Recent zbMATH articles in MSC 03C50https://zbmath.org/atom/cc/03C502024-03-13T18:33:02.981707ZWerkzeugSymmetry and the union of saturated models in superstable abstract elementary classeshttps://zbmath.org/1528.031572024-03-13T18:33:02.981707Z"VanDieren, M. M."https://zbmath.org/authors/?q=ai:vandieren.monica-mSummary: Our main result (Theorem 1) suggests a possible dividing line (\(\mu\)-superstable + \(\mu\)-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line.
\textbf{Theorem 1.} Let \(\mathcal{K}\) be an abstract elementary class with no maximal models of cardinality \(\mu^+\) which satisfies the joint embedding and amalgamation properties. Suppose \(\mu \geq \operatorname{LS}(\mathcal{K})\). If \(\mathcal{K}\) is \(\mu\)- and \(\mu^+\)-superstable and satisfies \(\mu^+\)-symmetry, then for any increasing sequence \(\langle M_i \in \mathcal{K}_{\geq \mu^+} \mid i < \theta <(\sup \| M_i \|)^+ \rangle\) of \(\mu^+\)-saturated models, \(\bigcup_{i < \theta} M_i\) is \(\mu^+\)-saturated.
We also apply results of [the author, Ann. Pure Appl. Logic 167, No. 12, 1171--1183 (2016; Zbl 1432.03057)] and use towers to transfer symmetry from \(\mu^+\) down to \(\mu\) in abstract elementary classes which are both \(\mu\)- and \(\mu^+\)-superstable:
\textbf{Theorem 2.} Suppose \(\mathcal{K}\) is an abstract elementary class satisfying the amalgamation and joint embedding properties and that \(\mathcal{K}\) is both \(\mu\)- and \(\mu^+\)-superstable. If \(\mathcal{K}\) has symmetry for non-\(\mu^+\)-splitting, then \(\mathcal{K}\) has symmetry for non-\(\mu\)-splitting.