Fusion of structures of finite Morleyrank^{∗}

Martin Ziegler

May 18, 2007

Abstract Let T_{1} and T_{2} be two countable complete theories in disjoint languages, of finite Morley rank, the same Morley degree, with definable Morley rank and degree. Let N be a common multiple of the ranks of T_{1} and T_{2}. We show that T_{1} ∪ T_{2} has a nice complete expansion of Morley rank N.

1 Introduction

We call a countable complete L–theory T good if it has finite definable rank^{1} n > 0 and definable degree^{2}. A conservative expansion T′ of T is a complete expansion of T, whose rank n′ is a multiple of n, such that for all L–formulas ϕ(x, b).

MR_{T′} ϕ(x, b) = \frac{n′}{n} MR_{T} ϕ(x, b)
MD_{T′} ϕ(x, b) = MD_{T} ϕ(x, b).

We call the quotient \frac{n′}{n} the index of the expansion.

In this note we will prove the following theorem.

Theorem 1.1. Let T_{1} and T_{2} be two good theories in disjoint languages of the same degree e and let N be a common multiple of their ranks. Then T_{1} and T_{2} have a common good conservative expansion T of rank N.

Furthermore, if in T_{i} the predicates P_{i}^{1}, …, P_{i}^{e} define a partition of the universe into sets of degree 1, T can be chosen to imply P_{1}^{j} = P_{2}^{j} for j = 1, …, e.

If both, T_{1} and T_{2}, have rank and degree 1, this is Hrushovski’s fusion [5], except that we allow the language of T to be larger than the union of the languages of T_{1} and T_{2}. The core of our proof is an adaption of the exposition of Hrushovski’s fusion given in [3] and (in Section 2.2) of ideas from Poizat’s [6].

As an immediate application we get an explanation of the title of Poizat’s [6]:

Corollary 1.2 ([6],[1]). In any characteristic there is an algebraically closed field K with a subset N such that (K, N) has rank 2.

∗finiterank.tex,v 2.5, May 18, 2007

1By “rank” we always mean “Morley rank”, “degree” is “Morley degree”.

2I.e. the DMP, the definable multiplicity property.

Proof. Apply 1.1 for T_{1} the theory of algebraically closed fields of some fixed characteristic and for T_{2} any good theory of rank 2 and degree 1, e.g. the “square of the identity”.

For another account of 1.2 see [2].

Theorem 1.1 was motivated by the following surprising result of A. Hasson:

Corollary 1.3 ([4]). Every good theory can be interpreted in a good strongly minimal set.

Proof. Let T_{1} be a good theory of rank n and degree e. Consider any good theory T_{2} of rank n and degree e which can be interpreted in a strongly minimal set X defined in T_{2}. Use 1.1 to obtain a good theory T of rank n which conservatively expands T_{1} and T_{2}. T_{2} is then interpreted in X, which is still strongly minimal in T.

The simplest example of a theory T_{2} as used in the above proof is the “disjoint union of e–copies of the n–th power of the identity”: Let X be an infinite set, Y_{1},…,Y_{e} be disjoint of copies of X^{n} and Δ the diagonal of Y_{1}. Then consider the structure

(M, Y_{1}, …, Y_{e}, Δ, f_{1}, …, f_{e})

where M is the disjoint union of the Y_{j} and f_{j} is the canonical bijection between Δ^{n} and Y_{j}.

The above proof shows that every good theory of rank n and degree e with a partition P_{1} ∪ ⋯ ∪ P_{e} into definable sets of degree 1 has a good conservative expansion of index 1 which contains a strongly minimal set X such that each P_{j} is in definable bijection with X^{n}. This yields

Corollary 1.4. Let T be a good theory and X and Y be two sets of maximal rank and the same degree. Then T has a good conservative expansion of index 1 with a definable bijection between X and Y.

Let T be a good theory of rank N with a definable bijection between the universe and the N–th power of a strongly minimal set X. Then the rank of every good expansion of T is a multiple of N. This shows that in Theorem 1.1 one has to assume that N is a common multiple of the ranks of T_{1} and T_{2}, even if one is not interested in the conservativeness of the expansions. A contrasting example is the case where the languages of the T_{i} have only unary predicates. Then the rank of a completion of T_{1} ∪ T_{2} is bounded by MR(T_{1}) + MR(T_{2}) − 1. So, in 1.1, one has in general to increase the language to find an expansion whose rank is a common multiple of the ranks of T_{1} and T_{2}.

I don’t know if the last corollary remains true, if one assumes only that X and Y have the same rank (and degree). The following theorem can be used to prove a weaker result.

Theorem 1.5. Let T be a two-sorted theory with sorts Σ_{1} and Σ_{2}. Let T_{1} = T ↾ Σ_{1} be the theory of the full induced structure on Σ_{1} and T_{1}^{∗} a conservative expansion of T_{1} of index 1. Assume that T and T_{1}^{∗} have definable finite rank. Then T^{∗} = T_{1}^{∗} ∪ T is a conservative expansion of T of index 1 which has again definable rank.

There are examples where T and T_{1}^{∗} have the DMP, but T^{∗} has not.

Corollary 1.6. Let T be a good theory and X and Y be two sets of the same rank and the same degree. Then T has a conservative expansion of T^{∗} of index 1 with a definable bijection between X and Y. T^{∗} has definable rank.

Proof. Let T′ be the following (good) theory with sorts Σ_{1} and Σ_{2}: Σ_{2} is a model of T; Σ_{1} is a disjoint union of two predicates X′ and Y′; there are bijections between X and X′ and between Y and Y′. In T′_{1} = T′ ↾ Σ_{1}, X′ and Y′ have maximal rank and same degree. By 1.4 T′_{1} has a good conservative expansion T_{1}^{′∗} of index 1 with a definable bijection between X′ and Y′. T^{∗} = (T′∪T_{1}^{′∗}) ↾ Σ_{2} has the required properties.

