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


The Hyper-Weak Distributive Law and Related Infinitary Games in
Boolean Algebras


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)".