which is a contradiction. Choose now a system of right representatives A1 , . . . , Ar of Gc in Glnc (F ) and set Ci = Ci−1 ∪ {cA1 , . . . , cAr }. 3. Difference sequences As in the previous section, T denotes a countable strongly minimal extension of T0 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 e0 and e1 two generic B–independent realizations. If H ∈ Gln (F ) and e0 ^ | B e0 − He1 , then φ(x) is a coset formula and H ∈ Inv(φ(x)). Proof. It follows from MR(He1 /B, He1 − e0 ) = MR(e0 /B, He1 − e0 ) = MR(e0 /B) ≥ MR(He1 /B) that e0 , He1 and He1 − e0 are pairwise independent over B. By [9] e0 , He1 and He1 − e0 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 mc ≥ 0 such that for no φc (x, b) there is a Morley sequence (ei ) of length mc and some b0 from the same sort as b with ei ^ 6 | b b0 for all i. Theorem 3.2. For every code c and any number µ > mc there exists a parameter free formula Ψc (x0 , . . . , xµ ), whose realizations are called difference sequences (of length µ), with the following properties.

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P(i) If e00 , . . . , e0µ , f is a Morley sequence of φc (x, b), then e00 − f, . . . , e0µ − f is a difference sequence.g P(ii) For every difference sequence e0 , . . . , eµ there is a unique b with |= φc (ei , b) for all i (we call the base of the sequence). Furthermore, b is uniquely determined if φc (ei , b) holds for at least mc many i’s.h P(iii) If e0 , . . . , eµ is a difference sequence then so is e0 − ei , . . . , ei−1 − ei , −ei , ei+1 − ei , . . . , eµ − ei . P(iv) Let e0 , . . . , eµ be a difference sequence with base b. We distinguish two cases: Suppose c is not a coset code: a) If ei is generic in φc (x, b), then ei ^ 6 | b ei − Hej for all H ∈ Glnc (F ) and i 6= j. Suppose c is a coset code: b) c) d) e)

φc (x, b) is a group formula. Ψc (e0 , . . . , ei−1 , ei − ej , ei+1 , . . . , eµ ) for all i 6= j.i Ψc (e0 , . . . , ei−1 , Hei , ei+1 , . . . , eµ ) for all H ∈ Inv(c).i If ei is a generic realization of φc (x, b), then ei ^ 6 | b ei − Hej for all i 6= j and H ∈ Glnc (F ) \ Inv(c).

P(v) For all H ∈ Gc Ψc (x0 , . . . , xµ ) ≡ Ψc (Hx0 , . . . , Hxµ ). The derived sequences of of (ei ) consist of all difference sequences obtained from (ei ) 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(e0 , . . . , eµ ): There is some b0 and a Morley sequence e00 , . . . , e0µ , f 0 of φc (x, b0 ) such that ei = e0i − f 0 . This is clearly a partial type. Claim: DS has all properties of Ψc . Proof: Assume ei = e0i − f 0 for a Morley sequence (e0i ), f 0 of φc (x, b0 ). Then (ei ) is a Morley sequence of φc (x + f 0 , b0 ) over b0 , f 0 . If φc (x + f 0 , b0 ) ∼ φc (x, b), then (ei ) general b will not be the base of (e0i ) in the sense of P(ii). follows that b ∈ dcl(ei1 , . . . , eimc ) for all 0 ≤ i1 < · · · imc ≤ µ. i By P(ii) and µ > m this new sequence has also base b. c g In h It