In [4, Theorem 18] it is proved that for any m and n, any two good strongly minimal sets can be glued together to form a two–sorted structure, where both sets have rank one and there is a definable m-to-n function between them. By Remark 3 of [4] the proof “generalizes to finite-rank”. A. Hasson has told me that the generalized proof shows that the union of two good theories of finite rank has a completion of finite rank. Since here the theories may have different degree, the expansions are in general not conservative.

2 Proof of Theorem 1.1

Theorem 1.1 follows from the next theorem, which we will prove in this section.

Theorem 2.1. Let T_{1} and T_{2} be to good theories in disjoint languages L_{1} and L_{2} with ranks N_{1} ≤ N_{2} and of degree e, and N be the least common multiple of N_{1} and N_{2}. In T_{i} let the predicates P_{i}^{1}, …, P_{i}^{e} define a partition of the universe into sets of degree 1. Assume also that T_{1} satisfies

(*) If N_{1} divides N_{2} = N, then each non-algebraic element is interalgebraic with infinitely many other elements. Otherwise, the universe is is a union of infinite ∅–definable ℚ–vector spaces V_{0}, …, V_{l}.

Then T_{1} ∪ T_{2} has a completion T of rank N which implies P_{1}^{j} = P_{2}^{j} and is a good conservative expansion of T_{1} and T_{2}.

Proof of 1.1. Denote the construction in 2.1 by T_{1} + T_{2}. Let now T_{1} and T_{2} be as in 1.1. By adding constants we may assume that the predicates P_{i}^{j} are present. Let T_{0} be the theory of the disjoint union of e infinite ℚ–vector spaces. T_{0} has rank 1 and degree e. Let N′ be the least common multiple of the ranks of T_{1} and T_{2}. Then

T′ = (T_{0} + T_{1}) + T_{2}

is a good conservative expansion of T_{1} ∪ T_{2} of rank N′. Finally set T = T′ + T_{3} for any good theory T_{3} of rank N and degree e.

Actually we need the proposition only in the case that N_{1} divides N_{2}. We have stated it in stronger form, since the proof can be given by a direct application of Hrushovski’s fusion machinery to T_{1} and T_{2}.

It is easy to see that, by naming parameters^{3}, we may assume the following.

(**) If N_{1} = N_{2}, for each j, the theory T_{2} has infinitely many 1–types over ∅ of rank N_{2} − 1 which contain P_{2}^{j}(x).

2.1 Hrushovki’s machinery

In this section we will develop the theory without using the assumptions (*) and (**). This is a straightforward^{4} generalization of sections 2–6 of [3]. We will omit most of the proofs.

2.1.1 Codes (see [3], Section 2)

Let T be a good theory of degree e with predicates P^{1}, …, P^{e} which define a partition of the universe in sets of degree 1. We call a formula χ(x, b) simple, if

• it has degree 1,

• the components of a generic realization are pairwise different and not algebraic over b.

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.

(i) ϕ_{c}(x, b) is either empty^{5} or simple. Furthermore there are indices e_{c,i} such that ϕ_{c}(x, y) implies that the x_{i} are pairwise different and P^{e_{c,1}}(x_{1})∧⋯∧ P^{e_{c,nc}} (x_{n_{c}}).

(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)

MR(a/ba_{s}) ≤ k_{c,s},

and equality holds for generic a.^{6}

(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′)^{7}, then b = b′.

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).

3We can forget the new constants after the construction of T. So, the language is not increased.

4 For the convenience of the reader many definition and statements are copied verbatim from [3].

5We assume that ϕ_{c}(x, b) is non-empty for some b.

6a_{s} = {a_{i}|i ∈ s}

7This means that the Morley rank of the symmetric difference of ϕ_{c}(x, b) and ϕ_{c}(x, b′) is smaller than k_{c}.

Proof. As the proof of [3, 2.2]. Note is that, by definability of rank, the rank is additive

MR(ab/B) = MR(a/Bb) + MR(b/B).

(see e.g. [7, 4.4]).

Let c be a code, ϕ_{c}(x, b) non-empty and p ∈ S(b) the (stationary) type of rank k_{c} determined by ϕ_{c}(x, b). (iv) implies that b is the canonical base of p. Hence, b lies in the definable closure of a sufficiently large segment of a Morley sequence of p (which we call a Morley sequence of ϕ_{c}(x, b).) Let m_{c} be some upper bound for the length of such a segment. Note that one can always bound m_{c} by the rank of the sort of y in ϕ_{c}(x, y). 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 of the e_{i}’s. We say for Ψ_{c}(x_{0}, …, x_{μ}, y) that “x_{0}, …, x_{μ} is a pseudo Morley sequence of

c over y”. Proof. As the proof of [3, 2.3].

We choose for every code (and every μ) 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 call two codes c and c′ 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, c^{σ} is equivalent to a code in C. Proof. As the proof of [3, 2.4]. Note that we may have to change the Ψ_{c}.

2.1.2 The δ–function (see [3], Section 3)

Let T_{1} and T_{2} be two good theories as in Theorem 1.1. We assume that the T_{i} has quantifier elimination in the relational language L_{i}. To deal with the predicates P_{i}^{j} in an effective way we replace both P_{1}^{j} and P_{2}^{j} by P^{j}. Then L_{1} and L_{2} intersect in L_{0} = {P_{1}, …, P_{e}} and T_{1} and T_{2} intersect in the theory of a partition of the universe into e infinite sets.

Define K to be the class of all models of T_{1,∀} ∪ T_{2,∀}. We allow also ∅ to be in K.

Let N_{i} be rank of T_{i}, N = lcm(N_{1}, N_{2}) and N = ν_{1}N_{1} = ν_{2}N_{2}. We define for finite A ∈ K

(2.1) δ(A) = ν_{1} MR_{1}(A) + ν_{2} MR_{2}(A) − N · |A|.

By additivity of rank δ has the following properties.

