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