Symmetry and the union of saturated models in superstable abstract elementary classes

Our main result (Theorem 1) suggests a possible dividing line (μ-superstable + μ-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. Theorem 1Let K be an abstract elementary class with no maximal models of cardinality μ+ which satisfies the joint embedding and amalgamation properties. Suppose μ≥LS(K). If K is μ- and μ+-superstable and satisfies μ+-symmetry, then for any increasing sequence 〈Mi∈K≥μ+|i<θ<(sup⁡‖Mi‖)+〉 of μ+-saturated models, ⋃i<θMi is μ+-saturated. We also apply results of [18] and use towers to transfer symmetry from μ+ down to μ in abstract elementary classes which are both μ- and μ+-superstable: Theorem 2Suppose K is an abstract elementary class satisfying the amalgamation and joint embedding properties and that K is both μ- and μ+-superstable. If K has symmetry for non-μ+-splitting, then K has symmetry for non-μ-splitting.