A note on stable sets, groups, and theories with NIP

✍ Scribed by Alf Onshuus; Ya'acov Peterzil

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 152 KB
• Volume: 53
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200610046

✦ Synopsis

Abstract

Let M be an arbitrary structure. Then we say that an M ‐formula φ (x) defines a stable set in M if every formula φ (x) ∧ α (x, y) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G /H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure.

More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)