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is a Morley sequence of φc (x, b).j P(ii) Suppose |= φc (ei , b00 ) for mc –many i’s. Then there exists such an i with ei ^ | b b00 . Hence MR(φc (x, b) ∧ φc (x, b00 )) = kc and therefore b = b00 . P(iii) Fix i ∈ {0, . . . , µ} and note that e00 , . . . , e0i−1 , f 0 , e0i+1 , . . . , e0µ , e0i is again a Morley sequence for φc (x, b0 ). Hence, the sequence e00 − e0i , . . . , e0i−1 − e0i , f 0 − e0i , e0i+1 − e0i , . . . , e0µ − e0i = e0 − ei , . . . , ei−1 − ei , −ei , ei+1 − ei , . . . , eµ − ei also satisfies DS. P(iva) If c is not a coset code, then φc (x, b) is not a coset formula and the claim follows from Lemma 3.1. P(ivb) If c is a coset code, then φc (x, b0 ) is a coset formula. Since f 0 is a generic realization, φc (x, b) ∼ φc (x + f 0 , b0 ) is a group formula. P(ivc) Extend the Morley sequence e0 , . . . , eµ of φc (x, b) by f . If φc (x, b) is a group formula, and i 6= j, then e0 + f, . . . , ei−1 + f, ei − ej + f, ei+1 + f, . . . , eµ + f, f is again a Morley sequence of φc (x, b). It follows that e0 , . . . , ei−1 , ei − ej , ei+1 , . . . , eµ realizes DS. P(ivd) Choose f as above. If H ∈ Inv(c), then e0 + f, . . . , ei−1 + f, Hei + f, ei+1 + f, . . . , eµ + f, f is also a Morley sequence of φc (x, b). It follows that e0 , . . . , ei−1 , Hei , ei+1 , . . . , eµ realizes DS. P(ive) Immediate from Lemma 3.1. P(v) If φc (Hx, b0 ) ≡ φc (x, b00 ), then He00 , . . . , He0µ , Hf is a Morley sequence of φc (x, b00 ) and (Hei ) = (He0i − Hf ) satisfies DS.

j Since

b is canonical.

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This proves the claim. We will take for Ψc 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 Ψ0c . Assume c to be a non–coset code. Write Vi (x0 , . . . , xµ ) = (x0 − xi , . . . , xi−1 − xi , −xi , xi+1 − xi , . . . , xµ − xi ) and VH (x0 , . . . , xµ ) = (Hx0 , . . . , Hxµ ). Let V be the finite group generated by V0 , . . . , Vµ and VH for H ∈ Gc . The formula ^ Ψ(¯ x) = Ψ0c (V (¯ x)) V ∈V

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 {VH }H∈Gc and the operations described in P(ivc) and P(ivd), and define. Ψc analogously. It satisfy then P(ivc) and P(ivd) and P(v), and thereforek also P(iii). We choose an appropriate Ψc (depending on µ) for every code c in such a way that ΨcH (x0 , . . . ) = Ψc (Hx0 , . . . ). For two codes c and c0 to be equivalent we also impose that Ψc ≡ Ψc0 . Corollary 3.3. Lemma 2.3 remains true if Ψc 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 theoriesl T1 and T2 which intersect in T0 , the theory of infinite F –vector spaces. By considering their morleyization, we may assume that : k Note l In

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

(I)

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QE-Assumption . Both theories Ti have quantifier elimination. Their languages Li are relational, except for the function symbols in L0 . We may also assume that codes φc and formulas Ψc for T1 and T2 are quantifier free, as well as Ti –types tpi (a/B). This assumption will be dropped only in section 9. Let K be the class of all models A of T1∀ ∪ T2∀ . So, A is an F –vector space, which occurs at the same time as a subspace of C1 and as a subspace of C2 , where Ci the monster model of Ti . For finite A ∈ K, define δ(A) = tr1 (A) + tr2 (A) − dim A. We have that: δ(0) = 0

(4.1)

δ(hai) ≤ 1

(4.2)

δ(A + B) + δ(A ∩ B) ≤ δ(A) + δ(B)

(4.3)

Moreover, if dim(A/B) is finitem , then we also set δ(A/B) = tr1 (A/B) + tr2 (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(A0 /B) ≥ 0 for all finite A0 ⊂ A and denote this by B ≤ A. A proper strong extension B ≤ A is minimal, if there is no A0 properly contained between B and A such that B ≤ A0 ≤ 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 T1 or of T2 . 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/A0 ) < 0 for all A0 which lie properly between B and A. Proof. One direction is clear, since A0 ≤ A implies δ(A/A0 ) ≥ 0. Conversely, if δ(A/A0 ) ≥ 0 for some A0 , we may assume that δ(A/A0 ) is maximal. Then A0 ≤ A and A is not minimal over B. m We

do not assume B ⊂ A. that B is strong in all A0 ⊂ A.

