Abstract

Let M be an arbitrary structure. Then we say that an M -formula φ defines a stable set inM if every formula φ ∧ α is stable. We prove: If G is an M -definable group and every definable stable subset of G has U -rank at most n , 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