Elements of Geometric Stability Theory

December 17, 1999

1 Completeness and quantifier elimination for some classical theories

We denote ACF_{p} the theory of algebraically closed fields of characteristic p.

Theorem 1 ACF_{p} is complete and allows quantifier elimination in the language (+, ·, 0, 1)

First we prove

Lemma 1.1 (weak form of Steinitz Theorem) Let B and C of the same uncountable cardinality μ be in ACF_{p}. Then any isomorphism α_{0} : B_{0} → C_{0} between subfields B_{0} < B and C_{0} < C of cardinality less than μ can be extended to an isomorphism α : B → C.

Proof We enumerate the fields and proceed back-and-forth constructing α_{i} : B_{i} → C_{i} of cardinality less than μ. Suppose B_{i} and C_{i} are isomorphic and of cardinality less than μ. Take the first b ∈ B not in B_{i}. If b is transcendental over B_{i} then by the cardinality considerations we can find c transcendental over C_{i}. Then

B_{i}(b) ≅ B_{i}(x) ≅ C_{i}(x) ≅ C_{i}(c).

If b is a root of f_{B}(x) over B_{i}, irreducible, then the corres pondent f_{C}(x) has a root c in C and

B_{i}(b) ≅ B_{i}[x]/{f_{B}(x)} ≅ C_{i}[x]/{f_{C}(x) ≅ C_{i}[c].

□

Corollary 1 ACF_{p} is μ-categorical for μ > ℵ_{0} and complete.

Lemma 1.2 For any A ⊆ F, any two n-tuples b̄ and c̄

qftp(b̄/A) = qftp(c̄/A) iff b̄ conj with c̄ over A iff tp(b̄/A) = tp(c̄/A).

Proof First consider n = 1. Denote (A) the subfield generated by A. If b is transcendental over (A) then so is c and

(A)(b) ≅ (A)(x) ≅ (A)(c) over A.

If b is a root of f(x) over (A), minimal for b, then so is c and

(A)(b) ≅ (A)[x]/{f(x)} ≅ (A)[c] over A.

If b̄ = ⟨b_{1} …, b_{n}⟩ and qftp(b̄/A) = qftp(c̄/A) then qftp(b_{1}/A) = qftp(c_{1}/A) thus by induction there is an isomorphism

α : (Ab_{1}) → (Ac_{1}).

Let ⟨b′_{2} …, b′_{n}⟩ be the image of ⟨b_{2} …, b_{n}⟩ under α. Then

qftp(⟨c_{1}, b′_{2} …, b′_{n}⟩/A) = qftp(b̄/A) = qftp(c̄/A),

hence

⟨b′_{2} …, b′_{n}⟩ conj with ⟨c_{2}, …, c_{n}⟩ over Ac_{1}.

Finally b̄ conj with c̄ over A.□

Proof of QE. Let φ(x̄) be any formula in the field language and

Φ(x̄) = {ψ(x̄) qfree : F ⊧ φ(x) → ψ(x̄)}.

If Φ&¬φ is consistent then in monster F′ ≻ F there is a realization b̄ of the type. qftp(b̄) must be consistent with φ for otherwise ¬ξ(x̄) is in Φ for some ξ ∈ qftp(b̄). Then there exists c̄ realizing qftp(b̄)&φ. A contradiction.

Thus ⊧ Φ → φ and so Φ is equivalent to its finite part and Φ ≡ φ.□

Ex. Prove the corresponding theorems for 1.Vector spaces 2.Nested equivalence relations.

2 Algebraic closure in abstract structures

Definition

cl(A) = {b ∈ M : there are a ∈ A^{n}, m ∈ N and φ(u, v) such that
M ⊧ φ(a, b)&∃^{≤m}vφ(a, v)}.

Lemma 2.1 The following properties of cl hold in any structure:

A ⊆ B implies A ⊆ cl(A) ⊆ cl(B)  (1)
cl(cl(A)) = cl(A).  (2)

Proof Easy.□

Remark For any field F and A ⊆ F:

cl(A) ⊇ acl(A)

for algebraic closure of A in F.

Lemma 2.2 Any elementary monomorphism α between A, A′ ⊆ M can be extended to cl(A) → cl(A′).

Proof Enumerate cl(A) and go by transfinite induction extending α to A ∪ {a_{i} : i < γ} finding for a_{γ} corresponding element a′_{γ} ∈ cl(A′) as a realisation of type

α(tp(a_{γ}/(A ∪ {a_{i} : i < γ})

obtained by replacing elements of cl(A) by corresponding elements of cl(A′) in tp(a_{γ}/(A ∪ {a_{i} : i < γ}). Both types are principal and algebraic since a_{γ} is algebraic over A ∪ {a_{i} : i < γ}. Notice that after exhausting the process on cl(A) the other part, cl(A′), will be exhausted too, since going back from cl(A′) to cl(A) we would find elements in cl(A). □

Minimal structures.

Definition A structure M is said to be minimal if any subset of M definable using parameters is either finite or a complement of a finite. We shall assume everywhere that the language of M is countable.

