A 0-definable set $ D $ in a structure $ M $ is said to be stably embedded if every $ M $-definable subset of $ D^n $, for any $ n $, is $ D $-definable.

(More generally, one can allow $ D $ to be type-definable, possibly over parameters. What is the definition in this case?)

In a stable theory, every definable set is stably embedded. This fact is sometimes called the Parameter Separation Theorem, e.g., by Bruno Poizat. More generally, in an arbitrary theory, the set of realizations of any stable type is stably embedded (right?). For example, any strongly minimal set, or set of Morley rank less than $ \infty $, is stably embedded.

An analogous result exists for o-minimal sets. Theorem 2 in Hasson and Onshuus's paper Embedded O-minimal Structures implies that if $ S $ is a 0-definable ordered set which, in the monster model, has the property that every definable subset of $ S^1 $ is a finite union of points and intervals, then $ S $ is stably embedded.

(Is every 0-definable set in an o-minimal theory stably embedded?)

In ACVF, the value group and the residue field are both stably embedded.

In the appendix to Chatzidakis and Hrushovski's paper Model theory of difference fields, several conditions equivalent to stable embeddedness are listed. Here is an incomplete list:

  • $ D $ is stably embedded
  • In a monster model $ M $, every elementary map from $ D(M) $ onto $ D(M) $ can be extended to an automorphism of $ M $.
  • Every complete type over $ D $ is definable over a small subset of $ D $.
  • Every complete type over $ D $ is implied by a partial type over a small subset of $ D $.

The assumption that enough sets are stably embedded plays a key role in the general means of getting binding groups from internality, as explained in Hrushovski's paper Groupoids, Imaginaries, and Internal Covers.