Stability in

Abstract Elementary Classes

Monica VanDieren

Carnegie Mellon University

www.math.cmu.edu/~monicav

Abstract elementary classes

Definition. Let K be a class of structures all in the same similarity type L(K), and let ≺_{K} be a partial order on K. The ordered pair ⟨K, ≺_{K}⟩ is an abstract elementary class iff

A0 (Closure under isomorphism)

(a) For every M ∈ K and every L(K)-structure N if M ≅ N then N ∈ K.

(b) Let N_{1}, N_{2} ∈ K and M_{1}, M_{2} ∈ K such that there exist f_{l} : N_{l} ≅ M_{l} (for l = 1, 2) satisfying f_{1} ⊆ f_{2} then N_{1} ≺_{K} N_{2} implies that M_{1} ≺_{K} M_{2}.

A1 For all M, N ∈ K if M ≺_{K} N then M ⊆ N.

A2 Let M, N, M^{∗} be L(K)-structures. If M ⊆ N, M ≺_{K} M^{∗} and N ≺_{K} M^{∗} then M ≺_{K} N.

A3 (Löwenheim-Skolem) There exists a cardinal LS(K) ≥ +ℵ_{0} + |L(K)| such that for every M ∈ K and for every A ⊆ |M| there exists N ∈ K such that N ≺_{K} M, |N| ⊇ A and ∥N∥ ≤ |A| + LS(K).

A4 (Tarski-Vaught Chain)

(a) For every regular cardinal μ and every N ∈ K if {M_{i} ≺_{K} N : i < μ} ⊆ K is increasing (i.e. i < j ⟹ M_{i} ≺_{K} M_{j}) then ⋃_{i<μ} M_{i} ∈ K and ⋃_{i<μ} M_{i} ≺_{K} N.

(b) For every regular μ, if {M_{i} : i < μ} ⊆ K is increasing then ⋃_{i<μ} M_{i} ∈ K and M_{0} ≺_{K} ⋃_{i<μ} M_{i}.

In pure model theory, through solutions to �ǒs Conjecture, central concepts such as prime models, minimal types, super-stability, etc. were identified.

�ǒs conjecture.(1956) Let T be a first-order theory. If there exists λ > |T | + ℵ_{0} such that I(λ, T) = 1 then I(μ, T) = 1 holds for every μ > |T | + ℵ_{0}.

According to Shelah [Sh702] one of the most important directions of pure model theory is the classification of non-elementary classes. To that end, a good test-question should be proposed.

Shelah’s conjecture.(about 1977) Let T be a countable theory in L_{ω_{1},ω}. If there exists λ ≥ ℶ_{ω_{1}} such that I(λ, T) = 1 then I(μ, T) = 1 holds for every μ ≥ ℶ_{ω_{1}}.

Status - very open. Partial results:

1. Keisler (1971), under the additional assumption of existence sequentially homogeneous model.

2. Shelah (1984) [87a] and [87b] for excellent classes.

3. Lessmann (1998) for countable finite diagrams.

4. Makkai and Shelah [Sh285] under the additional assumption that both λ and μ are above a strongly compact cardinal, AND λ = χ^{+} .

* 5. Kolman-Shelah [Sh362] and Shelah [Sh 472] partial going down results for λ above measurable cardinal.

* 6. Shelah [Sh394] partial going down result for Abstract Elementary Classes with the Amalgamation Property.

Towards Stability Theory

1. Pillay for existentially closed structures.

2. Hrushovski for Robinson theories.

3. Buechler-Lessmann for sequentially homogeneous structures.

* 4. Shelah [Sh300] for universal classes.

* 5. Shelah [Sh600] for good frames.

Amalgamation Property

Notation. For M, N ∈ K a monomorphism f : M → N is called a ≺_{K}-embedding iff f[M] ≺_{K} N. Denote this by f : M ↪_{K} N.

K_{λ} := {M ∈ K | ∥M∥ = λ}.

Definition. Let (K, ≺_{K}) be an abstract elementary class, and let λ ≥ LS(K) be a cardinal.

1. We say that M_{0} ∈ K_{λ} is an amalgamation base iff for every M_{1}, M_{2} ∈ K_{λ} and every f_{ℓ} : M_{0} ↪_{K} M_{ℓ} for ℓ = 1, 2, there exists N ∈ K_{λ} so that there are g_{1} : M_{1} ↪_{K} N and g_{2} : M_{2} ↪_{K} N so that g_{1} ∘ f_{1} ↾ |M_{0}| = g_{2} ∘ f_{2} ↾ |M_{0}|. Namely, the following diagram is commutative:

\xymatrix{M_{1} \ar[r]^{g_{1}} & N \\ M_{0} \ar[u]^{f_{1}} \ar[r]_{f_{2}} & M_{2} \ar[u]_{g_{2}}}

2. We say that K satisfies the λ amalgamation property (λ- AP) iff for every M_{0} ∈ K_{λ}, M_{0} is an amalgamation base.

3. K satisfies the amalgamation property (AP) iff K satisfies the λ-AP for every λ > LS(K).

The notion of Galois types

Definition. K_{λ}^{3} := {(M, N, a) | a ∈ M; M ≺_{K} N; M, N ∈ K_{λ}}. Definition. Let M ∈ K_{λ}, We say that (M, N_{1}, a_{1}) and (M, N_{2}, a_{2}) ∈ K_{λ}^{3}, are ∼-related, written

(M, N_{1}, a_{1}) ∼ (M, N_{2}, a_{2}),

if there exists N ∈ K_{λ} and ≺_{K}-embeddings

h_{1} : N_{1} → N and h_{2} : N_{2} → N, such that

h_{1}(a_{1}) = h_{2}(a_{2})

and the following diagram commutes:

\xymatrix{N_{1} \ar[r]^{h_{1}} & N \\ M \ar[u]^{id} \ar[r]_{id} & N_{2} \ar[u]_{h_{2}}}

Lemma. Suppose that K_{λ} has the amalgamation property. For M ∈ K_{λ} the relation ∼ is an equivalence relation on K_{λ}^{3}.