Lemma 2.3 In minimal structures the following exchange principle holds:

For any A ⊆ M, b, c ∈ M : b ∈ cl(A, c) \ cl(A) → c ∈ cl(A, b) (3)

Proof Suppose b ∈ cl(A, c) \ cl(A). Then for some φ(x, y) over A and some m

M ⊧ φ(b, c)&∃^{≤m}φ(x, c).  (4)

W.l.o.g. we assume

M ⊧ φ(x, y) → ∃^{≤m}φ(x, y).  (5)

Suppose, towards a contradiction, that φ(b, M) is infinite. Then card(¬φ(b, M)) ≤ k for some k, i.e. M ⊧ ∃^{≤k}y¬φ(b, y) and

B = {b′ ∈ M : M ⊧ ∃^{≤k}y¬φ(b′, y)}

is infinite, since b ∉ cl(A). Choose distinct b_{1}, …, b_{m+1} ∈ B. Then

φ(b_{1}, M) ∩ … ∩ φ(b_{m+1}, M)

is infinite and thus contains a point c′. It contradicts (5).□

Pregeometry and geometry of a minimal structure.

Definition An [abstract] pregeometry is a set M with an operator

cl : 2^{M} → 2^{M}

of finite character, i.e. for any A ⊆ M : cl(A) = {cl(A′) : A′ ⊆ A finite} and satisfying the conditions (1)-(3) above.

A pregeometry is said to be a geometry if

for any a ⊆ M cl({a}) = {a}  (6)

bf Getting a geometry from a pregeometry

Lemma 2.4 The relation ∼ on M \ cl(∅) defined as

x ∼ y iff cl(x) = cl(y)

is an equivalence relation.

Proof Follows from the exchange principle.□

Define for a pregeometry M the set

M̂ = (M \ cl(∅))/ ∼

Then any point in M̂ is of the form â = cl(a) \ cl(∅) for a corresponding a ∈ M \ cl(∅). For a subset Â = {â : a ∈ A} ⊆ M̂ define

cl(Â) = {b̂: b ∈ cl(A)}.

The operator cl on M̂ satisfies then (1)-(4) and thus M̂ is a geometry.

A pregeometry M with a fixed D ⊆ M gives rise to another pregeometry M_{D} the localisation of M with respect to D: the set of M_{D} is just M and cl_{D}(A) = cl(D ∪ A). Subspaces of a pregeometry are subset of the form cl(A). Pregeometry is said to be locally finite if cl(A) is finite whenever A is.

Examples: Vector spaces over division rings are pregeometries if we let

cl(A) = span(A).

Projective space associated with a vector space M is defined exactly as the geometry M̂.

Affine space associated with a vector space M is defined on the same set M by the new closure-operator:

cl_{aff}(A) = A + span(A − A)

where A − A = {a_{1} − a_{2} : a_{1}, a_{2} ∈ A}.

Exercise Show that an affine space is a geometry and its localisation with respect to any point is isomorphic to the initial vector space pregeometry.

A set A is said to be independent if cl(A) ≠ cl(A′) for any proper subset A′ ⊂ A. A maximal independent subset of a set A is said to be a basis of A.

Lemma 2.5 Any two bases B and C of a set A are of the same cardinality.

Proof First consider the case when, say B, is finite and consists of n elements b_{1}, … b_{n}. There is a c ∈ C such that

c ∈ cl(b_{1}, … b_{n}) \ cl(b_{1}, … b_{n−1})

for otherwise B is not independent. By the exchange principle {c, b_{1}, … b_{n−1}} is a basis of A. In the localisation M_{c} the sets {b_{1}, … b_{n−1}} and C \ {c} are bases of A. By induction on n the statement follows. Consider now the case when both B and C are infinite. It follows from the finite character of cl that for any b ∈ B there is a minimal finite C_{b} ⊂ C such that b ∈ cl(C_{b}). Thus there is a mapping of B into P_{fin}(C), the set of all finite subsets of C. The mapping is finite-to-one, since by the above proved the set

{d ∈ B : C_{d} = C_{b}}

is an independent subset of cl(C_{b}) and by the above proved its size is not bigger than the size of C_{b}. It follows cardB ≤ cardC. By the symmetry cardB = cardC. □

Now we can give

Definition For any subset A of a pregeometry define dim A to be the cardinality of a basis of A. If also B ⊆ A then dim(A/B) is the dimension of A in the pregeometry M_{B}.

Lemma 2.6 (the additive formula)

dim(A/B) + dim(B) = dim(A).

Proof One can construct a basis of A by adjoining to a basis of B a basis of A in M_{B}.□

Example The transcendence degree of a subset A of an algebraically closed field F is defined as dim A, which is well defined since F is a minimal structure (with cl = acl).) Since any field is a subfield of an algebraically closed one, the definition is applicable for subsets of any field.

Lemma 2.7 For X, Y ⊆ M subspaces of a pregometry

dim(X ∪ Y) ≤ dim X + dim Y − dim(X ∩ Y).

Proof Let Z be a basis of X ∩ Y. Let Z ∪ X_{0} and Z ∪ Y_{0} be bases of X and Y, correspondingly. Then cl(X_{0} ∪ Z ∪ Y_{0}) = cl(X ∪ Y) and thus dim(X ∪ Y) ≤ |X_{0} ∪ Z| + |Z ∪ Y_{0}| − |Z|. □

Definition A subset A of a structure M is said to be indiscernible over B if tp(ā/B) = tp(ā′/B) for any two n-tuples of distinct elements of A for any finite n.

Proposition 1 Let M be a minimal structure, A, B ⊆ M and A independent over B (in the pregeometry of M.) Then A is indiscernible over B. Moreover, the n-type tp(ā/B) for ā ∈ A^{n} does not depend on B.