n Note

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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 = hB, ai 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 = hB, ai, 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(hB, ai/B) = 0. Therefore hB, ai = A. δ(A/B) > 0. For each a ∈ A\B it follows that δ(hB, ai/B) 6= 0. Hence δ(hB, ai/B) = 1 and therefore hB, ai ≤ A. By minimality hB, ai = A. We define the class K0 ⊂ K as K0 = {M ∈ K | 0 ≤ M }. It is easy to see that K0 can be axiomatized by a set of universal L1 ∪ L2 –sentences. The following results are also easy. Lemma 4.3. Fix M in K 0 and define d(A) =

min

A⊂A0 ⊂M

δ(A0 )

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 K0 and A a finite subspace. Let A0 be an extension of A, minimal with δ(A0 ) = d(A). Then A0 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) = δhXi and d(X) = dhXi. Note that δ(A) ≤ dim(A).

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5. Prealgebraic codes From now on, T1 and T2 are two countable strongly minimal extensions of T0 with the DMP. We assume the QE-Assumption of section 4, as in the next three sections 6, 7 and 8. Choose for each Ti a set Ci of codes as in Corollary 3.3. A prealgebraic code c = (c1 , c2 ) consists of two codes c1 ∈ C1 and c2 ∈ C2 with the following properties: • nc := nc1 = nc2 = kc1 + kc2 • For all proper, non–zero subspaces U of F nc kc1 ,U + kc2 ,U + dim U < nc .

(5.1)

Set mc = max(mc1 , mc2 ). Note that simplicity of the φci (x, b) implies that nc ≥ 2. Note also that for every H ∈ Glnc (F ) H cH = (cH 1 , c2 )

is a prealgebraic code. Notation Unless otherwise stated, independence (a ^ | b c) means independent both in the sense of T1 and T2 . If c is a prealgebraic code, a (generic) realization of φc (x, b) is a (generic) realization of both φc1 (x, b1 ) and φc2 (x, b2 ). A Morley sequence of φc (x, b) is a Morley sequence for both φc1 (x, b1 ) and φc2 (x, b2 ). Similarly, for a set X of real elements, one defines X–generic realization of φc (x, b) and Morley sequence of φc (x, b) over X. A difference sequence for c with basis b = (b1 , b2 ) is a difference sequence for ci with basis bi for each i = 1, 2. We say c is a coset code if c1 and c2 are. We define then Inv(c) = Inv(c1 )∩Inv(c2 ). T1eq and T2eq have only the home sort in common. So b ∈ dcleq (A) (resp. acleq (A)) means that b is a pair consisting of an element in dcleq 1 (A) (resp. acleq 1 (A)) and an element in dcleq 2 (A) (resp. acleq 2 (A)). If M is a model of T1 ∪ T2 , then M eq consists of imaginary elements in the sense of T1 and in the sense of T2 . Lemma 5.1. Let B ≤ A be a prealgebraic minimal extension and a = (a1 , . . . , an ) a basis for A over B. Then there is a prealgebraic code c and b ∈ acleq (B) such that a is a generic realization of φc (x, b). Proof. Fix i ∈ {1, 2}. Choose di ∈ acleq i (B) such that tpi (a/Bdi ) is stationary. Since A/B is transcendental, we have dim(a/ acli (B)) = n. So we can find an Li –formula χi (x) ∈ tpi (a/Bdi ) of Morley rank ki = MRi (a/Bdi ). Since A/B is transcendental, χ(x) is simple. By 2.3 there is a Ti –code ci ∈ Ci and bi ∈ dcleq i (Bdi ) with χi (x) ∼ki φci (x, bi ).

