A categoricity theorem for quasi-minimal excellent classes

Boris Zilber University of Oxford, UK zilber@maths.ox.ac.uk

6 July 2003^{∗}

Introduction This work stemmed from attempts to prove a canonicity of a certain type of structures, which we called pseudo-analytic (see [Z1]). These attempts, even for a particular class of structures, fields with pseudoexponentiation, involved a substantial work in field arithmetic, and even after most of the algebraic problems were sorted out in [Z2], a problem remained, which we realized was of a pure model-theoretic nature. This is a categoricity statement for certain classes of structures which we now call quasi-minimal excellent. Initially we formulated the assumptions for the theorem as sufficient conditions applicable for some, and hopefully for all, pseudo-analytic structures, without being aware of Shelah’s theory of excellent classes. Later, after discussing this with B.Hart, R.Grossberg and O.Lessmann, we realized that the assumptions are essentially a special case of Shelah’s definition of excellent classes but formulated in the spirit of the Baldwin-Lachlan categoricity proof. There is no doubt that the proofs of two our main theorems follow Shelah’s [S], yet (as was pointed out by the referee ) their statements can not be derived directly from Shelah’s particularly because they treat classes more general than L_{ω_{1},ω}. This special form of a categoricity theorem has a much easier formulation customized for applications, and there is a rich and mathematically meaningful class of examples satisfying this special definition.

∗With last correction 06.04.2005

The proof does not use any previous theory, though the technique is essentially Shelah’s. The reader is encouraged to compare the present setting with Shelah’s original work and carry out a further analysis, especially with regards to pseudo-analytic and analytic examples. We discuss shortly the applications in the second section of the paper. Since the first version of the paper (July 2001) we have realized that a richer class of examples requires a slightly stronger version of the main theorem, this is a categoricity theorem for almost quasi-minimal excellent classes presented in the last section.

1 Quasi-minimal excellent structures

Let C be a class of structures closed under isomorphisms and satisfying the following assumptions 1-3:

Assumption 1 (pregeometry) For any H ∈ C for any X ⊆ H there is a superset X ⊆ cl(X) ⊆ H satisfying the following:

(i) cl(X) ∈ C as a substructure of H;

(ii) cl(Y) = ⋃{cl(X) : X ⊆ Y, X finite};

(iii) X → cl(X) is a monotone idempotent operator.

We are also often interested in two extra conditions:

the countable closure property: cl(X) is countable for X finite;

the exchange principle: z ∈ cl(Xy) \ cl(X) ⇒ y ∈ cl(Xz).

Definition Let H, H′ ∈ C and G ⊆ H, G ⊆ H′. Then a (partial) mapping, identical on G, φ : H → H′ is called a G-monomorphism if it preserves quantifier-free formulas over G.

Definition A non-empty subset X ⊆ H is called cl-independent over G if

cl(G) ∩ X = ∅

and for any Y_{1}, Y_{2} ⊆ X

cl(Y_{1} ∪ G) ∩ cl(Y_{2} ∪ G) = cl(Y_{1} ∩ Y_{2} ∪ G).

X is said to be just cl-independent, if it is cl-independent over ∅.

Assumption 2 (ω-homogeneity over a submodel) ^{1} Let G ⊆ H, G ⊆ H′, G ∈ C or G = ∅. Then

(i) if X and X′ are cl-independent subsets over G in some H, H′ ∈ C, correspondingly, then any bijection φ : X → X′ is a G-monomorphism;

(ii) if a partial φ : H → H′ is a G-monomorphism, Dom φ = X, with X finite, then for any y ∈ cl(X ∪ G) there is an extension φ′ of φ with Dom φ′ = X ∪ {y}.

(iii) if φ : X ∪ {y} → X′ ∪ {y′} is a monomorphism, then

y ∈ cl(X) iff y′ ∈ cl(X′).

Definition Given X, C ⊆ H we say that the type of X over C is defined over C_{0} if any φ : X → H′ which is a C_{0}-monomorphism is also a C-monomorphism.

Definition A subset C ⊆ H will be called special if there is a cl-independent A ⊆ H and A_{1}, … A_{k} ⊆ A such that

C = ⋃_{i} cl(A_{i}).

Assumption 3 Suppose C ⊆ H is special and X is a finite subset of cl(C). Then the type of X over C is defined over a finite subset C_{0} ⊆ C.

1with a correction in (ii) suggested by O.Roche, 3.11.04

Remark It follows from assumptions 1 and 2, by the Ehrenfeucht-Fraisse criterion, that, given finite X ⊆ H, any two points in H \ cl(X) are L_{ω_{1},ω}- equivalent. If also an uncountable structure H ∈ C satisfies the countable closure property then any L_{ω_{1},ω}-definable subset of H is either countable or has a countable complement.

A class C of structures satisfying assumptions 1-3 will be called in this paper quasi-minimal excellent. Also, each member of the class is said to be quasi-minimal excellent.

Below we usually use letters A, A_{0} … to denote infinite independent sets, while A, A_{0}, … will stand for their finite subsets.

Lemma 1.1 Let A_{0} ⊆ H, A′_{0} ⊆ H′ be independent subsets and suppose G = cl(A_{0}) and G′ = cl(A′_{0}) are countable. Then, given a bijection φ_{0} : A_{0} → A′_{0}, there is an isomorphism φ : G → G′, extending φ_{0}.

Proof Assuming A_{0} = {a_{i} : i ∈ ℕ}, A′_{0} = {a′_{i} : i ∈ ℕ} and φ_{0}(a_{i}) = a′_{i}, by induction on i assume that there is a isomorphism

ψ_{i} : cl(a_{1}, …, a_{i}) → cl(a′_{1}, …, a′_{i}),

sending a_{j} to a′_{j} for all j ≤ i. Then by definition {a_{i+1}} is cl-independent over cl(a_{1}, …, a_{i}) and so is {a′_{i+1}} over cl(a′_{1}, …, a′_{i}). It follows from the ωhomogeneity over a submodel and the countability assumption that ψ_{i} can be extended to

ψ_{i+1} : cl(a_{1}, …, a_{i+1}) → cl(a′_{1}, …, a′_{i+1}).

Then φ = ⋃_{i} ψ_{i} satisfies the requrements.□

Lemma 1.2 Let A be a cl-independent subset in H, A_{0} ⊆ A infinite and A_{0}, …, A_{m} ⊆ A finite subsets. For X ⊆ H denote cl^{∗}(X) = cl(A_{0} ∪ X), and let B_{0} ⊆ cl^{∗}(A_{0}) finite, b_{i} ∈ cl^{∗}(A_{i}), i = 1, …, m, and p(x_{1}, …, x_{m}) a quantifier-free type over B_{0}, such that ⊧ p(b_{1}, …, b_{m}). Then there is a mapping

π : ⋃ A_{i} ∪ B_{0} ∪ {b_{1}, …, b_{m}} → cl^{∗}(A_{0}),

which is identity on B_{0}, b_{i}^{π} ∈ cl^{∗}(A_{i}^{π}) ⊆ cl^{∗}(A_{i} ∩ A_{0}) and ⊧ p(b_{1}^{π}, …, b_{m}^{π}).

Proof Let A^{∗} be a finite subset of A_{0}, such that B_{0} ⊆ cl(A_{0} ∪ A^{∗}) and b_{i} ∈ cl(A_{i} ∪ A^{∗}) for all i = 1, …, m. Let G_{0} = cl(A^{∗} ∪ A_{0}) and π_{0} be the identity map on G_{0}. Then the extension of π_{0} to an injection

π_{1} : G_{0} ∪ ⋃_{1≤i≤m} A_{i} \ (A_{0} ∪ A^{∗}) → G_{0} ∪ A_{0} \ (A_{0} ∪ A^{∗})

is a monomorphism by the assumption 2(i). By ω-homogeneity extend π_{1} to a monomorphism π with a domain covering {b_{1}, …, b_{m}}. Then ⊧ p(b_{1}^{π}, …, b_{m}^{π}), since π is a monomorphism over B_{0}. Also, b_{i}^{π} ∈ cl(A_{i}^{π} ∪ A^{∗}) ⊆ cl^{∗}(A_{i}^{π}) by the construction, and cl^{∗}(A_{i}^{π}) ⊆ cl^{∗}(A_{i} ∩ A_{0}) by the definition of π_{1}.□

Theorem 1 Let C be a quasi-minimal excellent class, H, H′ ∈ C, both with the countable closure property, A ⊆ H, A′ ⊆ H′, independent and cl(A) = H, cl(A′) = H′. Suppose that there is a bijection ψ_{0} : A → A′. Then ψ_{0} extends to an isomorphism ψ : H → H′.

Proof If A is finite or countable, the statement is proved in Lemma 1.1. So we assume A is infinite and uncountable, and so contains a proper countable subset A_{0}. Let ψ_{0}(A_{0}) = A′_{0} ⊆ A′. W.l.o.g. we may assume that ψ_{0} is identity on A_{0}. By Lemma 1.1 we may also identify cl_{H}(A_{0}) = cl_{H′}(A′_{0}) = G. We continue the proof of the theorem through a series of intermediate statements.

We now change the language by fixing G pointwise and so “monomorphism” from now on means “monomorphism over G” and cl(X) means cl(X∪A_{0}), the same as cl^{∗} in Lemma 1.2 above. Notice that in the new language assumptions 1-3 still hold for structures containing A_{0}.

Lemma 1.3 ψ_{0} is a monomorphism.

Proof First notice that A \ A_{0} along with A′ \ A_{0} are independent. Then apply assumption 2(i).□

Lemma 1.4 If X ⊆ A is finite, then there is a isomorphism cl(X) → cl(ψ_{0}(X)) extending ψ_{0}|_{X}.

Proof Immediate from the countable closure property and the assumptions 2 by the back-and-forth method.□

Notation For X ⊆ H denote

cl^{−}(X) = ∪_{Y ⊊X}cl(Y).

Lemma 1.5 Let A_{0}, …, A_{m} ⊆ A ⊆ A, A finite and A_{0} ⊈ A_{i} for i > 0. Then any two tuples D, D′ in cl(A_{0}) which are isomorphic over cl^{−}(A_{0}), are also isomorphic over cl^{−}(A_{0}) ∪ ⋃_{1≤i≤m} cl(A_{i}).

Proof Assume not, and D, D′ provide a counterexample. Then there is a quantifier-free formula q(x, y) over cl^{−}(A_{0}), such that for some B ⊆ ⋃_{1≤i≤m} cl(A_{i}) both q(D, B) and ¬q(D′, B) hold in H. By assumption 1.2 there is a map

π : ⋃_{1≤i≤m} A_{i} ∪ D ∪ D′ ∪ B → cl(A_{0}),

identical on D ∪ D′, such that

⊧ ¬q(D′, B^{π}) & q(D, B^{π})

and

π(B ∩ cl(A_{i})) ⊆ cl(A_{0}) ∩ cl(A_{i}^{π}) ⊆ cl(A_{0} ∩ A_{i}) ⊆ cl^{−}(A_{0}).

Thus B^{π} ⊆ cl^{−}(A_{0}). This contradicts the assumptions on D, D′. □

Definition Given a finite A ⊆ A \ A_{0} we call a subset D ⊆ cl(A) perfect over A in H if (an inductive definition)

(i) A = ∅, or (ii) for any X ⊊ A the sets D_{X} = D ∩ cl(X) are perfect over X, and the type of D over cl^{−}(A) is defined over ⋃_{X⊊A} D_{X}.

Lemma 1.6 Let A ⊆ A \ A_{0} and D ⊆ cl(A) be finite, n ≥ 0 and assume that for each X ⊆ A with |X| ≤ n + 1 D_{X} = D ∩ cl(X) is perfect over X and φ_{X} : D_{X} → H′ is a monomorphism. Denote φ = ⋃_{X⊆A,|X|=n+1} φ_{X}, and suppose also that the restriction of φ on ⋃_{Y ⊆A,|Y |=n} D_{Y} is a monomorphism. Then φ is a monomorphism.

Proof Let

{X ⊆ A, |X| = n + 1} = {A_{i} : i = 0, …, m}, distinct.

Notice that since, by cl-independence, for any i ≠ j

D ∩ cl(A_{i}) ∩ cl(A_{j}) ⊆ ⋃_{Y ⊆A,|Y |=n} D_{Y} ,

φ is a well-defined mapping. Remember that

⋃_{Y ⊆A,|Y |=n} D_{Y} = ⋃_{i≤m} D_{A_{i}} ∩ cl^{−}(A_{i}),

and apply Lemma 1.5 to see that φ preserves the quantifier-free formulas on

D_{A_{0}} ∪ ⋃_{0<i≤m} D_{A_{i}} ∩ cl^{−}(A_{i}).

The same lemma by induction on l ≤ m yields that φ preserves the quantifierfree formulas on

⋃_{i≤l} D_{A_{i}} ∪ ⋃_{l<i≤m} D_{A_{i}} ∩ cl^{−}(A_{i}).

For l = m we have the statement of the present lemma. □

Lemma 1.7 Given a finite A ⊆ A \ A_{0} for any finite D ⊆ cl(A) there is a finite D̄, D ⊆ D̄ ⊆ cl(A), perfect over A. More precisely,

D̄ = D ∪ ⋃_{X⊊A} D̄_{X}, where D ∩ cl(X) ⊆ D̄_{X} ⊆ cl(X).

Moreover, if we relpace D̄_{X} by D̄′_{X} ⊇ D̄_{X}, perfect over X, and D by D′ ⊆ D, then

D̄′ = D′ ∪ ⋃_{X⊊A} D̄′_{X},

is still perfect over A.

Proof (i) Induction on |A|. The statement is obvious for A = ∅. In the general case by assumption 3 the type of D over G ∪ cl^{−}(A) is defined over a finite subset, say ⋃_{X⊊A} D_{X}, and G with each D_{X} ⊆ cl(X). By induction there are finite D̄_{X} perfect over X. Put

D̄ = D ∪ ⋃_{X⊊A} D̄_{X},

which is perfect over A by the definition. The ’moreover’ statement is immediate by the definition. □

Proof of the theorem. We proceed by induction on n assuming that there is a family

D_{n} = {D_{A,i} ⊆ cl(A) : A ⊆ A \ A_{0}, |A| ≤ n, i ∈ ℕ}

of finite sets perfect over A, such that

cl(A) = ⋃_{i∈ℕ}D_{A,i},

for each B ⊆ A

D_{A,i} ∩ cl(B) = D_{B,i},

and D_{m} ⊆ D_{n} for m ≤ n.

We also assume that there are monomorphisms for all k ∈ ℕ :

ψ_{n,k} : ⋃_{A⊊A,|A|≤n}D_{A,k} → H′, ψ_{0} ⊆ ψ_{n,k} ⊆ ψ_{n,k+1}

and for each A ⊆ A of size ≤ n

ψ_{A} = ⋃_{k∈ℕ}ψ_{n,k}|_{cl(A)} : cl(A) → cl(ψ_{0}(A)) is a bijection.

