Equivalent Definitions of Superstability in Tame Abstract Elementary Classes

Abstract

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:Corollary 0.1. 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 λ, K has no long splitting chains.(2) For all high-enough λ, there exists a good λ-frame on a skeleton of K λ .(3) For all high-enough λ, K has a unique limit model of cardinality λ.(4) For all high-enough λ, K has a superlimit model of cardinality λ.(5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated. (6) There exists µ such that for all high-enough λ, K is (λ, µ)-solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Symmetry in abstract elementary classes with amalgamation

“…For the rest of this section, we will work with global non-weak tameness. Superstability has been studied together with amalgamation and tameness in works from the second author [Vas16b,Vas16a,BVa,GV]. We will use the following upward transfer of superstability:…”

“…However beyond Fact 6.8, the arguments of [BVa] (in particular the use of averages) have other applications (see for example the proof of solvability in [GV,Theorem 4.9]).…”

“…The definition below is already implicit in [SV99] and has since then been studied in many papers, e.g. [Van06,GVV16,Vas16a,BVa,GV]. We will use the definition from [Vas16a, Definition 10.1]:…”

Saturation and solvability in abstract elementary classes with amalgamation

“…We also introduce semisolvability, which only asks for the EM model to be universal (instead of superlimit). Both variations are equivalent to superstability in the first-order case (see [BGVV] and [GV,Corollary 5.3]). Shelah writes that solvability is perhaps the true analog of superstability in abstract elementary classes [She09, N §4(B)].…”

“…We can generalize Fact 2.5 using a weakening of categoricity called solvability (see Definition 3.1 here). Solvability was introduced by Shelah in [She09, Chapter IV] as a possible definition of superstability in the AEC framework (it is equivalent to superstability in the first-order case [GV,Corollary 5.3]). Shelah has asked [She09,Question N.4.4] whether the solvability spectrum satisfies an analog of the categoricity conjecture.…”