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Set c = (c1 , c2 ) and b = (b1 , b2 ). It follows from k1 + k2 − n = tr1 (a/B) + tr2 (a/B) − dim(A/B) = δ(A/B) = 0 that nc = kc1 + kc2 . Inequality (5.1) follows from Lemma 4.1: kc1 ,U + kc2 ,U − (n − dim U ) = tr1 (a/b, U a) + tr2 (a/b, U a) − dim(F n /U ) =δ(A/B + U a) < 0. Lemma 5.2. Let B ∈ K, b ∈ acleq (B), c be a prealgebraic code, and a a B–generic realization of φc (x, b). Then hB, ai 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 + U a for some subspace U of F nc . Lemma 5.3. Let B ⊂ A be in K, c a prealgebraic code, b in acleq (B) and a ∈ A a realization of φc (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 φc (x, b). Proof. Let U a = hai ∩ B. Let U a = hai ∩ B. Since a is not contained in B, it follows that U is a proper subspace of F nc . Therefore δ(a/B) = tr1 (a/B) + tr2 (a/B) − (n − dim U ) ≤ kc1 ,U + kc2 ,U + dim U − n. If U 6= 0 the right hand side is negative. If U = 0, we have δ(a/B) = tr1 (a/B) + tr2 (a/B) − n ≤ kc1 + kc2 − n = 0. So δ(a/B) = 0 implies tri (a/B) = kci . 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 e0 , . . . , eµ in N , with basis b, and λ ≤ µ, either • the basis of some λ–derived sequence of e0 , . . . , eµ lies in dcleq (M ), or • for every subset A of M 0 with dim A ≤ ε the sequence e0 , . . . , eµ contains a Morley sequence of φc (x, b) over M, A of length r. Proof. By adding e0 , . . . , emc −1 to A, we may assume that b ∈ dcleq (M ∪ A). If at least (mc +1) many of the ei lie in the same class of N nc /M nc , we subtract one of these elements from the others and obtain a derived sequence with mc many elements in M , which then has a base in dcleq (M ). Therefore, we may assume that each class of N nc /M nc contains at most mc many ei ’s.

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Fix an A of dimension ε and set d = dim(e0 , . . . , eµ /hM, Ai). Then dim(e0 , . . . , eµ /M ) ≤ d + ε. Thus by our assumption µ + 1 ≤ mc |F |(d+ε)nc . Consider the following sets of indices. X1 = {i ≤ µ | ei generic over M, A, e0 , . . . , ei−1 } X2 = {i ≤ µ | i 6∈ X1 ∧ dim(ei /M, A, e0 , . . . , ei−1 ) > 0} It is clear that d ≤ (|X1 | + |X2 |) nc . With the notation δ(i) = δ(ei /M, A, e0 , . . . , ei−1 ), Lemma 5.3 implies that δ(i) < 0 if x ∈ X2 , and δ(i) = 0 otherwise. Since M ≤ N we have 0 ≤ δ(A, e0 , . . . , eµ /M ) = δ(A/M ) +

µ X

δ(i) ≤ ε − |X2 |.

i=1

If we put the three inequalities together, we obtain µ + 1 ≤ mc |F |(|X1 |nc +ε nc +ε)nc . If µ is large enough, |X1 | ≥ r and (ei )i∈X1 is our Morley sequence. 6. The class Kµ 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 nc = 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) M(iii) M(iv) M(v)

µ(c) ≥ λ(nc , 1, c) + 1 µ(c) ≥ λ(0, λ(0, mc + 1, c) + 1, c) µ(c) ≥ λ(0, µ∗ (c) + 1, c) µ(c) = µ(d), if c is equivalent to some dH .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 dH , and (ei ) is a difference sequence for d, then (Hei ) is a difference sequence for c. o Note

that every dH can be equivalent to only one prealgebraic c.

