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

Abstract

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 1LetKbe an abstract elementary class with no maximal models of cardinalityμ+which satisfies the joint embedding and amalgamation properties. Supposeμ≥LS(K). IfKis μ- andμ+-superstable and satisfiesμ+-symmetry, then for any increasing sequence〈Mi∈K≥μ+|i<θ<(sup⁡‖Mi‖)+〉ofμ+-saturated models,⋃i<θMiisμ+-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 2SupposeKis an abstract elementary class satisfying the amalgamation and joint embedding properties and thatKis both μ- andμ+-superstable. IfKhas symmetry for non-μ+-splitting, thenKhas symmetry for non-μ-splitting.