Local homogeneity

We study the expansion of stable structures by adding predicates for arbitrary subsets. Generalizing work of Poizat-Bouscaren on the one hand and Baldwin-BenediktCasanovas-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). Let M be a stable structure in a language L and form an L(P ) = L∗-structure (M, A) by interpreting a new predicate P as the set A. When is the new structure stable? Clearly the structure induced on A (called Aind) must be stable and so it is natural to assume Aind is stable. But stability of Aind is not sufficient [12]. Poizat [12] and Bouscaren [6] restrict the question by assuming the set A is the universe of a submodel. Baldwin and Benedikt [3] restrict the question by assuming the set A is a set of L-indiscernibles. Casanovas and Ziegler [8] generalize ‘model’ and ‘indiscernible’ to ‘nfcp over A’. (nfcp is short for ‘does ∗Partially supported by CDRF grant KM2-2246. †Partially supported by NSF grant DMS-0100594 and CDRF grant KM2-2246.