Examples of non-locality

We use κ-free but not Whitehead Abelian groups to construct Abstract Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC’s which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is an AEC with amalgamation which is not (א0,א1)-tame but is (2א0 ,∞)tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ א2,≤ א2)-compact. A primary object of study in first order model theory is a syntactic type: the type of a over B in a model N is the collection of formulas φ(x,b) which are true of a in N . Usually the N is suppressed because a preliminary construction has established a universal domain for the investigation. In such a homogeneous universal domain one can identify the type of a over B as the orbit of a under automorphisms which fix B pointwise. An abstract elementary class is a pair (K,≺K ), a collection of structures of a fixed similarity type and a partial order on K which refines substructure and satisfies natural axioms which are enumerated in many places such as ([She87, Bal00, She99, Gro02, GV06b]. In this case, ‘Galois’ types (introduced in [She87] and named in [Gro02]) are defined as equivalence classes of triples (M,a,N) where a ∈ N −M under the equivalence relation generated by (M1, a1, N1) ∼ ∗Partially supported by NSF-0500841. †This is paper 862 in Shelah’s bibliography. The authors thank the Binational Science Foundation for partial support of this research.