Series: Penn State Logic Seminar Date: Tuesday, April 6, 2004 Time: 2:30 - 3:45 PM Place: 307 Boucke Building Speaker: Natasha Dobrinen, Mathematics, Penn State Title: Kappa-Club Sets and Games in Boolean Algebras Abstract: We continue our work investigating infinitary games related to generalized distributive laws in Boolean algebras. Let B denote a complete Boolean algebra. In answer to an open problem of Jech [84], Kamburelis [94] proved that $B$ is weakly $(\omega,\lambda)$-distributive and preserves stationarity of $[\check{\lambda}]^{\le\omega}\cap V$ iff Player I does not have a winning strategy for the game $\mathcal{G}^{\omega}_{<\omega}(\lambda)$. As the cardinality of the allowable size of subsets of $\lambda$ increases, the generalization uses a property stronger than stationarity. We call a set $C\subset[\lambda]^{\le\kappa}$ $\kappa$-club if it is unbounded in $[\lambda]^{\le\kappa}$ and is closed under increasing chains of order type $\kappa$. A set $S\subset[\lambda]^{\le\kappa}$ is called $\kappa$-stationary if it meets every $\kappa$-club set. Generalizing and improving on the aforementioned result of Kamburelis, we show that (assuming $\mu\le\kappa=\kappa^{<\kappa}\le\lambda$ and $B$ is $(<\kappa,\kappa)$-distributive) $B$ is $(\kappa,\kappa,<\mu)$-distributive and preserves $\kappa$-stationarity of $[\check{\lambda}]^{\le\kappa}$ iff Player I does not have a winning strategy for the game $\mathcal{G}^{\kappa}_{<\mu}(\lambda)$.