Notice that for D_{∅,k} we can take any ascending chain of finite subsets covering cl(∅), which is a substructure of common G, and ψ_{0,k} be the identity mappings on the appropriate subsets. Suppose that a chain D_{X,i} has been constructed for each X ⊆ A \ A_{0} of size ≤ n as well as monomorphisms ψ_{n,k} for all k ∈ ℕ. For each |A| = n + 1 put D_{A,0} = ⋃_{X⊊A} D_{X,0}, cl^{0}(A) = cl(A) \ cl^{−}(A), and for each i ∈ ℕ by induction on i :

D_{A,i+1} = \{ \begin{matrix}D_{A,i} ∪ ⋃_{X⊊A} D_{X,i+1} ∪ min\{cl^{0}(A) \ D_{A,i}\} if this is perfect over A \\ D_{A,i} ∪ ⋃_{X⊊A} D_{X,i+1} otherwise \end{matrix}

Since, given a ∈ cl^{0}(A), for some j > i the type of {a} ∪ D_{A,i} over G ∪ cl^{−}(A) is defined over G ∪ ⋃_{X⊊A} D_{X,j}, we get by Lemma 1.7 a ∈ D_{A,j}, and thus ⋃_{i} D_{A,i} = cl(A). Also for all j D_{A,j} \⋃_{X⊊A} D_{X,j} ⊆ cl^{0}(A), so for any B ⊆ A, by the induction hypothesis, D_{A,j} ∩ cl(B) = D_{B,j}. By the ω-homogeneity for each k ∈ ℕ we can extend ψ_{n,k}|_{cl(A)}, whose domain is obviously a subset of ⋃_{X⊆A,|X|=n} D_{X,k}, to

ψ_{A,k} : D_{A,k} → H′.

More precisely, when ψ_{A,k} is constructed, A′ = ψ_{0}(A), and a′ is the first element in cl^{0}(A′), of quantifier free type, say p, over ψ_{A,k}(D_{A,k}), let j ≥ k be such that the corresponding type ψ_{A,k}^{−1}(p) over D_{A,k} is realised in D_{A,j}, say by a. Extend by the ω-homogeneity ψ_{A,k} to ψ_{A,j} : D_{A,j} → H′ so that ψ_{A,j}(a) = a′. By doing this we have also defined ψ_{A,m} for k ≤ m ≤ j as the corresponding restrictions of ψ_{A,j}. Also, this process guarantees that the image of ⋃_{i∈ℕ} ψ_{A,i} is the whole of cl(A′). Define

ψ_{n+1,k} = ⋃_{A⊆A,|A|=n+1} ψ_{A,k}.

Since for any distinct A_{1}, A_{2} ⊆ A of size n + 1

D_{A_{1},i} ∩ D_{A_{2},i} = D_{A_{1}∩A_{2},i},

and |A_{1} ∩ A_{2}| ≤ n, the definition is consistent.

Given a finite B ⊆ A \ A_{0}, the mapping φ = ψ_{n+1,k}|_{cl(B)}, by Lemma 1.6, is a monomorphism. So ψ_{n+1,k} is a a monomorphism.

Finally, put ψ = ⋃_{n∈ℕ} ψ_{n,n}. Obviously, the domain of ψ is cl(A) = H and the range is

⋃_{A′⊆A′ finite} cl(A′) = H′.

It finishes the construction of an isomorphism.□

Corollary 1 If under the assumptions of the theorem cl satisfies also the exchange principle, then C is categorical in every uncountable cardinality.

Proof Under the exchange principle cl-independence is equivalent to the independence in the usual pregeometry sence. In other words A is just a basis of H, which exists for any structure H, and by assumptions 1(ii) and 1(iv) has the same cardinality as H modulo ℵ_{0}. It follows from the theorem that any H and H′ in C of the same uncountable cardinality are isomorphic.□