Proof Consider ā = ⟨a_{1}, … a_{n}⟩, ā′ = ⟨a′_{1}, …, a′_{n}⟩ all with distinct coordinates from A. In case the size n = 1 tp(a/B) is just the set of those formulas φ(x) over B which have φ(M) infinite. The same characterises tp(a′/B). Thus the types are equal. For n > 1 suppose, as an inductive hypothesis, the tuples have the same type over B. Then for a_{n+1} ∈ A \ {a_{1}, … a_{n}} and any formula φ(x̄, y) over B

⊧ φ(ā, a_{n+1}) iff φ(ā, M) is infinite .

By the equality of types the latter is equivalent to φ(ā′, M) is infinite which gives ⊧ φ(ā′, a′_{n+1}) for any a′_{n+1} ∈ A distinct from the coordinates of ā′.□

Lemma 2.8 Any minimal structure in a countable language is homogeneous.

Proof Suppose B, B′ ⊆ M are of cardinality less than cardM and there is an elementary monomorphism α : B → B′. It follows dim B = dim B′. From the assumptions on cardinalities it follows also that dim B < cardM. Thus from the additive formula dim M/B = dim M/B′. Let A ⊇ B and A′ ⊇ B′ be extensions to bases of M. Since

cardA \ B = cardA′ \ B′

there is a bijection β : A → A′ extending α. The description of types of n-tuples in bases above shows that β is an elementary monomorphism. Now Lemma 2.2 finishes the proof.□

3 Dimension notion in strongly minimal structures

Theorem 2 Minimal structures of infinite dimension are saturated.

Proof Notice that in this case dim M = cardM. Let φ(x) be a formula over A, cardA < cardM. Either φ(M) is finite, or M \ cl(A) ⊆ φ(M). Thus for any consistent set of such formulas

⋂_{i} φ_{i}(M)

either contains the nonempty set M \ cl(A) or is a nonempty subset of some finite φ_{i}(M).□

Corollary 2 Any structure e.e.equivalent to a minimal structure of infinite dimension is minimal too. It also satisfies f.c.p.: for any φ(x, ȳ) there is a natural number m such that cardφ(M, ā) > m implies φ(M, ā) is infinite.

Definition A minimal structure is said to be strongly minimal if it is elementarily equivalent to a minimal structure of infinite dimension.

Rank notion for sets definable in s.m.structures.

We assume below M is strongly minimal of infinite dimension.

Definition Let A ⊆ M be finite, M saturated minimal. For an A-definable subset S ⊆ M^{n} put the Morley rank to be

rk S = max_{⟨s_{1},...s_{n}⟩∈S} dim({s_{1}, … s_{n}}/A)

Lemma 3.1 rk φ(M) has the same value in every saturated structure elementary equivalent to a given s.m. one and does not depend on A.

Proof Immediate from saturatedness of M.□

Definition For an arbitrary s.m.structure M rk φ(M) is defined as the rank in saturated elementary extensions of M.

Lemma 3.2 (Basic Rank Properties) For any s.m.structure M

(i) rk M^{n} = n; (ii) rk S = 0 iff S is finite (iii) rk (S_{1} ∪ S_{2}) = max{rk S_{1}, rk S_{2}} (iv) For the projection pr : M^{n} → M^{k}

rk S ≤ rk pr(S) + max_{t∈pr(S)} rk pr^{−1}(t).

(v) Suppose rk pr^{−1}(t) is the same for all t ∈ pr(S). Then

rk S = rk pr(S) + rk pr^{−1}(t).

Proof (i)-(iii) are immediate from the definition. (iv) Let ⟨s_{1}, … s_{n}⟩ ∈ S be of maximal dimension in S. Then

rk S = dim({s_{1}, … s_{n}}/A) = dim({s_{1}, … s_{n}}/{s_{1}, … s_{k}} ∪ A)+

dim({s_{1}, … s_{k}}/A) ≤ rk pr^{−1}(⟨s_{1}, … s_{k}⟩) + rk prS.

(v) If one chooses first a generic ⟨s_{1}, … s_{k}⟩ ∈ prS and then extends it to ⟨s_{1}, … s_{n}⟩ ∈ S of maximal possible dimension over {s_{1}, … s_{k}} ∪ A, then

dim({s_{1}, … s_{n}}/{s_{1}, … s_{k}} ∪ A) = rk pr^{−1}(⟨s_{1}, … s_{k}⟩),
dim({s_{1}, … s_{k}}/A) = rk prS

and thus

rk S ≥ rk prS + rk pr^{−1}(⟨s_{1}, … s_{k}⟩).

□

Lemma 3.3 For any definable S ⊆ M^{n} there is an upper bound for m ∈ N such that S can be partitioned into k disjoint subsets

S = S_{1} ∪ … ∪ S_{m}

each of rank equal to rk S.

Proof We use induction on n. For n = 1 the statement follows from the definitions.

For arbitrary n let rk S = k. This means there is a point ⟨s_{1}, … s_{n}⟩ ∈ S of dimension k and thus some {s_{i_{1}}, … s_{i_{k}}} are independent. Let us consider the case ⟨i_{1}, … i_{k}⟩ = ⟨1, … k⟩. Then s_{j} ∈ cl{s_{1}, … s_{k}} for all j = 1, …, n}, thus for some natural number l = l_{i_{1},...i_{k}}

⊧ ∃^{=l}⟨x_{1}, … x_{n}⟩ ∈ S : ⟨x_{1}, … x_{k}⟩ = ⟨s_{1}, … s_{k}⟩.

