Tameness and frames revisited

Abstract

We study the problem of extending an abstract independence notion for types of singletons (what Shelah calls a good frame) to longer types. Working in the framework of tame abstract elementary classes, we show that good frames can always be extended to types of independent sequences. As an application, we show that tameness and a good frame imply Shelah's notion of dimension is well-behaved, complementing previous work of Jarden and Sitton. We also improve a result of the first author on extending a frame to larger models.

36 pages

References in corpus (6)

• Canonical forking in AECs
• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Forking and superstability in tame AECs
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Downward categoricity from a successor inside a good frame

Cited by in corpus (11)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Forking independence from the categorical point of view
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Universal classes near ${\aleph }_1$
• Abstract elementary classes stable in ${\aleph }_0$
• On categoricity in successive cardinals
• Tameness from two successive good frames
• Non-forking w-good frames
• Tameness, Uniqueness and amalgamation
• Density of uniqueness triples from the diamond axiom