Decomposition Theorem for Abstract Elementary Classes

elementary classes (AEC) were introduced by Shelah to generate a common framework to treat those classes of models that, in spite of being non-elementary, behave similarly to elementary classes. Part of the motivation was the study of classes of models for theories in infinitary languages. As stated in the introduction, Rami Grossberg and Olivier Lessmann presented an axiomatic framework which generalizes the decomposition theorem to AEC in [7]. This theorem states that for a class of models K satisfying the postulated axioms, every M ∈ K can be decomposed in a tree of small submodels such that M is prime and minimal over their union. This section is devoted to prove that theorem. There are two differences between [7] and the approach here outlined which are important to mention. Firstly, the choice of axioms is slightly different. In [7], the authors present three kinds of axioms postulating: the existence of an independence relation which is well-behaved over models; the existence— over certain sets—of a special kind of prime models, called primary, which are unique modulo isomorphism and behave well with respect to the independence relation; and finally, the existence of certain types, called regular, which also behave well with respect to the independence relation. The axioms proposed below will still postulate the existence of prime models over certain sets, but instead of assuming as an axiom the uniqueness of primary or prime models, an alternative axiom—related to types—will be posited. The motivation for this new axiom will be clear from the proofs. Secondly, an additional condition is added to the definition of decomposition. Although this additional condition does not change the main idea of the proposed decomposition theorem proof, it simplifies some of the argumentation in section 3. This axiom was suggested to me by Tapani Hyttinen. When primary models are not bona fide(see [7], p. 6), the new axiom is used to prove their uniqueness, so our choice for it still complies with the axiomatic framework suggested in [7]. It is important to notice that to relativize all the axioms from primary models to prime models makes essentially no changes to the proofs.