Chains of saturated models in AECs

TL;DR

This paper investigates the conditions under which unions of saturated models remain saturated within tame AECs with amalgamation, establishing key properties like the existence of good frames and unique limit models.

Contribution

It proves that in tame AECs with amalgamation and superstability, unions of saturated models are saturated, and constructs a type-full good frame with unique limit models.

Findings

Unions of increasing chains of saturated models are saturated in tame AECs.

Existence of a type-full good λ-frame with saturated models.

Uniqueness of limit models of size λ.

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.