Theorem 2 Suppose a quasi-minimal excellent class C^{#} is axiomatizable by an L_{ω_{1},ω}-sentence Σ and the relations y ∈ cl(x_{1}, …, x_{n}) are L_{ω_{1},ω}-definable for all n. Suppose also that there is an H ∈ C^{#} containing an infinite clindependent subset A. Then for any infinite cardinal κ there is a structure H_{κ} ∈ C^{#} of cardinality κ, satisfying the countable closure property. A structure with these properties is unique in C^{#}, provided the class satisfies the exchange principle.^{2}

Proof Let C be the subclass of C^{#} satisfying the countable closure property. Choose A_{0} ⊆ A countable. By assumption 1 cl(A_{0}) ∈ C^{#}. By the downward Lowenheim-Skolem theorem choose a countable submodel H_{0} ⊆ cl(A_{0}) of Σ.

2The uniqueness statement was formulated incorrectly in the previous version of the article. I am grateful to O.Roche for pointing out the error to me.

H_{0} obviously satisfies the countable closure property. So, we may assume cl(A_{0}) = H_{0} ∈ C. Suppose now cl(A) = H ∈ C for some A of cardinality λ ≥ ℵ_{0}. Notice that by assumption 2 any cl-independent set is indiscernible. Let a be a point in A, A′ = A \ {a} and H′ = cl(A′). Obviously, H′ ⊊ H and by Theorem 1 a bijection between A′ and A extends to an isomorphism H′ ≅ H. This gives rise to a sequence

H′ = H_{0} ⊆ H_{1} ⊆ … H_{α} ⊆ …

of isomorphic structures and embeddings of the form H_{α} = cl(A_{α}), for α ordinals of cardinality λ, and

A′ = A_{0} ⊆ A_{1} ⊆ … A_{α} ⊆ …

with unions on limit steps. Taking

H_{λ^{+}} = ⋃_{α<λ^{+}} H_{α},

we get a structure of cardinality λ^{+}, which is still in C^{#}, since by the ωhomogeneity and the Ehrenfeucht-Fraisse criterion

H ≡_{L_{ω_{1},ω}} H_{λ^{+}}.

Also the countable closure property is satisfied for H_{λ^{+}} since

H_{λ^{+}} = ⋃_{α<λ^{+}} cl(A_{α}).

Thus H_{λ^{+}} ∈ C. We can then carry on the same process up to κ. □

Remark The countable closure property is not L_{ω_{1},ω}-axiomatizable (yet, it is L_{ω_{1},ω}(Q) in many cases) so, the theorem proves categoricity for not necessarily L_{ω_{1},ω}-axiomatizable subclass of C^{♯}.

The following question has been suggested by the referee.

Question Does Theorem 2 hold when Σ is an L_{ω_{1},ω}(Q)-sentence?

2 Examples and applications

Let F be an algebraically closed field of characteristic 0 and F^{×} its multiplicative group. Then, by obvious group theoretic reasons there exists an abelian divisible torsion free group H and a homomorphism ex such that the sequence

0 → ℤ →^{i} H →^{ex} F^{×} → 0  (1)

is exact. We call this sequence a cover of the multiplicative group of the field F. In logical terms the sequence is a two-sorted structure (H, F) with a group structure on sort H and a field structure on F and the mapping ex between the sorts. It is more convenient in the context of the present paper to represent the sequence of the form (1) as a one sorted structure

H = (H, +, E(x, y), S(x, y, z), P (x, y, z))

with domain H, a group operation +, a binary equivalence relation E interpreted as the pullback of the equality on F by ex, and ternary relations S and P which are pullbacks of the basic ternary additive and multiplicative equations on F. In fact, P is redundant, since ex(x) · ex(y) = ex(z) is a consequence of x + y = z in the group H, so we omit it. In particular, for the classical structure, the complex numbers with exponentiation: H = F = ℂ and ex = exp, we have

E(x, y) ≡ x − y ∈ 2πiℤ and S(x, y, z) ≡ e^{x} + e^{y} = e^{z}.