(2.2) δ(∅) = 0
(2.3) δ({a}) ≤ N for single elements a
(2.4) δ(A ∪ B) + δ(A ∩ B) ≤ δ(A) + δ(B)

(2.3) is a special case of

(2.5) δ(a/B) ≤ ν_{i} MR_{i}(a/B), (i = 1, 2),

which holds for arbitrary tuples a.

If A \ B is finite, we set

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

For finite B, it follows that δ(A/B) = δ(A ∪ B) − δ(B).

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

B ≤ A.

B ≨ A is minimal if B ≤ A′ ≤ A for no A′ properly contained between B and A. a is algebraic over B, if a/B is algebraic in the sense of T_{1} or T_{2}. A/B is non-algebraic if no a ∈ A \ B is algebraic over B.

Lemma 2.5. B ≤ A is minimal iff δ(A/A′) < 0 for all A’ which lie properly between B and A.

Proof. As the proof of [3, 3.1].

Lemma [3, 3.2] is not longer true, instead we have

Lemma 2.6. Let B ≤ A be a minimal extension. There are three cases

(I) δ(A/B) = 0, A = B ∪ {a} for an element a ∈ A \ B, which is algebraic over B. ( algebraic simple extension)

(II) δ(A/B) = 0, A/B is non-algebraic. ( prealgebraic extension)

(III) A/B is non-algebraic and 1 ≤ δ(A/B) ≤ N, ( transcendental extension). If δ(A/B) = N, we have A = B ∪{a} for an element a with MR_{i}(a/B) = N_{i} for i = 1, 2. ( transcendental simple extension^{8})

Proof. Assume first that A/B is algebraic. That means that some element a ∈ A \ B is algebraic over B. This implies δ(a/B) = 0 and B ∪ {a} ≤ A. So we are in case (I).

Now assume that A/B is transcendental and δ(A/B) ≥ N. Since δ(a/B) ≤ N for all elements a ∈ A \ B, Lemma 2.5 implies B ∪ {a} = A.

Note that, unlike the situation in [3], there may be prealgebraic extensions A/B by single elements if N_{1} and N_{2} are not relatively prime. We do not call these extensions “simple”.

Remark. If N_{1} and N_{2} are relatively prime, each strong extension by a single element is simple.

Proof. Let A = B∪{a} be a strong extension of B. If δ(A/B) > 0, the extension is transcendental simple. Otherwise

ν_{1} MR_{1}(a/A) + ν_{2} MR_{2}(a/A) = N_{2} MR_{1}(a/A) + N_{1} MR_{2}(a/A) = N.

It follows that MR_{1}(a/A) is divisible by N_{1} and MR_{2}(a/A) is divisible by N_{2}. Whence either MR_{1}(a/A) or MR_{2}(a/A) must be zero. So A/B is algebraic simple.

We will work in the class

K^{0} = {M ∈ K|∅ ≤ M}.

Fix an element M of K^{0}. We define for finite subsets of M.

d(A) = min_{A⊂A′⊂M} δ(A′).

d satisfies (2.2), (2.3), (2.4) and

(2.6) d(A) ≥ 0
(2.7) A ⊂ B ⇒ d(A) ≤ d(B)

We define

d(A/B) = d(AB) − d(B) = δ(cl(AB)/ cl(B)),

where cl(A), the closure of A, is the smallest strong subset of M which extends A. Note that the closure of a finite set is again finite (cf. [3, 3.4]).

8A transcendental simple extension is a transcendental extension by a single element. Note that simple extensions are not related to simple formulas.

2.1.3 Prealgebraic codes (see [3], Section 4)

For each T_{i} fix a set C_{i} of codes as in 2.4. We may assume that all ϕ_{c} and Ψ_{c} are quantifier free.

A prealgebraic code is a pair c ∈ C_{1} × C_{2} such that

• n_{c} = n_{c_{1}} = n_{c_{2}}

• e_{c_{1},j} = e_{c_{2},j} for all j ∈ {1, …, n_{c}}.

• ν_{1}k_{c_{1}} + ν_{2}k_{c_{2}} − N · n_{c} = 0

• ν_{1}k_{c_{1},s} + ν_{2}k_{c_{2},s} − N(n_{c} − |s|) < 0 for all non-empty proper subsets s of {1, …, n_{c}}.

Set m_{c} = max(m_{c_{1}}, m_{c_{2}}) and for each permutation σ c^{σ} = (c_{1}^{σ}, c_{2}^{σ}). c^{σ} is again 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}.

The following three lemmas are proved as Lemmas 4.1, 4.2 and 4.3 in [3].

Lemma 2.7. 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).

Lemma 2.8. 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 2.9. 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.

The next Lemma is the analogue of [3, 4.4].

Lemma 2.10. Let M ≤ N an extension in K and e_{0}, …, e_{μ} ∈ N a pseudo Morley sequence of c over b. Then one of the following holds:

• b ∈ dcl^{eq}(M)

• more than μ − m_{c} · (N(n_{c} − 1) + 1) many of the e_{i} lie in N \ M.

Proof. If b is not in dcl^{eq}(M), less than m_{c} many of the e_{i} lie in M. Let r be the number of elements not in N \ M. We change the indexing so that e_{i} ∈ N \ M implies i ≥ r and e_{i} ∈ M implies i < (m_{c} − 1). By Lemma 2.9 we have δ(e_{i}/Me_{0}, …, e_{i−1}) < 0 for all i ∈ [m_{c}, r). This implies, for m = min(m_{c}, r),

0 ≤ δ(e_{0}, …, e_{r−1}/M) ≤ δ(e_{0}, …, e_{m−1}/M) − (r − m_{c}).

On the other hand we have δ(e_{0}, …, e_{m−1}/M) ≤ N · m · (n_{c} − 1), which implies

r ≤ N · m · (n_{c} − 1) + m_{c} ≤ N · m_{c} · (n_{c} − 1) + m_{c}.