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The class Kµ consists of all elements A of K0 which do not contain a difference sequence for any prealgebraic code. Lemma 6.1. Let B ≤ M ∈ Kµ 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 ∈ acleq (B) such that the types tpi (a/Bdi ) are stationary. It suffices to show that for all such d the partial type tp1 (a/Bd1 ) ∪ tp2 (a/Bd2 ) has only finitely many realizations in M . For this we choose a prealgebraic code c and b ∈ acleq (B) with |= φc (a, b) by 5.1. We now show that φc (x, b) has only finitely many realizations in M . If not, there is an infinite sequence e0 , . . . of realizations such that ei is not contained in hB, e0 , . . . , ei−1 i (since the latter set is finite). Strongness of B in M yields that e0 is a B–generic realization by 5.3. From δ(e0 /B) = 0 we conclude that hB, e0 i ≤ M . If we proceed in this way, we see that e0 , . . . is a Morley sequence of φc (x, b) over B. Now P(i) yields that e1 − e0 , . . . , eµ(c)+1 − e0 is a difference sequence of c. Contradiction. Corollary 6.2. Let B ≤ M ∈ Kµ and B ⊂ A finite with δ(A/B) = 0. Then there are only finitely many B ≤ A0 ⊂ M , which are isomorphic to A over B. Note that automatically A0 ≤ M . Proof. Decompose the extension A/B into a sequence of minimal extensions. Corollary 6.3. Let X be a finite subset of M ∈ Kµ . Then the d–closure of X: cld (X) = {x ∈ M | d(Xx) = d(X)} is at most countable. Proof. Note that cld (X) is the union of all A0 ⊂ M with cl(X) ⊂ A0 and δ(A0 / cl(X)) = 0. Lemma 6.4. Let M ∈ Kµ , M ≤ M 0 a minimal extension and (ei ) a difference sequence for a prealgebraic code c with base b ∈ acleq (M ). Then c has a difference sequence (e0i ) with the same base b such that M contains e00 , . . . , e0µ(c)−1 . In particular, e0µ(c) is an M –generic realization of φc (b), which generates M 0 over M as a vector space. Also b must be in dcleq (M ). Proof. Let ei be any element which does not lie in M . By strongness of M in M 0 and Lemma 5.3, it follows that ei is an M –generic realization of φc (x, b). We have δ(hM, ei i/M ) = 0 and whence hM, ei i ≤ M 0 . By minimality hM, ei i = M 0 . After permutation we may assume that e0 , . . . , eν−1 are in M and eν , . . . , eµ(c) are not. Since M ∈ Kµ , it follows that ν ≤ µ(c). As above, for i ≥ ν, ei is an M –generic realization of φc (x, b) which generates M 0 /M , so ei − Hi eµ(c) ∈ M for some Hi ∈ Glnc (F ). Therefore ei ^ | b ei − Hi eµ(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 Hi ∈ Inv(c) by P(ive). By P(ivc) and P(ivd) the difference sequence e0 , . . . , eν−1 , eν − Hν eµ(c) , . . . , eµ(c)−1 − Hµ(c)−1 eµ(c) , eµ(c) is as stated in the claim. Note that the above sequence has same base b. 7. Amalgamation Theorem 7.1. Kµ (and therefore also the class of all finite elements of Kµ ) has the amalgamation property with respect to strong embeddings. Proof. Consider B ≤ M and B ≤ A in Kµ . We want to find a strong extension M 0 ∈ Kµ of M and a B ≤ A0 ≤ M 0 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 0 of M and A is in Kµ or that M and A are isomorphic over B. Case 1: A/B is algebraic. Then A = hB, ai for an element a which is (e.g.) algebraic over B in the sense of T1 and transcendental over B in the sense of T2 . There are two (non exclusive) subcases. Subcase 1.1: tp1 (a/B) is realized in M . Choose some realization a0 in M . Hence, a0 /B is transcendental in the sense of T2 and a0 7→ a defines an isomorphism between M = hB, a0 i and A over B. Subcase 1.2: There is some a0 6∈ M , which realizes tp1 (a/B) (in the sense of T1 ). Define the structure M 0 = hM, ai by setting a to have the same T1 –type over M as a0 and being transcendental over M in the sense of T2 i.e. M 0 is a free amalgam of A and M over B in the sense that M are A are independent over B and linearly independentp over B. It is easy to see that, in free amalgams, M ≤ M 0 and A ≤ M 0 . By Lemma 7.2 below, M 0 belongs to Kµ . Case 2: A/B is transcendental. We may assume that M ∩ A = B. Since A/B is transcendental, we find M 0 = M + A in K, such that M and A are independent over B. So M 0 is a free amalgam of M and A, and M 0 is a minimal extension of M and of A. If M 0 ∈ Kµ , we are done. Otherwise, 7.3 shows that, by symmetry, we may assume that M 0 contains a difference sequence (ei ) of a prealgebraic code c with base b ∈ acleq (M ). Also by Lemma 7.2 , dim(M 0 /M ) > 1 and A/B is prealgebraic. By minimality and Lemma 6.4, we may also assume that e0 , . . . , eµ(c)−1 are in M and eµ(c) is an M –generic realization of φc (x, b), which generates M 0 over M . Write eµ(c) = m+a for m ∈ M and a ∈ A. Therefore δ(a/B) = δ(a/M ) = δ(eµ(c) /M ) = 0. p I.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 0 /A) and m and obtain two subcases: Subcase 2.1: There is a (µ(c) − 1)–derived difference sequence (e0i ) with basis b0 ∈ dcleq (A). Since e0i ∈ M for i ≤ µ(c) − 1, the base b0 is in dcleq (M ) ∩ dcleq (A) ⊂ acleq (B). Hence e0µ(c) is an M –generic realization of φc (x, b0 ) which generates M 0 over M . Again there are two cases. Subsubcase 2.1.1: e0µ(c) ∈ A. Since A ∈ Kµ , there is an e0i ∈ M not in A. By minimality e0i generates M over B and e0µ(c) 7→ e0i defines a B-isomorphism between A and M . Subsubcase 2.1.2: e0µ(c) 6∈ A. Then e0µ(c) is an A–generic realization of φc (x, b0 ). Write e0µ(c) = m0 + a0 for m0 ∈ M and a0 ∈ A. Since e0µ(c) , m0 and a0 are pairwise independent over b0 , then, for i = 1, 2, φci (x, b0i ) is a coset formula by [9] and whence a group formula by C(v) and P(ivb). It follows that −m0 and a0 are generics of the same Bb0i –definable coset of a Bb0i –definable connected group. Thus they have the same type over B. As above m0 generates M over B and a0 generates A over B. So the map a0 7→ −m0 defines an isomorphism between A and M over B. Subcase 2.2: e0 , . . . , eµ(c)−1 contains a B, m–generic realization of φc (x, b), say e0 . For i = 1, 2, e0 and eµ(c) have the same Ti –type over B, m, bi . Whence e0 − m and a have the same Ti –type over B, m, bi , a forteriori over B. Whence a 7→ e0 − m defines a B–isomorphism between A and M .