Denote the formula ψ(⟨s_{1}, … s_{k}⟩) and notice that ψ(M) ⊆ M^{k} is of rank k. Let

S^{0} = {⟨s_{1}, … s_{n}⟩ ∈ S : ψ(⟨s_{1}, … s_{k}⟩)}

By formulas above rk S^{0} = rk ψ(M) = rk S. Suppose

S^{0} = S_{1}^{0} ∪ … ∪ S_{m}^{0}

is a partition and all the summands are A′-definable of rank k. Then necessarily for any j ≤ m there is ⟨s_{j,1}, … s_{j,n}⟩ ∈ S_{j}^{0} with the first k coordinates independent over A′. By indiscernibility we can choose ⟨s_{j,1}, … s_{j,n}⟩ ∈ S_{j}^{0} so, that

⟨s_{j,1}, … s_{j,k}⟩ = ⟨s_{1,1}, … s_{1,n}⟩

for all j. It follows immediately that m ≤ l. Taking into account all possibilities for ⟨i_{1}, … i_{k}⟩ we get

m ≤ ∑_{{i_{1},...i_{k}}} l_{i_{1},...i_{k}}.

□

Definition The exact upper bound for the equirank partition of S is said to be the Morley degree of S. A definable set of Morley degree one is called irreducible.

Definition For a type p(x̄) definable over A Morley rank of type is defined as

rk (p) = min{rk φ(x̄) : φ ∈ p}.

For a point s̄ ∈ M^{n} and a subset A ⊆ M the Morley rank of the point over A is defined as

rk (s̄/A) = rk (tp(s̄/A)).

A point s̄ ∈ S ⊆ M^{n} for an irreducible subset S defined over A is said to be generic over A if

rk (s̄/A) = rk S.

Lemma 3.4 For an irreducible S over A there is a unique complete type p over A containing S. More exactly,

p = tp(s/A)

for s generic in S. In particular, any two generic points have the same type over A.

Proof

p = {φ(x̄) over A : rk (φ(M) ∩ S) = rk S}.

The rest follows from definitions. □

Lemma 3.5 (The additive formula for tuples)

rk (b̄c̄/A) = rk (b̄/Ac̄) + rk (c̄/A)

Here Ac̄ = A ∪ |c̄|, |c̄| is the set of the coordinates of c̄.

Proof Follows from the additive formula for dimensions taking into account that rk (b̄/A) = dim(|b̄|/A), which follows immediately from the definitions.□

Definition Two points b̄ ∈ M^{k} and c̄ ∈ M^{n} are said to be independent over A if

rk (b̄/Ac̄) = rk (b̄/A).

Lemma 3.6 The independence relation is symmetric

Proof rk (b̄c̄/A) = rk (b̄/Ac̄) + rk (c̄/A) = rk (c̄/Ab̄) + rk (b̄/A) by the additive formula. Then if rk (b̄/Ac̄) = rk (b̄/A) so rk (c̄/Ab̄) + rk (c̄/A).□

Lemma 3.7 (Definability of Morley Rank) For any formula φ(x̄, ȳ) with length(x̄) = k, length(ȳ) = n, and any m the set

{ā ∈ M^{k} : rk φ(ā, M) ≥ m}

is definable.

Proof By induction on n. For n = 1 rk φ(a, M) ≥ 0 iff φ(a, M) ≠ ∅, and rk φ(a, M) ≥ 1 iff φ(a, M) is infinite iff cardφ(a, M) ≥ n_{φ} by f.c.p. For arbitrary n rk φ(a, y_{1}, …, y_{n}) ≥ m iff {b ∈ M : rk φ(a, b, y_{2}, …, y_{n}) ≥ m−1} is infinite or {b ∈ M : rk φ(a, b, y_{2}, …, y_{n}) ≥ m} ≠ ∅ by the additive formula for ranks. The both conditions on the right hand side are definable by induction hypothesis.□

Sets definable in M.

We shall consider Morley rank for sets definable in strongly minimal M. Recall that any such set is of the form U = S/E, where S ⊆ M^{n} is a definable subset and E ⊆ S^{2} ⊆ M^{2n} is a definable subset wich is an equivalence relation. We consider only U such that E is equirank, i.e. rk E(s, M) is of the same value for all s ∈ S.

Definition

rk U = rk S − rk E(s, M) for s ∈ S.

Lemma 3.8 The definition is invariant under definable bijections, i.e. if there is a bijection

f : S_{1}/E_{1} → S_{2}/E_{2}

and f is a definable function, then rk S_{1}/E_{1} = rk S_{2}/E_{2}.

Proof By definition f = F/E, where F ⊆ S_{1} × S_{2}, E = E_{1} × E_{2} and the following hold for any s_{1}, s′_{1} ∈ S_{1}, s_{2}, s′_{2} ∈ S_{2}

F (s_{1}, s_{2})&F (s′_{1}, s′_{2}) → (E_{1}(s_{1}, s′_{1}) ↔ E_{2}(s_{2}, s′_{2})),

pr_{S_{1}}F = S_{1} and pr_{S_{2}}F = S_{2}.

From the additive formula, projecting on S_{1}, we get

rk F = rk S_{1} + rk E_{2}(s_{2}, M)

and projecting on S_{2}

rk F = rk S_{2} + rk E_{1}(s_{1}, M).

It follows

rk S_{1} − rk E_{1}(s_{1}, M) = rk S_{2} − rk E_{2}(s_{2}, M).

