We study the expansion of stable
structures by adding predicates for arbitrary subsets.
Generalizing work of Poizat-Bouscaren on the one hand and
Baldwin-Benedikt-Casanovas-Ziegler on the other we provide a
sufficient condition (Theorem 4.7) for such an expansion to be
stable. This generalization weakens the original definitions in
two ways: dealing with arbitrary subsets rather than just
submodels and removing the ‘small' or ‘belles paires' hypothesis.
We use this generalization to characterize in terms of pairs, the
‘triviality' of the geometry on a strongly minimal set (Theorem
2.5).
Call a set A benign if any type over A in the
expanded language is determined by its restriction to the base
language. We characterize the notion of benign as a kind of local
homogenity (Theorem 1.7). Answering a question
of [8] we characterize the property that M has
the finite cover property over A (Theorem 3.9).