Lemma 7.2. Let M ∈ Kµ , M ≤ M 0 and dim(M 0 /M ) = 1. Then, M 0 ∈ Kµ . Proof. Assume M 0 6∈ Kµ and (ei ) is a difference sequence in M 0 for a prealgebraic code c with base b witnessing this fact. Since dim(M 0 /M ) = 1 and nc ≥ 2, no ei is an M –generic realization. By the choice of µ(c) and Lemma 5.4 we may assume that b ∈ dcleq (M ). By Lemma 5.3 we conclude that all ei lie in M . Contradiction. Lemma 7.3. Let M 0 be a free amalgam of M and A over B and (ei ) a difference sequence in M 0 . Then there is a derived sequence with base in acleq (M ) or a derived sequence with base in acleq (A). Actually we find the base in dcleq (M ), dcleq (A) or acleq (B). Proof. Let b be the base of s = (ei ). If no derivation has a base in dcleq (M ), Lemma 5.4 and M(iii) yield a subsequence s0 of length λ(0, mc + 1, c) + 1 which is a Morley sequence of φc (x, b) over M . Again by 5.4, applied to M 0 /A, if there is no derivation with base in dcleq (A), there is a subsequence s00 of s0 of length

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mc + 1, say e0 , . . . , emc , which is also a Morley sequence of φc (x, b) over A. Set E = {e0 , . . . , emc −1 }. Hence, b ∈ dcleq (E) and emc ^ | M, E ,

emc ^ | A, E .

b

b

Write every e ∈ E as the sum of an element of M and an element of A. Define EM to be the set of all elements in M which occur as summands, and likewise EA , and set E 0 = EM ∪ EA . Then also b ∈ dcleq (E 0 ) and, since E 0 and E are interdefinable over M and as well as over A, we have emc ^ | M, E 0 ,

