Series: Penn State Logic Seminar Date: Tuesday, January 22, 2002 Time: 2:30 - 3:45 PM Place: 113 McAllister Building Speaker: Natasha Dobrinen, Mathematics, Penn State Title: General Infinitary Distributive Laws and Related Games in Boolean Algebras, part 1 Abstract: General infinitary distributive laws generalize the basic finitary distributive law in Boolean algebras: (x_0 + y_0)(x_1 + y_1)=x_0*y_0 + x_1*y_0 + x_0*y_1 + x_1*y_1. Distributive laws are of interest in their own right, and also for their equivalences to some useful forcing properties. Jech pioneered the connections between the (\kappa,\infty)-d.l., (\omega,\lambda)-d.l., the weak(\omega,\lambda)-d.l., and the (\omega,\omega,\lambda)-d.l. and related infintary games between two players. Foreman, Kamburelis, and Shelah soon added to this body of knowledege. We generalize some of their work to distributive laws with with first entry an arbitrary cardinal \kappa. Specifically, we show that for many pairs and triples of cardinals, each of the general distributive laws is equivalent to the non-existence of a winning strategy for Player 1 in a related game. Then for all regular cardinals \kappa (and certain \lambda and \eta), using the principle diamond_{\kappa^+} we construct a \kappa^+ -Suslin algebra in which neither player has a winning strategy in the games related to the (\kappa,\infty)-d.l., (\kappa,\lambda)-d.l., weak(\kappa,\lambda)-d.l., and the (\kappa,<\eta,\lambda)-d.l. This implies that for these cardinal pairs and triples, under \diamond_{\kappa^+}, the existence of a winning strategy for Player 2 in the related game is strictly stronger than the non-existence of a winning strategy for Player 1.