Equivalent definitions of superstability in tame abstract elementary classes

TL;DR

This paper proves the equivalence of multiple definitions of superstability in tame abstract elementary classes with a monster model, clarifying the concept's consistency and providing a unified framework for understanding superstability.

Contribution

It establishes the equivalence of various superstability notions in tame AECs with a monster model under stability assumptions, unifying previous disparate definitions.

Findings

No long splitting chains implies solvability.

Uniqueness of limit models implies no long splitting chains.

All definitions of superstability are equivalent in this context.

Abstract

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: $\mathbf{Corollary}$ Let $K$ be a tame AEC with a monster model. Assume that $K$ is stable in a proper class of cardinals. The following are equivalent: 1) For all high-enough $\lambda$, $K$ has no long splitting chains. 2) For all high-enough $\lambda$, there exists a good $\lambda$-frame on a skeleton of $K_\lambda$. 3) For all high-enough $\lambda$, $K$ has a unique limit model of cardinality $\lambda$. 4) For all high-enough $\lambda$, $K$ has a superlimit model of cardinality $\lambda$. 5) For all high-enough $\lambda$, the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 6) There exists $\mu$ such that for all high-enough $\lambda$, $K$ is $(\lambda, \mu)$-solvable. This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.