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
$\mathcal{G}^{\kappa}_{<\mu}(\lambda)$.