emc ^ | A, E 0 ,

b

b

which implies emc ^ | M , 0

emc ^ | A. 0

B,E

B,E

Furthermore M ^ | A. 0 B,E

Write emc = m + a for m ∈ M and a ∈ A. Then emc , m, and a are pairwise independent over B, E 0 . Fix i = 1, 2. Then φci (x, bi ) is a group formula for a definable group Gi and bi is the canonical parameter of Gi . Moreover, a is a generic element of an acleq i (B, E 0 )–definable coset of Gi and bi is definable from the canonical base of p = tpi (a/ acleq i (B, E 0 )). Note that a ^ | B,E E 0 . So the canonical base of p is in A eq eq acl i (A), hence b ∈ acl (A). By symmetry b ∈ acleq (M ), and since M and A are independent over B, this yields b ∈ acleq (B). We call M ∈ Kµ rich, if for all finite B ≤ M and all finite B ≤ A ∈ Kµ there is an B ≤ A0 ≤ M , which is B–isomorphic to A. We will show in the next section (8.3) that rich structures are models of T1 ∪ T2 . Corollary 7.4. There is a unique countable rich structure K µ . All rich structures are (L1 ∪ L2 )∞,ω –equivalent. 8. The theory T µ Lemma 8.1. Let M ∈ Kµ , b ∈ dcleq (M ), c a prealgebraic code and M 0 a prealgebraic minimal extension of M , generated by an M –generic realization a of φc (x, b) as in 5.2. If M 0 does not belong to Kµ , one of the following is true. (a) M 0 contains a difference sequence (ei ) for c whose elements but one lie in M . (b) M 0 contains a difference sequence for a prealgebraic code c0 with base b0 which contains a Morley sequence of φc0 (x, b0 ) over M of length µ∗ (c0 ) + 1.

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Proof. If M 0 6∈ Kµ there is a difference sequence (e0i ) in M 0 for a prealgebraic code c0 with base b0 . If case (b) does not occur, by M(iv) and Lemma 5.4 we may assume that b0 ∈ dcleq (M ) and furthermore that (e0i ) is as in Lemma 6.4. So nc0 = nc = dim(M 0 /M ) and we have He0µ(c0 ) + m = a for some H ∈ Glnc (F ) and m ∈ M . By C(vi) there is a d ∈ dcleq (M ) with φci (x + m, bi ) ∼kci φci (x, di ) (i = 1, 2). Then He0µ(c0 ) is an M –generic realization of φc (x, d), i.e. e0µ(c0 ) is an M –generic realization of φcH (x, d). By C(ix) there is a prealgebraic code c00 which is equivalent to cH . We have φcH (x, d) ≡ φc00 (x, b00 ) for some b00 ∈ dcleq (M ). By C(viii) and C(iv) we conclude c00 = c0 and b00 = b0 . Finally note that (e0i ) is a difference sequence for cH . So (ei ) = (He0i ) 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 αc . 2. Let c be a prealgebraic code, M ∈ Kµ a model of T1 ∪ T2 . That no extension of M in Kµ is generated by a generic realization of some φc (x, b) with b ∈ dcleq (M ) can be expressed by an sentence βc . 3. Let M ∈ Kµ be a model of T1 ∪T2 . That M has no prealgebraic minimal extension in Kµ can be expressed by a set of sentences. Proof. 1. Let αc = ¬∃x0 , . . . , xµ(c) Ψc1 (x0 , . . . , xµ(c) ) ∧ Ψc2 (x0 , . . . , xµ(c) ) . 2. Fix i = 1, 2 and let M be a submodel of Ci . Let m ∈ M , φ(x, m) an Li –formula of Morley rank k and degree 1, and a ∈ Ci 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) = MRx φ(x, y) ∧ ψ(x, y) = k This shows that, if M ∈ K and a is an M –generic realization of φc (x, b), then any L1 ∪ L2 –sentence α about hM, ai can be translated into an L1 ∪ L2 –sentence αc (b) about M . Now there is only a finite set Cc of codes c0 which can occur in (b) of 8.1 since ∗ 0 (µ (c ) + 1)nc0 ≤ dim(M 0 /M ) = nc . So set βc = ∀yc αcc (yc ) ∧

^

∀yc0 αcc0 (yc0 ).

c0 ∈Cc

The variables yc , yc0 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 ∈ Kµ (ii) M is a model of T1 ∪ T2 (iii) No prealgebraic minimal extension of M belongs to Kµ .

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 ∈ Kµ of B. We want to find a copy B ≤ A0 ≤ M of A/B. case (I): A/B is algebraic. Since M is a model of T1 ∪ T2 , it has no proper algebraic extension in K. So A0 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 a0 ∈ M which is transcendental over B such that hB, a0 i ≤ M . Since this equivalent to realize a partial type, and since M is ω– saturated, it suffices to find a0 in an elementary extension M 0 of M . Choose M ’ uncountable. By 6.3 cld (B) ≤ M 0 is countable. For every a0 ∈ M 0 \ cld (B), we have δ(a0 /B) = 1 and hB, a0 i ≤ M 0 . 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 Kµ of arbitrary finite dimension. So M is infinite and we need only show that M is algebraically closed in the sense of T1 and of T2 . Let a be an element in acl1 (M ) and transcendental over M in the sense of T2 . Therefore, a is 1–algebraic over a finite subset B of M . We may assume that B ≤ M . Since (by Lemma 7.2) B ≤ hB, ai ∈ Kµ , there is a copy of a over B in M . This implies that M acl1 –closed. Likewise M is algebraically closed in the sense of T2 . Axiom (iii): Let M 0 be a prealgebraic minimal extension generated by an M –generic realization a of φc (x, b). Assume M 0 ∈ Kµ . Choose a finite subspace C0 ≤ M with b ∈ dcleq (C0 ). Then C0 ≤ hC0 , ai. Since M is rich, M contains a copy e0 of a over C0 with C1 = hC0 , e0 i ≤ M . Continuing this way we obtain an infinite Morley sequence