□

Proposition 2 Basic Rank Properties (i)-(v) hold for definable sets as well as Lemma 3.3 and the definability of rank.

Proof By appropriately using the same arguments as in the proofs of the statements mentioned.□

Proposition 3 (Finite Equivalence Relation Theorem) For any A-definable set S of rank k there is an A-definable subset S^{0} ⊆ S and an equivalence relation E on S^{0} such that S^{0} ⊏⊐ S, S^{0}/E is finite and each equivalence class is of rank k and irreducible.

Proof Let Morley degree of S be m and

S = ⋃_{i≤m} Ψ_{i}(b_{i}, M)

be the partition of S into irreducible subsets of rank k with Ψ_{i}(z, x) be Adefinable formulas and b_{i} ranging in M^{n} for some n. Claim 1. We may choose Ψ_{i} = Ψ be independent on i. To get this consider formula Ψ of parameters b_{1}, …, b_{m} and c_{1}, … c_{m}, where c_{1}, … c_{m} are some distinct elements in M :

Ψ(y, x) ≡ ⋀_{i≤m}(y = c_{i} → Ψ(b_{i}, x)).

Evidently, Ψ(c_{i}, x) ≡ Ψ_{i}(b_{i}, x), which proves the claim. Put

P = {b ∈ M^{n} : Ψ(b, M) ⊆ S & rk Ψ(b, M) = k &
∀c ∈ M^{n}(rk (Ψ(b, M) ∩ Ψ(c, M)) = k → (Ψ(b, M) ⊏ Ψ(c, M)))}.

This set is A-definable, b_{i} ∈ P for all i ≤ m and for any b ∈ P Ψ(b, M) is an irreducible subset of S of rank k. Define an equivalence relation on P

F (b, b′) iff Ψ(b, M) ⊏⊐ Ψ(b′, M).

There are exactly m equivalence classes of F with b_{1}, …, b_{m} representatives of the classes. Claim 2. Let b ∈ P, s ∈ Ψ(b, M) and rk (s/Ab) = k. Then

F (b, M) ⊏ Ψ(M, s).

Proof of Claim. Let b′ ∈ F (b, M) be of maximal rank, i.e. rk (b′/Abs) = rk F (b, M). Then rk (b′/Abs) = rk (b′/Ab), which means b′ and s are independent over Ab. Then rk (s/Abb′) = rk (s/Ab) = k. Since Ψ(b, M) ⊏⊐ Ψ(b′, M) it follows s ∈ Ψ(b′, M). In other words Ψ(b′, s) holds for all such b′ and hence F (b, M) ⊏ Ψ(M, s). Claim 3. For any s ∈ S with rk (s/Ab_{1}, …, b_{m}) = k there is exactly one class F (b, M) such that F (b, M) ⊏ Ψ(M, s). Indeed, suppose F (b, M) ⊏ Ψ(M, s) for b = b_{1} and b = b_{2}. Choose ⟨b′_{1}, b′_{2}⟩ ∈ F (b_{1}, M) × F (b_{2}, M) of maximal rank over Ab_{1}b_{2}s. Then rk (b′_{1}b′_{2}/Ab_{1}b_{2}s) = rk F (b_{1}, M) + rk F (b_{2}, M) = rk (b′_{1}b′_{2}/Ab_{1}b_{2}). It follows rk (s/Ab_{1}b_{2}b′_{1}b′_{2}) = rk (s/Ab_{1}b_{2}), and Ψ(b′_{1}, s)&Ψ(b′_{2}, s) holds. Hence rk (Ψ(b′_{1}, M)&Ψ(b′_{1}, M)) = k, contradicting ¬F (b′_{1}, b′_{2}). Claim proved. Define

S^{0} = {s ∈ S : there is exactly one class F (b, M) ⊏ Ψ(M, s)}

and E on S^{0} to be

E(s_{1}, s_{2}) iff ∃b ∈ P F (b, M) ⊏ Ψ(M, s_{1}) ∩ Ψ(M, s_{2}).

By definition and claim 3 there is a one-to-one correspondence between Eclasses and F -classes. By Claim 2 Ψ(b_{i}, M) ⊏ E(s_{i}, M) for any generic s_{i} ∈ Ψ(b_{i}, M) for all i ≤ m. Thus Ψ(b_{i}, M) ⊏⊐ E(s_{i}, M). □

Proposition 4 Let S, P, Ψ ⊆ P × S be A-definable and

P = {b ∈ M^{n} : Ψ(b, M) ⊆ S & rk Ψ(b, M) = k &

∀c ∈ M^{n}(rk (Ψ(b, M) ∩ Ψ(c, M)) = k → (Ψ(b, M) ⊏ Ψ(c, M)))}.

Then there is an A-definable set L and an incidence relation I such that

Corollary 3 For A-definable S as above there is a finite A-definable set S_{0} such that any irreducible equirank subset of S almost coinsides with an Abdefinable subset for b ∈ S_{0}.

4 Trichotomy Theorem

Macro- and micro-geometries on a s.m.structure.

Proposition 5 Suppose there are a_{1}, a_{2}, b_{1}, b_{2}, c ∈ M every four of which are independent, c ∈ cl(a_{1}, a_{2}, b_{1}, b_{2}) and

cl(a_{1}, a_{2}, c) ∩ cl(b_{1}, b_{2}, c) = cl(c).