Remark. Shelah and Villaveces are working towards Shelah’s Conjecture with only density of amalgamation bases [Sh635].

Types and Stability

Assume that K has the amalgamation property. Definition

1. For (M, N, a) ∈ K_{λ}^{3}, we let

tp(a/M, N) = (M, N, a)/ ∼ . This is the type of a over M in N.

2. For M ∈ K_{λ}, we let S(M) = { tp(a′/M, N′) : (M, N′, a′) ∈ K_{λ}^{3}}.

Definition. K is stable in λ iff |S(M)| ≤ λ for every M ∈ K_{λ}.

Dependence Relation: μ-splitting

Definition. A type p ∈ S(M) μ-splits over N iff there exists a ≺_{K}-mapping h and models N_{1}, N_{2} ∈ K_{μ} such that

1. N ≺_{K} N_{1}, N_{2} ≺_{K} M,

2. h : N_{1} ≅ N_{2},

3. h ↾ N = id_{N} and

4. p ↾ N_{2} ≠ h(p ↾ N_{1}).

Theorem. [Sh 394] Suppose that K is stable in μ and ⟨M_{i} | i ≤ κ⟩ is an ≺_{K}-increasing and continuous chain of models in K_{μ}. If p ∈ S(M_{κ}) and 2^{κ} > μ, then there exists i < κ such that p ↾ M_{i+1} does not μ-split over M_{i}.

Definition.[G-V] Define κ_{μ}(K) to be the minimal regular cardinal κ such that for every ⟨M_{i} ∈ K_{μ} | i ≤ κ⟩, ≺_{K}-increasing and continuous, and every p ∈ S(M_{κ}), there exists i < κ such that p does not μ-split over M_{i}. Otherwise, let κ_{μ}(K) = ∞.

Theorem.(Shelah-Villaveces) Under GCH, categoricity in some λ > μ > LS(K) implies that κ_{μ}(K) = ω.

Proposition.[Sh 394] Stability in μ implies that κ_{μ}(K) ≤ μ^{+} .

Question. Does stability, or even superstability, imply that κ_{μ}(K) < μ^{+} ?

Definition. K satisfies E_{μ}-equivalence iff for every M ∈ K and every p, q ∈ S(M),

p = q ⇔ [∀N ∈ K_{μ}, N ≺_{K} M](p ↾ N = q ↾ N).

Morley Sequences

Definition.[G-V] Let K be an AEC. The class is called μ-superstable iff there exists μ > LS(K) satisfying 1. K is stable in μ and

2. κ_{μ}(K) < μ^{+} .

Theorem.[G-V] Suppose K is μ-superstable for some μ ≥ LS(K) and K has the amagamation property. Let M ∈ K_{>μ}, A, I ⊂ M be given such that |I| ≥ μ^{+} > |A|. Then there exists J ⊂ I of cardinality μ^{+} , indiscernible over A. Moreover J can be chosen to be a Morley sequence over A.

Where

Definition {a_{i} | i < μ^{+} } is a Morley sequence over M_{0} iff there exists an increasing ≺_{K}-chain {M_{i} | i < μ^{+} } such that

1. p_{i} := tp(a_{i}/M_{i}, M_{i+1} ) does not μ-split over M_{0} and

2. ⟨a_{i} | i ∈ λ⟩ is an indiscernible sequence over M_{0}.

Outline of Proof. Given {a_{i} | i < μ^{+} ⟩ ⊆ I, define ⟨M_{i} ∈ K_{μ} | i < μ^{+} ⟩, a ≺_{K}-increasing and continuous chain of models such that

1. A ⊆ M_{0},

2. M_{i+1} is (μ, ω)-limit over M_{i} (substitute for relatively saturated) and

3. a_{i} ∈ M_{i+1} .

Define p_{i} := tp(a_{i}/M_{i}, M_{i+1} ).

Let f : S_{κ_{μ}(K)}^{μ^{+}} → μ^{+} be the mapping

f(i) := min{j < i | p_{i} does not μ − split over M_{j}}.

By Fodor’s Lemma, there exists a stationary S ⊆ S_{κ_{μ}(K)}^{μ^{+}}

and j_{0} < μ^{+} such that

i ∈ S ⇒ f(i) = j_{0}.

Since K is stable in μ, we can apply the pigeon-hole principle and find S^{∗} ⊆ S of cardinality μ^{+} and p^{∗} such that

i ∈ S^{∗} ⇒ p_{i} ↾ M_{j_{0}} = p^{∗}.

Enumerate and rename S^{∗}. Let M^{∗} := M_{1}. Again, by stability we can find J ⊂ S^{∗} of cardinality μ^{+} such that for every i ∈ S^{∗}, p^{∗∗} = p_{i} ↾ M^{∗}. Enumerate in increasing order and rename the elements in J.

Claim 1. For i < j ∈ J, p_{i} = p_{j} ↾ M_{i}.

Claim 2. {a_{i} | i ∈ J} are indiscernible over M_{0}.