2.1.4 The class K^{μ} (see [3], Section 5)

Choose a function μ^{∗} from prealgebraic codes to natural numbers similar to section 5 of [3]. μ^{∗} must satisfy μ^{∗}(c) ≥ m_{c} − 1 and be finite-to-one for every fixed n_{c}. Also we must have μ^{∗}(c) = μ^{∗}(d), if c is equivalent to a permutation of d. Then set

μ(c) = m_{c} · (N(n_{c} − 1) + 1) + μ^{∗}(c).

From now on, a pseudo Morley sequence denotes a pseudo Morley sequence of length μ(c) + 1 for a prealgebraic code c.

The class K^{μ} consists of the all structures in K^{0} which do not contain any pseudo Morley sequence.

The following lemma and its corollary have the same proofs as their analogues [3, 5.1] and [3, 5.2].

Lemma 2.11. 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.

Corollary 2.12. 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.

Lemma [3, 5.4] may be wrong here. We have instead:

Lemma 2.13. Let M ∈ K^{μ} and N a simple extension of M. Then N ∈ K^{μ}.

Proof. Let (e_{i}) ∈ N a pseudo Morley sequence of c over b. At least μ(c) of the e_{i} lie in M. Since μ(c) ≥ m_{c}, we have b ∈ dcl^{eq}(M). Since M belongs to K^{μ}, one e_{i} does not lie in M. By 2.9 we conclude that e_{i} is disjoint from M and a generic realization of ϕ_{c}(x, b). So n_{c} = 1 and N/M is prealgebraic, i.e. not simple.

Proposition 2.14. K^{μ} has the amalgamation property with respect to strong embeddings.

Proof. The proof is the same as the proof of [3, 5.5], the main ingredient being Lemma 2.10. Only one point has to be checked: If A/B is strong and a ∈ A is algebraic over b, say in the sense of T_{1}, then tp_{2}(a/B) is uniquely determined. This is the case, since 0 ≤ δ(a/B) = ν_{2} MR_{2}(a/B) − N ≤ ν_{2}N_{2} − N = 0 implies that MR_{2}(a/B) = N_{2}. On the other hand, tp_{1}(a/B) implies a ∈ P^{j} for some j. So the T_{2}–type of a/B is uniquely determined since P^{j} has degree 1 in T_{2}.

The proof has the following corollary.

Corollary 2.15. Two strong extensions B ≤ M and B ≤ A in K^{μ} can be amalgamated in M, A ≤ M′ ∈ K^{μ} such that δ(M′/M) = δ(A/B) and δ(M′/A) = δ(M/B).

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

2.1.5 The theory T^{μ} (see [3], Section 6)

Lemma 2.17. 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.

Proof. As in the proof of [3, 6.1], this follows from 2.9 and 2.10.

As in [3], Lemmas 2.7, 2.8 and 2.17 imply that we can describe all M with the following properties by an elementary theory T^{μ}.

Axioms of T^{μ}.

(a) M ∈ K^{μ}

(b) T_{1} ∪ T_{2}

(c) M has no prealgebraic minimal extension in K^{μ}.

To prove the analogue of Theorem [3, 6.3], which says that the rich structures are the ω–saturated models of T^{μ} we need the assumptions (*) and (**). Whithout this we can only show^{9}

Lemma 2.18. Rich structures are models of T^{μ}.

Proof. Let K be rich. Consider an quantifier free L_{1}–formula χ(x) with parameters in K which is T_{1}–consistent. Let B be a finite strong subset of K which contains the parameters. If χ(x) is not realized in B, realize χ(x) by a new element a and define the structure A = B ∪ {a} in such a way that MR_{2}(a/B) = N_{2}. Then δ(a/B) = ν_{1} MR_{1}(a/B), so B ≤ A and A/B is simple. So by 2.13 B belong to K^{μ}. Since K is rich, it contains a copy of A/B. This proves that χ(x) is realized in K. This shows that K is model of T_{1}. The same proof shows that K is also a model of T_{2}.

Axiom (c) is proved like in the proof of [3, 6.3].

9 It is conceivable that T^{μ} might be incomplete. We even do not know wether T^{μ} has an ω–stable completion. (This question was raised by the referee.)

2.2 Poizat’s argument

We assume now conditions (*) and (**) of Theorem 2.1. We want to show that ω–saturated models of T^{μ} are rich. We start with two lemmas.

Lemma 2.19. T_{1} has the following property. Let M_{1} > 0 and M_{2} be two natural numbers, a an element of an ∅–definable ℚ–vector space V_{α}. Let B be a set of parameters such that V_{α} contains elements which are of rank 1 over B. Then there are elements c_{1}, …, c_{M_{2}} of V_{α} such that for all s ⊂ {1, …, M_{2}}

(2.8)  min(M_{1}, |s|) ≤ MR_{1}(c_{s}/Ba) ≤ M_{1}

and, if |s| > M_{1}

(2.9) MR_{1}(c_{s}/B) = MR_{1}(c_{s}/Ba) + MR_{1}(a/B).

Proof. We start with a sequence v_{1}, …, v_{M_{2}} of elements of ℚ^{M_{1}} such that

• any M_{1} elements of the sequence are ℚ–linearly independent,

• any M_{1} + 1 elements of the sequence are linearly dependent, but affinely independent.

Then we choose any B–independent sequence ē = (e_{1}, …, e_{M_{1}}) of elements of V_{α} which have rank 1 over B, such that ē is independent from a over B We consider ē as a column vector and the v_{i} as a row vectors and define

c_{i} = v_{i} · ē + a.

Since all c_{i} are algebraic over Baē, it is clear that

MR_{1}(c_{s}/Ba) ≤ MR_{1}(ē/Ba) = M_{1}.

