Series: Penn State Logic Seminar Date: Tuesday, October 22, 2002 Time: 2:30 - 3:45 PM Place: 312 Boucke Building Speaker: Natasha Dobrinen, Mathematics, Penn State Title: Games and Generalized Distributive Laws in Boolean Algebras Abstract: Jech first introduced infinitary games played by two players in a Boolean algebra. He obtained a game theoretic characterization of the $(\omega,\infty)$-distributive law. He later generalized this work for other versions of distributivity, namely, the $(\omega,\kappa)$-d.l., the weak $(\omega,\kappa)$-d.l., and the $(\omega,\kappa,\omega)$-d.l., obtaining similar results. Kamburelis later solved an open problem of Jech regarding the weak distributive law, using a stationarity condition. Distributive laws in Boolean algebras are equivalent to important forcing properties of generic extensions of models of ZFC. For instance, the $(\kappa,\lambda)$-distributive law holds in a Boolean algebra B if and only if in each forcing extension of a model M of ZFC by B, for each function in M^B $f : \kappa -> \lambda$, there is a function in M $g :\kappa -> \lambda$ such that for each $\alpha < \kappa$, $f(\alpha) \in g(\alpha)$. We will present some of Jech's and Kamburelis' results along with some of our own. We will present a generalized notion of weak distributivity, namely the hyper-weak distributive laws, and show that for certain pairs of cardinal numbers, the $(\kappa,\lambda,\nu)$-d.l. and the hyper-weak $(\kappa,\lambda)$-d.l. are equivalent to the non-existence of a winning strategy for the first player in the appropriate games. Under GCH, this equivalence holds for all pairs $\kappa \geq \lambda$. We also will present a correction to a previous result. We previously had eroneously "showed" that for $\nu < min(\kappa,\lambda)$, assuming $\kappa$ regular and $\diamond_{\kappa^+}$ we can construct a $\kappa^+$-Suslin tree in which certain games are undetermined. Balcar showed us that this is false. However, Balcar also pointed out that $\kappa^{<\kappa}=\kappa$ and $\diamond_{\kappa^+}(E(\kappa))$ suffice to construct a $\kappa^+$-Suslin tree, and we have found that the previous construction in which the games are undetermined still can be carried out under these stronger assumptions. Similarly for the hyper-weak distributive laws.