Superstability in simple finitary AECs

Tapani Hyttinen¹ ; Meeri Kesälä¹

¹ Department of Mathematics and Statistics University of Helsinki P.O. Box 68 FI-00014 Helsinki, Finland

Fundamenta Mathematicae, Tome 195 (2007) no. 3, pp. 221-268.

Résumé

We continue the study of {finitary abstract elementary classes} beyond $\aleph_0$-stability. We suggest a possible notion of superstability for simple finitary AECs, and derive from this notion several good properties for independence. We also study constructible models and the behaviour of Galois types and {weak Lascar strong types} in this context.We show that superstability is implied by {a-categoricity} in a suitable cardinal. As an application we prove the following theorem: Assume that $(\mathbb{K},\preccurlyeq_\mathbb{K})$ is a simple, tame, finitary AEC, a-categorical in some cardinal $\kappa$ above the Hanf number such that $\mathop{\rm cf}\nolimits(\kappa)>\omega$. Then $(\mathbb{K},\preccurlyeq_\mathbb{K})$ is a-categorical in each cardinal above the Hanf number.

Détail

DOI : 10.4064/fm195-3-3

Keywords: continue study finitary abstract elementary classes beyond aleph stability suggest possible notion superstability simple finitary aecs derive notion several properties independence study constructible models behaviour galois types weak lascar strong types context superstability implied a categoricity suitable cardinal application prove following theorem assume mathbb preccurlyeq mathbb simple tame finitary aec a categorical cardinal kappa above hanf number mathop nolimits kappa omega mathbb preccurlyeq mathbb a categorical each cardinal above hanf number