Then an incidence system (S, P, I) is definable in M with properties:

rk (S) = 2, Mdeg(S) = 1 rk (L) ≥ 2
rk (Ip) = 1 for all p ∈ P
if p_{1}, p_{2} ∈ P, p_{1} ≠ p_{2} then Ip_{1} ∩ Ip_{2} is finite or empty .

Proof Put S_{0} = M × M, P_{0} = M × M × M, and let I_{0} ⊆ S_{0} × P_{0} be an ∅-definable relation such that

⟨b_{1}, b_{2}⟩I_{0}⟨a_{1}, a_{2}, c⟩
⟨x_{1}, x_{2}⟩I_{0}⟨y_{1}, y_{2}, z⟩ → z ∈ cl(x_{1}, x_{2}, y_{1}, y_{2}).

A relation witnessing dependence between a_{1}, a_{2}, b_{1}, b_{2}, c has these properties. Denote p_{0} = ⟨a_{1}, a_{2}, c⟩. Then I_{0}p_{0} is an p_{0}-definable set of Morley rank 1.By the Finite Equivalence Relation Theorem using p_{0} one can define an equivalence relation E_{p_{0}} on I_{0}p_{0} with finitely many classes, and, say, m of them of rank 1 irreducible. Denote

I_{1} = {⟨s, p⟩ ∈ I_{0} : E_{p} is an equivalence relation with exactly m infinite classes and

s in one of them}.

Denote a binary relation E on I_{1} :

⟨s, p⟩E⟨s′, p′⟩ iff p = p′ & sE_{p}s′.

Define

P_{1} = I_{1}/E

and for q ∈ P_{1}, s ∈ S_{0} write sI_{2}q iff q = ⟨s, p⟩ for some p ∈ P_{0}. By definitions there is a canonical mapping

α : P_{1} → P_{0},

corresponding to the projection I_{1} → P_{0}, which is exactly m-to-one mapping. Also,

for all q ∈ P_{1} rk (I_{2}q) = 1.

By definitions, for q_{0} corresponding to p_{0} via α I_{2}q_{0} is irreducible. Define

P_{2} = {q ∈ P_{1} : ∀q′ ∈ P_{1}rk (I_{2}q ∩ I_{2}q′) = 1 → I_{2}q ⊏ I_{2}q′}.

It follows from the above remark that q_{0} ∈ P_{2} and for all q ∈ P_{2} rk (I_{2}q) = 1. Define an equivalence relation on P_{2}

qF q′ iff I_{2}q ⊏⊐ I_{2}q′.

We are now in the situation of Claim 2 of the proof of the preceeding Theorem. It follows that

qF ⊏ sI_{2} whenever s is generic in I_{2}q over q and s is generic in S_{0}. Define

P_{3} = P_{2}/F

and for p̄ ∈ P_{3}, s ∈ S_{0}

sI_{3}p̄ iff p̄ ⊏ I_{2}s.

From the above proved s_{0}I_{3}p̄_{0} holds, where p̄_{0} is obtained throughout the construction from p_{0}, s_{0}. Also, by the construction p̄_{0} ∈ cl(p_{0}). Since s_{0} ∈ I_{3}p̄_{0} it follows rk (I_{3}p̄_{0}) ≥ 1. On the other hand, if s ∈ I_{3}p̄_{0} is of maximal rank over p̄_{0} and q ∈ p_{0}F is of maximal rank over s, p̄_{0} then by definition sI_{2}p holds and q and s are independent over p̄_{0}. It follows rk (s/p̄_{0}) = rk (s/p̄_{0}, q) ≤ 1. Thus

rk (I_{3}p̄_{0}) = 1.

Let

P = {p̄ ∈ P_{3} : rk (I_{3}p̄_{0}) = 1}, S = {s ∈ S_{0} : ∃p̄ ∈ P sI_{3}p̄},
I = I_{3} ∩ (S × P).

Now we need to show that for distinct p̄_{1}, p̄_{2} from P Ip̄_{1} ∩ Ip̄_{2} is finite. So, suppose s is a point in the intersection. Choose ⟨q_{1}, q_{2}⟩ ∈ p_{1}F × p_{2}F of maximal rank over s, p̄_{1}, p̄_{2}. Then s ∈ I_{2}q_{1} ∩ I_{2}q_{2} and s is independent with q_{1}, q_{2} over p̄_{1}, p̄_{2}. Then

rk (s/p̄_{1}, p̄_{2}) = rk (s/p̄_{1}, p̄_{2}, q_{1}, q_{2}) < 1

since ¬q_{1}F q_{2}. To finish the proof we need to show that rk (P) ≥ 2 which would follow from rk (p̄_{0}/∅) ≥ 2. Suppose towards the contradiction rk (p̄_{0}/∅) ≤ 1. Then, since rk (s_{0}/p̄_{0}) = 1 < rk (s_{0}/∅), we have rk (p̄_{0}/s_{0}) < rk (p̄_{0}/∅), i.e. p̄_{0} ∈ cl(s_{0}) = cl(b_{1}, b_{2}). Then, from the assumptions of the proposition c ∉ cl(p̄_{0}). On the other hand p̄_{0} ∈ cl(p_{0}) = cl(a_{1}, a_{2}, c). It follows b_{1} ∉ cl(p_{0}), b_{2} ∉ cl(p_{0}). Therefore there exists c′ ∈ M such that

tp(cc′/p̄_{0}) = tp(b_{1}b_{2}/p̄_{0}) = tp(s_{0}/p̄_{0}.

