Symmetry and the union of saturated models in superstable abstract elementary classes
Annals of Pure and Applied Logic
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. Theoerem 1: Let 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 〈 M_i∈K_≥μ^+| i<θ<(M_i)^+〉 of μ^+-saturated models, _i<θM_i is μ^+-saturated. We also apply results of VanDieren's Superstability and Symmetry paper and use towers to transfer symmetry from μ^+ down to μ in abstract elementary classes which are both μ- and μ^+-superstable: Theorem 2: Suppose 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.