CHAPTER 11 - CANONICAL BASES

Summary

Definition 11.1. The multiplicity of a type p(x) ∈ S(A) is the number Mlt(p) of its global nonforking extensions p(x) ∈ S(ℭ). If there is a proper class of global nonforking extensions of p, we say that p has unbounded multiplicity and we write Mlt(p) = ∞; otherwise we say that p has bounded multiplicity. A stationary type is a type of multiplicity 1. Thus over any B ⊇ A a stationary type p(x) ∈ S(A) has a unique nonforking extension q(x) ∈ S(B). We use the notation p|B for q.

Lemma 11.2. Let T be simple. If p ∈ S(A) is stationary, then its global nonforking extension is Definable over A.

Proof. Let p be the global nonforking extension of p, and let φ(x, y) ∈ L. We will show that p ↾ φ is A-definable. Let Δ_φ(y) and Δ_¬φ(y) be types over A given by Corollary 5.23 for p and φ and for p and ¬φ respectively. By compactness, the conjunction ψ(y) of a finite subset of Δ_φ(y) is inconsistent with Δ_¬φ(y). It is clear that ψ(y) defines p ↾ φ.

Corollary 11.3. Let T be simple. If types over models are stationary, then T is stable.

Proof. Lemma 11.2 implies that in this situation every global type is Definable.