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.