Submission date: 30 March 2015.

Abstract:

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. Under a natural superstability assumption (which follows from categoricity in a high-enough cardinal), we prove:
Theorem
If K is a tame superstable AEC with amalgamation, then for all high-enough λ:
* The union of an increasing chain of λ-saturated models is λ-saturated.
* There exists a type-full good λ-frame with underlying class the saturated models of size λ.
* There exists a unique limit model of size λ.

Our proofs use independence calculus and a generalization of averages to this non first-order context.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases:

Full text arXiv 1503.08781: pdf, ps.