To show min(M_{1}, |s|) ≤ MR_{1}(c_{s}/Ba), we may assume that |s| ≤ M_{1}. Since the v_{i}, i ∈ s are linearly independent there is a subsequence ē′ of ē of length M_{1} − |s| such that the elements of ē′ and v_{s} · ē span the same ℚ–vector space as the elements of ē. So we have

M_{1} = MR_{1}(ē/Ba) = MR_{1}(ē′, v_{s} · ē/Ba) ≤ (M_{1} − |s|) + MR_{1}(v_{s} · ē/Ba)

and hence

|s| ≤ MR_{1}(v_{s} · ē/Ba) = MR_{1}(c_{s}/Ba).

The last equation follows from the fact that each M_{1} + 1 many of the e_{i} span an affine subspace which contains a. The reason for this is that the according v_{i} are linearly dependent, but affinely independent, and therefore span an affine space which contains 0.

Lemma 2.20. If N_{1} = N_{2}, T_{2} has the following property. Let B be any set of parameters, and p be the type over B of an M_{2}–tuple of independent elements of rank N_{2} over B. Then p is the limit of types of tuples of independent elements of rank N_{2} − 1 over B.

Proof. We indicate the proof for M_{2} = 2. Let p = tp(a_{1}a_{2}/B) and ϕ(x_{1}, x_{2}) ∈ p. The formula ϕ_{1}(x_{1}) = “ MR_{x_{2}} ϕ(x_{1}, x_{2}) ≥ N′′_{2} has rank N_{2}. Therefore, by (**), there is a type q_{1} over B which has rank N_{2} − 1 and contains ϕ_{1}(x_{1}). Let b_{1} be a realization of q_{1}. By the open mapping theorem, and (**) again, ϕ(b_{1}, x_{2}) contains a type q_{2} over Bb_{1}, of rank N_{2} − 1 which does not fork over B. Realize q_{2} by b_{2}. The type of b_{1}b_{2} over B contains ϕ, b_{1} and b_{2} are independent and of rank N_{2} − 1 over B.

Proposition 2.21. The rich structures are exactly the ω–saturated models of T^{μ}.

Proof. That rich structure are models of T^{μ} was proved in 2.18. As in the proof of [3, 6.3] one sees that it suffices to prove that ω–saturated models of T^{μ} are rich. So let K be an ω–saturated model, B ≤ K finite and B ≤ A a minimal extension which belongs to K^{μ}. We show that A/B can be strongly embedded in K by induction over d = δ(A/B).

If d = 0 the extension is algebraic or prealgebraic and the claim follows from 2.14, since K has no algebraic or prealgebraic extensions. So we assume d > 0. All we use from the minimality of A/B in this case is that A ≠ B and δ(X/B) > 0 for all subsets of A, which are not contained in B.

We may assume that B is large enough to have, for each j, parameters for an L_{2}–formula in P^{j} which has rank N_{2} − 1 in T_{2}. Choose two numbers M_{1} and M_{2} such that

ν_{1}M_{1} − ν_{2}M_{2} = −1.

The M_{i} are uniquely determined if we impose the condition 0 ≤ M_{1} < ν_{2}. We have then

M_{1} = \frac{ν_{2}M_{2} − 1}{ν_{1}} < M_{2},

since ν_{2} ≤ ν_{1}.

Let a be an arbitrary element of A \ B. Since δ(a/B) > 0, a is not algebraic over B.

If N_{1} divides N_{2}, i.e. if ν_{2} = M_{2} = 1 and M_{1} = 0, we choose an element c_{1} ∉ A, which is in the sense of T_{1} interalgebraic with a and has rank N_{2} over A in the sense of T_{2}. We set C = A ∪ {c_{1}}. If N_{1} does not divide N_{2}, we have M_{1} > 0. We define then C = A∪{c_{1}, …, c_{M_{2}}} where the c_{i} are given by Lemma 2.19 and are – in the sense of T_{1} – independent from A over Ba. In the sense of T_{2} they are chosen to be A–independent and of rank N_{2} − 1 over A.

We compute

δ(C/A) = ν_{1}M_{1} + ν_{2}M_{2}(N_{2} − 1) − NM_{2} = ν_{1}M_{1} − ν_{2}M_{2} = −1.

Claim 1: B ≤ C. Proof: Let X be a set between B and A and Y be a subset of {c_{1}, …, c_{M_{2}}} of size y. Note that δ(XY/B) ≥ δ(Y/A) + δ(X/B) and by equation (2.8) we have

δ(Y/A) ≥ ν_{1} min(M_{1}, y) + ν_{2}y(N_{2} − 1) − Ny = ν_{1} min(M_{1}, y) − ν_{2}y.

Case 1: y ≤ M_{1}. Then δ(XY/B) ≥ δ(Y/A) ≥ (ν_{1} − ν_{2})y ≥ 0.

Case 2: M_{1} < y. Then we have δ(Y/A) = ν_{1}M_{1} − ν_{2}y ≥ ν_{1}M_{1} − ν_{2}M_{2} = −1 and distinguish two cases: If X = B, then, by (2.9), MR_{1}(Y/B) > MR_{1}(Y/A) and therefore δ(XY/B) = δ(Y/B) > δ(Y/A) ≥ −1. If X is different from B we have δ(XY/B) ≥ −1 + δ(X/B) ≥ 0. This proves the claim.

Claim 2: The closure of A in C equals C. Proof: Let Y be a proper subset of {c_{1}, …, c_{M_{2}}} of size y. We have to show that δ(Y/A) > −1. By the above this is clear if y ≤ M_{1}. Otherwise we have

δ(Y/A) = ν_{1}M_{1} − ν_{2}y > ν_{1}M_{1} − ν_{2}M_{2} = −1.

This proves the claim.

It follows (if N_{1} does not divide N_{2}, from the proof of Lemma 2.19) that one can produce a sequence of extensions A ⊂ C_{i} like above such that the types tp_{1}(C_{i}/A) converge against a type tp_{1}(D/A) where the elements d_{0}, …, d_{M_{2}} are of rank ≥ 1 and algebraically independent^{10} over A in the sense of T_{1}. If N_{1} < N_{2} we simply choose the types tp_{2}(C_{i}/A) and tp_{2}(D/A) to be all the same and with components of rank N_{2} − 1 independent over A in the sense of T_{2}. If N_{1} = N_{2}, it follows from Lemma 2.20 that we may assume that the types tp_{2}(C_{i}/A) converge to tp_{2}(D/A) and that the d_{i} have rank N_{2} over A and are independent over A in the sense of T_{2}.