Thus rk (cc′/p̄_{0}) = 1 and so

c′ ∈ cl(p̄_{0}, c) ⊆ cl(a_{1}, a_{2}, c) ∩ cl(b_{1}, b_{2}, c).

Hence cl(c′) = cl(c), contradicting cl(b_{1}) ≠ cl(b_{2}). □

Definition It is said, with a slight deviation from the standard terminology, that a pseudo-plane is definable in M if there is a two-sorted structure (S, P, I) definable in M_{A}, some A, with properties stated in the Proposition.

Definition An [abstract] projective geometry is a set of ’points’ and ’lines’ satisfying:

(i) through any two points there is a line; (ii) there are at least three points on every line; (iii) two distinct lines intersect in at most one point; (iv) for any distinct points a, b, c, d : if lines (a, b) and (c, d) intersect then lines (a, c) and (b, d) do. Any 3 points a, b, c of a projective geometry which do not lie on a common line generate a projective plane as the set of points

S(a, b, c) = ⋃{(a, z) : z ∈ (b, c)}.

By (iv) the plane generated by any non-collinear a′, b′, c′ ∈ S(a, b, c) coincides with S(a, b, c). The n-subspaces of a projective geometry are defined by induction as

S(a_{1}, …, a_{n+1}) = ⋃{(a_{n+1}, z) : z ∈ (a_{1}, …, a_{n})}

for a_{1}, …, a_{n+1} not in a (n − 1)-subspace. Again by axiom (iv) the definition is invariant on the choice of the points in the subspace.

Theorem 3 Any projective geometry of dimension greater than two (generated by no less than 4 points) is isomorphic to a projective geometry over a division ring.

Proof See [Se].□

Theorem 4 (weak form of Trichotomy Theorem) For any s.minimal M either a pseudo-plane is definable in M or one of the following hold:

(i) the geometry of M is trivial, i.e. for any X ⊆ M̂ cl(X) = (X) in M̂; (ii) the geometry of M is locally projective, i.e. for generic c ∈ M the geometry M̂_{c} is isomorphic to a projective geometry over a division ring.

Proof Assume no pseudo-plane is definable in M and c is a fixed generic element in M. Claim 1. For any x, y ∈ M and Z ⊆ M finite

x ∈ cl(y, c, Z) implies ∃z ∈ cl(c, Z) : x ∈ cl(y, z, c).

We may assume that Z is independent over c and proceed by induction on #Z. For #Z = 1 there is nothing to prove. Suppose Z = {z_{1}, z_{2}} ˙∪Z′, x ∈ cl(y, c, Z), y ∉ cl(c, Z). Then by the Proposition in M_{Z′} either

(i) some quadruple from x, y, z_{1}, z_{2}, c is dependent or (ii) ∃z ∈ M \ cl(c)

cl(z_{1}, z_{2}, c) ∪ cl(x, y, c) ⊇ cl(z, c).

In case (i) only x ∈ cl(y, z_{1}, z_{2}) is possible. Which means in M x ∈ cl(y, z_{1}, {z_{2}Z′}). Since #{z_{2}, Z′} < #Z by induction hypothesis there is z ∈ cl(z_{1}, z_{2}, Z′) : x ∈ cl(y, z_{1}, z). If then dim(y, z_{1}, z, c) = 3, we have x ∈ cl(y, z_{1}, c). or x ∈ cl(y, z, c) and we get the desired. Otherwise, there is a point z′_{1} ∈ cl(z_{1}, z, c) \ (cl(z_{1}, z) ∪ cl(c, z_{1}) ∪ cl(c, z)). Assuming x ∉ cl(y, z) we have then that in {x, y, z, z′_{1}, c} any four points are independent. Again, using the Proposition there must exist z′ ∈ M \ cl(c) such that

cl(z, z′_{1}, c) ∩ cl(x, y, c) ⊇ cl(z′, c).

Clearly z′ ∈ cl(c, Z) ∩ cl(x, y, c), so x ∈ cl(y, z′, c) and we are done. In case (ii) z ∈ cl(c, Z) and x ∈ cl_{Z′}(y, z, c), i.e. x ∈ cl(y, c{z, Z′}). By the induction hypothesis there is z′ ∈ cl(c, z, Z′) such that x ∈ cl(y, z′, c). Claim proved. Claim 2. If cl(x, y, c) = cl(x, c) = cl(y, c) for some x, y independent over c then the geometry M̂ is degenerate, i.e.

cl(x_{0}, …, x_{n}) = cl(x_{0}) ∪ … ∪ cl(x_{n})

for any x_{0}, …, x_{n} ∈ M. Indeed, under the assumption, cl(x_{0}, x_{1}, x_{2}) = cl(x_{0}, x_{1}) ∪ cl(x_{0}, x_{2}) for any independent triple. We show first that the claim is true for n = 1. Assume towards a contradiction y ∈ cl(x_{1}, x_{2})\(cl(x_{1})∪cl(x_{2})). Choose x_{0} ∉ cl(x_{1}, x_{2}). Then

y ∈ cl(x_{0}, x_{1}, x_{2}) = cl(x_{0}, x_{1}) ∪ cl(x_{0}, x_{2}).

