Chains of saturated models in AECs

Abstract

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: $\mathbf{Theorem}$ If $K$ is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $\lambda$: * The union of an increasing chain of $\lambda$-saturated models is $\lambda$-saturated. * There exists a type-full good $\lambda$-frame with underlying class the saturated models of size $\lambda$. * There exists a unique limit model of size $\lambda$. Our proofs use independence calculus and a generalization of averages to this non first-order context.

27 pages

References in corpus (2)

• Building independence relations in abstract elementary classes
• Infinitary stability theory

Cited by in corpus (8)

• Building independence relations in abstract elementary classes
• $\mu$-Abstract Elementary Classes and other generalizations
• Saturation and solvability in abstract elementary classes with amalgamation
• Toward a stability theory of tame abstract elementary classes
• Abstract elementary classes stable in ${\aleph }_0$
• Building prime models in fully good abstract elementary classes
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Approximations of superstability in concrete accessible categories