If N_{1} < N_{2}, we have

δ(d_{i}/Ad_{0} … d_{i−1}) ≥ ν_{1} · 1 + ν_{2}(N_{2} − 1) − N = ν_{1} − ν_{2} > 0.

If N_{1} = N_{2} we have for every i

δ(d_{i}/Ad_{0} … d_{i−1}) ≥ ν_{1} · 1 + ν_{2}N_{2} − N = ν_{1} > 0.

So D is a strong extension of A which splits into a sequence of transcendental simple extensions. So, by Lemma 2.13, D belongs to K^{μ}.

Claim: For large enough i we have C_{i} ∈ K^{μ}. Proof: Since the C_{i} have all the same size, if C_{i} does not belong to K^{μ} and μ is finite-to-1 for fixed n_{c}, there is a certain finite set of prealgebraic codes which can be responsible for this. Since D ∈ K^{μ}, almost all C_{i} belong to K^{μ}.

Now by induction for large enough i, C_{i} can be strongly embedded over B into K. Since K is ω–saturated this implies that D can be strongly embedded into K. Such an embedding also strongly embeds A, since A ≤ D.

Corollary 2.22. T^{μ} is complete. In models of T^{μ} two tuples have the same type iff they have isomorphic closures.

Proof. Same as the proof of [3, 7.1].

2.3 Rank computation

Proposition 2.23. In T^{μ} we have for tuples a

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

10It suffices that d_{i} is not in acl_{1}(Ad_{0} … d_{i−1}).

Proof. We prove first MR(a/B) ≤ d(a/B). Since the closure is algebraic we may assume that B and A = B ∪ {a} are closed. Then d(a/B) = δ(a/B), so it suffices to show that MR(a/B) ≤ δ(a/B) for all closed B and arbitrary a. We do this by induction on d = δ(a/B).

Let M be an ω–saturated model, which contains B such that the (a priori infinite) rank of a over M is the same as the rank of a over B. Then δ(a/M) ≤ δ(a/B) and by induction we may assume that δ(a/M) = d. Also we may assume that a is disjoint from M. Write a = (a_{1}, …, a_{n}).

Choose for i = 1, 2 an L_{i}(M)-formula ϕ_{i}(x) ∈ tp_{i}(a/M) with the following properties.

(i) ϕ_{i} has degree 1

If a′ is any realization of ϕ(x), then

(ii) the components of a′ are pairwise different

(iii) MR_{i}(a′/Ma′_{s}) ≤ k_{i,s}, where s is any subset of {1, …, n} and k_{i,s} = MR_{i}(a/Ma_{s}).

It follows that MR_{i} ϕ_{i} = k_{i,∅} = MR_{i}(a/M).

Let a′ be any realization of ϕ(x, b) = ϕ_{1}(x, b) ∧ ϕ_{2}(x, b). The inequality MR(a/M) ≤ d follows the from ω–saturation of M and the next claim.

Claim: Either MR(a′/M) < d or tp(a′/M) = tp(a/M).

Proof: Case 1. δ(a′/M) < d. Then MR(a′/M) < d by induction.

Case 2. δ(a′/M) ≥ d. Set s = {i | a′_{i} ∈ M} consider the inequality

δ(a′/M) = ν_{1} · MR_{1}(a′/Ma′_{s}) + ν_{2} MR_{2}(a′/Ma′_{s}) − N · (n − |s|)
≤ ν_{1} · k_{1,s} + ν_{2}k_{2,s} − N · (n − |s|)
= δ(a/Ma_{s}) ≤ δ(a/M).

Our assumption implies MR_{i}(a′/Ma′_{s}) = k_{i,s} and δ(a/Ma_{s}) = δ(a/M). The latter implies δ(a_{s}/M) = 0, so a_{s}/M is algebraic in the sense of T^{μ} (2.12), which is only possible if s is empty. So we have MR_{i}(a′/M) = MR_{i}(a/M), which implies that a′ and a are isomorphic over M, and δ(a′/M) = d.

Case 2.1 M ∪{a′} is not closed. Then a′ has an extension a′′ with δ(a′′/M) < d. It follows MR(a′/M) ≤ MR(a′′/M) < d by induction.

Case 2.2 M ∪ {a′} is closed. Then tp(a′/M) = tp(a/M).

Now we prove d(a/B) ≤ MR(a/B) by induction on d = d(a/B). We may we may assume that B is finite, that B and B ∪ {a} are closed and (using 2.15) that B has, for each j, parameters for an L_{2}–formula in P^{j} which has rank N_{2} − 1 in T_{2}. If d = 0, there is nothing to show. If d > 0, we decompose A/B into B ≤ B′ ≤ A, where B′ is maximal with δ(B′/B) = 0.

Now we can use the construction in proof of 2.21 to obtain a sequence of extensions A ⊂ C_{i} and A ≤ D, such that B′ ≤ C_{i}, δ(C_{i}/A) = d − 1, all in K^{μ}, such that C_{i} is the closure of A and the qf-types of the C_{i} over A converge against the qf-type of D over A. We may assume that D is closed (in the monster model). We also choose a copies C′_{i} of C_{i} over B′ which are closed. Let A′_{i} be the corresponding copy of A in C′_{i}. Since the types of the tp(C′_{i}/B) converge against tp(D/B), the types tp(A′_{i}/B) converge against tp(A/B). Now d(A′_{i}/B) = δ(C′_{i}/B) = d − 1, so by induction d − 1 ≤ MR(C′_{i}/B), which implies d ≤ MR(A/B).

