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

摘要

Let M be an arbitrary structure. Then we say that an M -formula (x) defines a stable set in M if every formula (x) (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.