Building prime models in fully good abstract elementary classes

TL;DR

This paper demonstrates how to construct prime models in saturated classes of an abstract elementary class with good independence properties, generalizing Shelah's earlier results to all sufficiently large cardinals.

Contribution

It extends the construction of prime models to a broader class of AECs with well-behaved independence, beyond successor cardinals, under certain categoricity and existence assumptions.

Findings

Prime models exist over any set of the form M ∪ {a} in the class of Galois saturated models.

Generalizes Shelah's result from successor cardinals to all larger cardinals.

Applicable to almost fully good AECs with categoricity and domination triple existence properties.

Abstract

We show how to build primes models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: $\mathbf{Theorem.}$ 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.