The referee has pointed out that our proof of MR(a/B) ≤ d(a/B) can be rephrased as follows: It it easy to see that d–independence defines a notion of independence. The claim in the proof of 2.23 shows that types over ω–saturated models are isolated among the types of at least the same rank. This implies the above inequality.

Lemma 2.24. Let ϕ(x) be an L_{i}–formula (with parameters). Then

MR ϕ = ν_{i} MR_{i} ϕ.

Proof. Consider i = 1, the case i = 2 works the same. Let ϕ(x) be defined over the closed set B. If a is any realization of ϕ, we have by (2.5)

MR(a/B) ≤ δ(a/B) ≤ ν_{1} MR_{1}(a/B) ≤ ν_{1} MR_{1} ϕ.

So MR ϕ ≤ ν_{1} MR_{1} ϕ. For the converse choose a generic realization a = (a_{1}, …, a_{n}) of ϕ. Choose tp_{2}(a/B) of maximal possible rank^{11}. Then clearly δ(a/B) = ν_{1} MR_{1}(a/B) = ν_{1} MR_{1} ϕ. Also, for every i, B ∪ {a_{1}, …, a_{i}} is equal to, or a simple extension of, B ∪ {a_{1}, …, a_{i−1}}. So, by 2.13, B ∪ {a} belongs to K^{μ}. We can therefore find B ∪ {a} as a closed subset of a model of T^{μ}. This implies MR(a/B) = δ(a/B) = ν_{1} MR_{1} ϕ.

Lemma 2.25. Let ϕ(x) be an L_{i}–formula (with parameters). Then

MD ϕ = MD_{i} ϕ.

Proof. Consider i = 1. Let ϕ(x) be defined over the closed set B. We may assume that ϕ is simple in the sense of T_{1}. Let a be a realization of ϕ(x) with MR(a/B) = MR ϕ. Then MR_{1}(a/B) = MR_{1} ϕ, which determines tp_{1}(a/B) uniquely, since MD_{1} ϕ = 1. In the sense of T_{2} the a_{i} are B–independent generic elements of certain P^{j}’s, so the type tp_{2}(a/B) is uniquely determined. Finally B ∪ {a} must be closed. This implies that tp(a/B) is uniquely determined and MD ϕ = 1.

2.4 Definable rank and degree

It remains to show that T^{μ} has definable rank and degree. If N_{1} does not divide N_{2} the definability of rank follows from the fact that the universe of T^{μ} is covered by a finite set of definable groups. We give a proof which works also for the case N_{1}|N_{2}.

We use the following observation, due to M. Hils. Call a formula ϕ(x, b) of rank n and degree 1 normal if b satisfies a formula θ(y) such that ϕ(x, b′) has rank n and degree 1 for all realizations b′ of θ. A type is normal if it contains a normal formula of the same rank. We have then Lemma 2.26. Let T be a complete theory of finite rank. Then

11This is N_{2} times the number of different a_{i}’s

1. T has definable rank and degree iff every type over a model M is normal.

2. If tp(a, a′/M) is normal, and a′ is algebraic over Ma, then also tp(a/M) is normal.

In 1. it suffices to consider ω–saturated models M. Also, if M is ω–saturated and b ∈ M, then ϕ(x, b) is normal iff there is a θ(y) defined over M such that ϕ(x, b′) has rank n and degree 1 for all b′ in θ(M).

Consider an ω–saturated model M of T^{μ} and a type p = tp(a/M) of rank d = d(a/M). We want to show that p is normal. By 2.26.2 we may assume that M ∪ {a} is closed, i.e. d = δ(a/M). We may also assume that a is disjoint from M and that all components of a are different. Choose for each i = 1, 2 formulas ϕ_{i}(x, m) ∈ tp_{i}(a/M) with properties (i), (ii), (iii) as in the first part of the proof of proposition 2.23. Choose a formula θ(x) over M, which is satisfied by m, such that for all m′ ∈ θ(M) the formulas ϕ(x, m′) satisfy (i), (ii), and (iii) and MR_{i} ϕ_{i}(x, m′) = k_{i,∅} for i = 1, 2. Let a′ be a generic realization of ϕ(x, m′), which has a unique qf-type over M. Then δ(a′_{s}/M) = δ(a_{s}/M) for all s ⊂ {1, …, n}, especially δ(a′/M) = d. This implies that M_{m′} = M ∪ {a′} is a strong extension of M. One sees easily, like in [3, 6.2], that we can strengthen θ to ensure that M_{m′} ∈ K^{μ}^{12}. So we can find a′ with M_{m′} closed in the universe. This implies MR(a′/M) = d.

The proof of 2.23 shows that for all realizations a′′ of ϕ(x, m′) either MR(a′′/M) < d or tp(a′′/M) = tp(a′/M). This shows that ϕ(x, m′) has rank d degree 1 and that ϕ(x, m) is normal.

This completes the proof of Theorem 2.1.

3 Proof of Theorem 1.5

We start with an easy lemma.

Lemma 3.1. Let T be a complete two–sorted theory with sorts Σ_{1} and Σ_{2}. Then the following are equivalent.

a) Σ_{1} is stably embedded.

b) Let T_{1}^{∗} be a one–sorted complete expansion of T_{1} = T ↾ Σ_{1}. Then T^{∗} = T_{1}^{∗}∪T is complete.

Proof. a)→b): Consider S = (S_{1}^{∗}, S_{2}) two saturated models S′ = (S_{1}^{′∗}, S′_{2}) of T^{∗} of the same cardinality. Since T and T_{1}^{∗} are complete, there are isomorphisms f : (S_{1}, S_{2}) → (S′_{1}, S′_{2}) and g : S_{1}^{∗} → S_{1}^{′∗}. f^{−1}g ↾ S_{1} is an automorphism of the structure induced on S_{1}. Since S_{1} is stably embedded, there is an extension of f^{−1}g ↾ S_{1} to an automorphism h of (S_{1}, S_{2}). Then fh is an isomorphism S → S′.

12The argument is as follows. Decompose the extension M ≤ M ∪{a} into a sequence of minimal extensions, where the prealgebraic extensions are given by codes c_{1}, …, c_{k}. Strengthen θ so that the extensions M ≤ M ∪ {a′} are also composed of prealgebraic extension coming from c_{1}, ⋯, c_{k}. The argument of [3, 6.2] shows now that “M ∪ {a′} ∈ K^{μ}” is an elementary property of m′.

b)→a): This is not used in this article and left to the reader.

We fix for the rest of the section T, T_{1}, T_{1}^{∗} and T^{∗} be as in 1.5. Let L, L_{1}, L_{1}^{∗} and L^{∗} = L_{1}^{∗} ∪ L be the respective languages. We may assume that T_{1} has elimination of imaginaries.^{13}

The following lemma is due to Anand Pillay. We need only that Σ_{1} is stably embedded.

Corollary 3.2. In T^{∗} every L^{∗}–formula Φ(x) is equivalent to a formula of the form

ψ^{∗}(t(x)),

where ψ^{∗}(y) is an L_{1}^{∗}–formula and t is a T –definable function with values in some power of Σ_{1}.

Proof. Let S = (S_{1}, S_{2}) be a model of T, where S_{1} is a model of T_{1} and S^{∗} be an expansion to a model of T^{∗}. Let a be a tuple from S. Since S_{1} is stably embedded and has elimination of imaginaries, every a–definable relation on S_{1} has a canonical parameter in S_{1}. B = dcl(a) ∩ S_{1} is the set of these parameters and (S_{1}, b)_{b∈B} is the structure induced by (S, a) on S_{1}.

By 3.1

Th(S^{∗}, a) = Th(S_{1}, b)_{b∈B} ∪ Th(S, a).

This means that tp^{∗}(a) is axiomatized by tp^{1}(B) ∪ tp(a), which implies the lemma.

Corollary 3.3. S_{1}^{∗} is the structure induced by S^{∗} on S_{1}.

Proof of Theorem 1.5: We prove the following claim by induction on k.

1) For every L–definable X with MR X ≤ k we have MRD^{∗} X = MRD X.

2) For all L^{∗}–formulas Φ(x, y) is “MR^{∗} Φ(x, b) = k” an L^{∗}–elementary property of b.

Case k = 0: Let Φ(x, b) be of the form ψ^{∗}(t(x)), where ψ^{∗} and t are defined from b. Consider t as a map S → S_{1}. Then ψ^{∗}(t(x)) is finite iff the L_{1}^{∗}–formula ∃x (y =^{·} t(x) ∧ ψ^{∗}(y)) and all the fibers t(x) = a for ⊧ ψ^{∗}(a) are finite. This can be elementarily expressed since finiteness can be expressed in T_{1}^{∗} and T. This proves 2). 1) is clear.

Case k + 1:

1): Assume MR X ≤ k+1. If all L^{∗}–definable subsets of X are L–definable, it is clear that MRD^{∗} X = MRD X. So assume that there is an L^{∗}–definable A ⊂ X which is not L–definable. By Corollary 3.2 there is an L–definable surjection t : X → Y ⊂ S_{1}^{n} and an L^{∗}–definable B ⊂ Y such that A = t^{−1}B. Since MR is definable in T we can partition Y into finitely many L-definable sets on each of which the ranks of the fibers t^{−1}y have constant rank. The inverse image of this partition is an L–definable partition of X. Since it is enough to prove 1) for each of the sets of the partition, we may assume that all fibers of t have the same rank f. Since A is not L–definable, Y must be infinite. So we have f = MR X − MR Y ≤ k. By induction all fibers have T^{∗}–rank f. Since, again by induction, all T^{∗}–ranks ≤ k are definable, it follows^{14} that MR^{∗} X = f + MR^{∗}(Y) = f + MR Y = MR X.

13For this we replace T_{1} by T_{1}^{eq}. Actually the sort Σ_{1} may be itself a many-sorted structure.

To prove that MD^{∗} X = MD X, we may assume that MD X = 1. We have to show that MR^{∗}(X \ A) < MR X for every L^{∗}–definable A ⊂ X of T^{∗}– rank MR X. This is clear if A is L–definable. If not, we choose Y, t and B as above. Again we may assume that all fibers have rank f. We have then MD^{∗} Y = MD Y = 1. Since f ≤ k, we have again by induction that MR^{∗} B = MR^{∗} A − f = MR X − f = MR Y = MR^{∗} Y. So MR^{∗}(X \ A) = f + MR^{∗}(Y \ B) < f + MR^{∗}(Y) = f + MR Y = MR(X).

2): Consider L^{∗}–definable sets A ⊂ S^{m}. Let N be the T –rank of S^{m}. MR^{∗} X ≥ k + 1 is ⋀–definable and ⋁–definable , since this is equivalent to “for all/some L–definable t : S^{m} → S_{1}^{n} with A = t^{−1}B for B = t(A) there is a number f ≤ N such that the T^{∗}–rank of C_{f} = {b ∈ B | MR(t^{−1}b) = f} is ≥ k + 1 − f”. In deed, if there is such a t and f, we have

MR^{∗} A ≥ f + MR^{∗} C_{f} ≥ k + 1.

If conversely MR^{∗} A ≥ k + 1 and t is such that A = t^{−1}B for B = t(A), there is a C_{f} such that MR^{∗} t^{−1}C_{f} ≥ k + 1. If MR^{∗} C_{f} ≤ k − f we would have f ≤ k and by definability of T^{∗}–ranks ≤ k we have MR^{∗} t^{−1}C_{f} = f + MR^{∗} C_{f} ≤ k. So MR^{∗} C_{f} ≥ k + 1 − f.

Finally let us state an open problem: Let T be a good theory with two sorts Σ_{1} and Σ_{2} and T′ be a conservative expansion of T ↾ Σ_{1}. Does T′ ∪T have finite Morley rank?

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