It can easily be seen, as in [Z2], that There is an L_{ω_{1},ω}-sentence Σ (which is a finite diagram in Shelah’s sense) such that any model H of Σ represents a sequence (1) with some algebraically closed field F, and conversely, any H corresponding to a sequence of the form (1) is a model of Σ. The main result of [Z2] is the following

Theorem 3 The class of models of Σ is quasi-minimal excellent. Moreover, the class satisfies the exchange principle and the countable closure property. Uncountable models of Σ are not homogeneous.

Corollary 2 L_{ω_{1},ω} theory of the structure

(ℂ, +, e^{x} + e^{y} = e^{z})

is categorical in all uncountable cardinalities.

Fields with pseudo-exponentiation. These are structures of the form (F, +, ·, ex), where (F, +, ·) is an algebraically closed field of characteristic zero. Further assumptions are:

1. ex is a homomorphism from the additive group onto the multiplicative group of the field, with cyclic kernel.

2. “The Schanuel conjecture” holds: for any additively independent x_{1}, …, x_{n} ∈ F the transcendence degree of x_{1}, …, x_{n}, ex(x_{1}), …, ex(x_{n}) is at least n.

3. Any normal, free system of equations over F has a generic solution in F. (We skip here the definitions of ’normal’ and ’free’.)

The class of fields with pseudo-exponentiation was studied in [Z3]. A closure operator cl was introduced, and the main result of the paper states that The class satisfying the L_{ω_{1},ω}-conditions (1)-(3) above is quasi-minimal excellent. The countable closure property (which can be formulated by a sentence of L_{ω_{1},ω}(Q)) does not follow from (1)-(3). But by Theorem 2 we get that in any uncountable cardinality there is a unique field with pseudo-exponentiation satisfying the countable closure property. The one of cardinality continuum is a very close analogue of the complex numbers with exponentiation (ℂ, +, ·, exp) and, conjecturally, is isomorphic to it.

3 Almost quasi-minimal classes

We slightly generalize here the main result of the paper by extending the main definition. A good reason for this is the generalizations of the example of the previous section studied in [Z4], group covers of semi-Abelian varieties. Under certain arithmetic assumptions group covers of semi-Abelian varieties are almost quasi-minimal in the sense below.

Let B be a class of structures with a closure operator cl satisfying Assumption 1 and with an extra unary predicate U such that for all H ∈ B the closure operator cl on H has the property

cl(U(H)) = H.

We call such a class almost quasi-minimal excellent if it satisfies Assumptions 1-3 with the following correction to Assumption 2(i) (with the same meaning of H, H′ and G):

(i) if X ⊆ U(H) and X′ ⊆ U(H′) are cl-independent over G in some H, H′ ∈ B, correspondingly, then any bijection φ : X → X′ is a G-monomorphism;

Part (ii) and (iii) of the assumption remain unchanged.

We also redefine

Definition A subset C ⊆ H is called special if there is a cl-independent A ⊆ U(H) and A_{1}, … A_{k} ⊂ A such that

C = ⋃_{i} cl(A_{i}).

With this definition in mind Assumption 3 stays unchanged.

Theorem 4 Let B be an almost quasi-minimal excellent class, H, H′ ∈ B, both with the countable closure property, A ⊆ U(H), A′ ⊆ U(H′), independent and cl(A) = H, cl(A′) = H′. Suppose that there is a bijection ψ_{0} : A → A′. Then ψ_{0} extends to an isomorphism ψ : H → H′.

Proof Same as for Theorem 1.□

Theorem 5 Suppose a quasi-minimal excellent class B^{#} is axiomatizable by an L_{ω_{1},ω}-sentence Σ and the relations y ∈ cl(x_{1}, …, x_{n}) are L_{ω_{1},ω}-definable for all n. Suppose also that there is an H ∈ B^{#} containing an infinite clindependent subset A ⊆ U(H). Then for any infinite cardinal κ there is a unique structure H_{κ} ∈ B^{#} of cardinality κ, satisfying the countable closure property.

Proof Same as for Theorem 2.□

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