Stability results assuming tameness, monster model and continuity of nonsplitting

Abstract

Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>LS({\bf K})$ be a regular stability cardinal and let $\chi$ be the local character of $\mu$-nonsplitting. The following holds: 1. When $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension and continuity. It also has local character $\chi$. This generalizes Vasey's result which assumed $\mu$-superstability to obtain same properties but with local character $\aleph_0$. 2. There is $\lambda\in[\mu,h(\mu))$ such that if ${\bf K}$ is stable in every cardinal between $\mu$ and $\lambda$, then ${\bf K}$ has $\mu$-symmetry while $\mu$-nonforking in (1) has symmetry. In this case (a) ${\bf K}$ has the uniqueness of $(\mu,\geq\chi)$-limit models: if $M_1,M_2$ are both $(\mu,\geq\chi)$-limit over some $M_0\in K_\mu$, then $M_1\cong_{M_0}M_2$; (b) any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a $\mu^+$-saturated union. These generalize VanDieren-Vasey's result and remove the symmetry assumption in Boney-VanDieren and Vasey's result. Under $(<\mu)$-tameness, the conclusions of (1), (2)(a)(b) are equivalent to ${\bf K}$ having the $\chi$-local character of $\mu$-nonsplitting. Grossberg and Vasey gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the $U$-rank.