But if y ∈ cl(x_{0}, x_{i}) for i = 1 or i = 2 then x_{0} ∈ cl(y, x_{i}) = cl(x_{1}, x_{2}), the contradiction. Now we proceed by induction on n. Suppose y ∈ cl(x_{0}, …, x_{n}). Then by Claim 1 there is x ∈ cl(x_{0}, …, x_{n−1}) such that y ∈ cl(x_{n}, x, x_{0}). From what is proved already y ∈ cl(x, x_{0}) ∪ cl(x_{n}, x_{0}). Hence y ∈ cl(x_{0}, …, x_{n−1}) ∪ cl(x_{0}, x_{n}) = cl(x_{0}, …, x_{n−1}) ∪ cl(x_{n}) ∪ cl(x_{n}) = cl(x_{0}) ∪ … ∪ cl(x_{n}). This finishes the proof of the claim and of the Theorem.□

Theorem 5 The geometry of a minimal locally finite structure is either trivial or isomorphic to an affine or a projective geometry over a finite field.

Proof First notice that the structure is saturated so s.minimal. Then we can use a result by Doyen and Hubaut which states that any finite locally projective geometry of dimension ≥ 4 and equal number of points on all lines is either affine or projective. Thus by the Trichotomy Theorem we need to prove only that there is no pseudo-plane in M. It is done by developing a combinatorial-geometric analysis of the pseudo-plane, assuming it exists. The main tool of the analysis is the notion of the ’degree of a line’ which is very similar to the degree of algebraic curve. See [Z] for the proof.□

Corollary 4 Any locally finite geometry satisfying homogeneity assumtion:any bijection between bases can be extended to an automorphism is either trivial or isomorphic to an affine or a projective geometry over a finite field.

Proof In an appropriate language such a geometry can be represented as a minimal locally finite structure.□

5 Categoricity in uncountable powers

Theorem 6 (Ehrenfeucht-Mostowski) If a countable theory T has infinite models than for any infinte cardinal κ there is a model M of T such that for any A ⊆ M the number of complete 1-types over A realised in M is of cardinality cardA + ℵ_{0}.

Proof See [CK].□

Definition A countable theory T is said to be ω-stable (or totally transcendental) if for any model M of T and any subset A ⊆ M the set S(A) of complete 1-types over A is of cardinality not bigger than cardA + ℵ_{0}.

Theorem 7 If a countable theory T is categorical in some uncountable cardinality κ then T is ω-stable.

Proof By Ehrenfeucht-Mostowski for any A ⊆ M with cardA < κ S(A) is of cardinality not bigger than cardA + ℵ_{0}. It follows also that M of cardinality κ is saturated. Consider A to be a countable saturated submodel A ≺ M ≺ M^{∗}, M^{∗} the monster model. Define for a topological space X the Cantor-Bendixson Derivative d(X) to be the subset of all the limit points in X. Then define by induction d^{0}(X) = X d^{α+1}(X) = d(d^{α}(X)) d^{λ}(X) = ⋂_{α<λ} d^{α}(X) for λ limit. For compact X d^{α}(X) is compact too. The α where the process is stabilised is called the Cantor-Bendixson rank of X. Then d^{α}(X) is empty or perfect (the perfect kernel), i.e. without isolated points. In our situation by cardinality arguments in the Stone topology the topological space S(A) must have Cantor-Bendixson rank less than ω_{1} and the perfect kernel empty. For an A-definable set U define CB(U) to be the Cantor-Bendixson rank of the Stone space

S_{U}(A) = {p ∈ S(A) : U ∈ P }.

Define for a complete type p ∈ S(A)

CB(p) = max{CB(U) : U ∈ p}.

By definition for any U there is only finite number of types of CB-rank equal to that of U and so there is a maximum for numbers m such that U has m disjoint A-definable subsets of the same CB-rank. We call U irreducible if m = 1. It follows from the definitions that for any U over A the definition of CB(U) can be given by induction as follows: CB(U) ≥ 1 iff U ≠ ∅ CB(U) ≥ α iff for any β < α there are infinitely many disjoint A-definable subsets of CB-rank greater or equal to β. This definition can be applied to any A ≺ M^{∗} and by induction on CB-rank of U it can be easily checked that it doesn’t depend on the choice of A, provided A is ω-saturated model containing parameters for U. On the other hand any complete type over such A is uniquely defined by U ∈ p if CB(p) = CB(U) and U is irreducible:

p = {ϕ(x) : CB(ϕ(M^{∗}) ∩ U) = CB(U)}.

It follows that the cardinality of S(A) is equal to cardA + ℵ_{0}.□

Remark The proof above gives properties of CB-rank in ω-stable theories.

Definition In a ω-stable theory Morley rank of definable subset U ⊆ M^{n} is defined as

rk (U) = CB(U) − 1

or equivalently by induction rk (U) ≥ 0 iff U ≠ ∅ rk (U) ≥ α iff for any β < α there are infinitely many disjoint A-definable subsets of Morley rank greater or equal to β.

Remark In any ω-stable theory there exists a definable set U of Morley rank 1 irreducible. Such a set with the induced structure is by definition strongly minimal.

Theorem 8 In any uncountably categorical theory Morley rank of any definable set is finite, and all properties of the rank proved for sets definable in s.minimal structures hold.

Proof In the literature

References

[CK] C.C.Chang, H.J.Kiesler,Model Theory, North Holland, Amsterdam, 1973

[Se] A.Seidenberg, Lectures in Projective Geometry, D.van Nostrand Co., 1962

[Z] B.Zilber,Uncountably Categorical Theories AMS Translations of Mathematical Monographs, v.117, Providence, R.I. 1993