Ernest Schimmerling

### Mathematical logic seminar - November 20, 2003

 Time: 12 - 1:20 p.m. Room: CFA 110 Speaker: Natasha Dobrinen Chowla Research Assistant Professor Department of Mathematics Pennsylvania State University Title: The hyper-weak distributive law and related infinitary games in Boolean algebras Abstract: The work we will present is joint with James Cummings. The hyper-weak distributive law in Boolean algebras, invented by Prikry, is a non-trivial generalization of the three-parameter distributive law. It fails in the Cohen algebra, but holds in many other Boolean algebras. We will define the hyper-weak distributive law and a related infinitary two-player game in Boolean algebras, and show some implications between the existance or non-existance of a winning strategy for either player and the hyper-weak distributive law. We will also show that it is consistent with ZFC that for all infinite cardinals $\kappa$, for each infinite regular carinal $\nu \le \kappa$ there is a $\kappa^+$-Suslin algebra containing a $<\nu$-closed dense subset in which many games of length $\ge \nu$ are all undetermined. To do this, we use $\square_{\kappa}$ and $\diamond_{\kappa^+}(S)$ for all stationary sets $S$ contained in $\kappa^+$. This improves on an earlier result of Dobrinen (03) which showed that for regular cardinals $\kappa$, there is consistently a large gap between the strengths of "$B$ satisfies the $(\kappa,\infty)$-d.l." and "player II has a winning strategy in the game $\mathcal{G}^{\kappa}_1(\infty)$. Organizer's note: