Series: Penn State Logic Seminar Date: Tuesday, November 18, 2003 Time: 2:30 - 3:45 PM Place: 324 Sackett Building Speaker: Natasha Dobrinen, Penn State University, Mathematics Title: The Hyper-Weak Distributive Law and Related Infinitary Games in Boolean Algebras Abstract: The work we will present is joint with James Cummings. The hyper-weak distributive law in Boolean algebras, invented by Prikry, is a non-trivial generalization of the three-parameter distributive law. It fails in the Cohen algebra, but holds in many other Boolean algebras. We will define the hyper-weak distributive law and a related infinitary two-player game in Boolean algebras, and show some implications between the existence or non-existence of a winning strategy for either player and the hyper-weak distributive law. We will also show that it is consistent with ZFC that for all infinite cardinals kappa, for each infinite regular carinal nu less than or equal kappa there is a kappa^+-Suslin algebra containing a nu-closed dense subset in which many games of length greater than or equal nu are all undetermined. To do this, we use square_kappa and diamond_kappa^+(S) for all stationary sets S contained in kappa^+. This improves on an earlier result of Dobrinen (03) which showed that for regular cardinals kappa, there is consistently a large gap between the strengths of "B satisfies the (kappa,infinity)-d.l." and "player II has a winning strategy in the game G^kappa_1(infinity)".