From dobrinen@math.psu.edu Thu Oct 30 16:04:05 2003
Date: Thu, 30 Oct 2003 15:07:22 -0500 (EST)
From: Natasha L Dobrinen
To: Ernest Schimmerling
Cc: Natasha L Dobrinen
Subject: Title and Abstract
Dear Ernest,
Sorry for the delay. Here is my title and abstract.
Best,
Natasha
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 existance or non-existance 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 $\nu \le \kappa$
there is a $\kappa^+$-Suslin algebra containing a $<\nu$-closed dense
subset in which many games of length $\ge \nu$ 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 $(\kappa,\infty)$-d.l." and "player II has a winning strategy in the
game $\mathcal{G}^{\kappa}_1(\infty)$.