Series: Penn State Logic Seminar

Date: Tuesday, January 29, 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 2


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.