Building prime models in fully good abstract elementary classes

Abstract

We show how to build primes models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: $Theorem.$ Let $K$ be an almost fully good AEC that is categorical in $\text{LS}(K)$ and has the $\text{LS}(K)$-existence property for domination triples. For any $\lambda >\text{LS}(K)$, the class of Galois saturated models of $K$ of size $\lambda$ has prime models over every set of the form $M\cup \{a\}$. This generalizes an argument of Shelah, who proved the result when $\lambda$ is a successor cardinal.

15 pages. This was previously called "On prime models in totally categorical abstract elementary classes"

References in corpus (6)

• Building independence relations in abstract elementary classes
• Infinitary stability theory
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Chains of saturated models in AECs
• Downward categoricity from a successor inside a good frame
• Symmetry in abstract elementary classes with amalgamation

Cited by in corpus (4)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Shelah-Villaveces revisited