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e0 , e1 , . . . of φc (x, b). By P(i), e1 − e0 , . . . , eµ(c)+1 − e0 is a difference sequence for c. Choose an ω–saturated M 0 ≡ M . By the above we know that M 0 is rich. Since M 0 ≡∞,ω M , this implies that M is ω–saturated. 9. Proof of the Theorem In this section quantifier elimination for T1 and T2 will no longer be required. Hence, replace in the class K embeddings by elementary maps in the sense of T1 and in the sense of T2 , which we call bi-elementary maps. Corollary 9.1. T µ is complete. Two tuples a and a0 in two models M and M 0 have the same type iff there is bi-elementary bijection f : cl(a) → cl(a0 ) which maps a to a0 . 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 0 ≡ M . So M 0 ≡∞,ω K µ , which proves completeness. To prove the second statement choose ω–saturated elementary extensions M ≺ N and M 0 ≺ N 0 . It is easy to seeq that M ≤ N and M 0 ≤ N 0 , so “cl” does not increase. Since M 0 and N 0 are rich, f is even ∞, ω–elementary. For the converse suppose that a and a0 have the same type. There is a bi-elementary map f : cl(a) → M 0 which maps a onto a0 . We write A0 for f (cl(a)). Then d(a) = δ(cl(a)) = δ(A0 ). It follows d(a0 ) ≤ d(a) and d(a0 ) = d(a) by symmetry. A0 has, like cl(a), no proper subset A00 which contains a0 and with δ(A00 ) = d(a0 ). This implies A0 = cl(a0 ). 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 q If M 6≤ 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 L1 ∪ L2 –formula φ(a, ¯b). However, φ(x, ¯b) is not satisfiable in M , whence M 6≺ N . r This is the type of elements a which are transcendental over cl(B) and for which hcl(B), ai 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 T1 and T2 . 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 = hK, ai 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 T1 (and T2 ), whence by Axiom (ii) we have a ∈ M . Contradiction. Corollary 9.3. If T1 and T2 are model-complete, then T µ is also model-complete. We now prove the last remark of the introduction. Let T1 and T2 be both the theory of algebraically closed fields of characteristic p formulated in L1 = {+, } and L2 = {+, ⊗}. Let T µ be a fusion over T0 , the theory of Fp –vector spaces. Let x be transcendental (in the sense of T µ ), xi the i–th power in the sense of T1 and X = {xi | i ∈ N}. Let S be any subset of X. Then dim(S) = |S| and tr1 (S) ≤ 1. It follows from Theorem 1.2, 1. that tr2 (S) ≥ |S| − 1. We claim that tr2 (S) = |S|, which is clear for S = {x0 }. Assume the contrary. Then, for some n > 0, we have tr2 (x1 . . . , xn /x0 ) < n. But xn+1 is also transcendental, therefore it has the same type as x. So tr2 (xn+1 , . . . , x(n+1)n /x0 ) < n. It follows tr2 (x1 , . . . , xn , xn+1 , . . . , x(n+1)n /x0 ) < 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. L¨ owe, and T. Schatz, editors, Festschrift f¨ ur Ulrich Felgner zum 65. Geburtstag, volume 4 of Studies in Logic, pages 15–31. College Publications, London, 2007. [4] Bruno Poizat. Le carr´e de l’egalit´e. 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. [8] Bruno Poizat. L’´egalit´e au cube. J. Symbolic Logic, 66:1647–1676, 2001. [9] Martin Ziegler. A note on generic types. (http://arxiv.org/math.LO/0608433), 2006.

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