Mathematical logic seminar - October 30, 2002

General infinitary distributive laws generalize the basic finitary distributive law in Boolean algebras:

$(x_0 + y_0)(x_1 + y_1) = (x_0 \times y_0) + (x_1 \times y_0) + (x_0 \times y_1) + (x_1 \times y_1)$.

Infinitary 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 $(\omega,\infty)$-, $(\omega,\lambda)$-, weak$(\omega,\lambda)$-, and $(\omega,\lambda,\omega)$-distributive laws and related infintary games between two players. Foreman, Kamburelis, and others have added to this body of knowledege.

We will present some results of Jech, Foreman, and Kamburelis along with some of our own. We will also 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 will also show that, assuming $\kappa^{<\kappa}=\kappa$ and $\diamond_{\kappa^+}(E(\kappa))$ and certain assumptions about the relationships between $\kappa$ and $\lambda$, there are $\kappa^+$-Suslin algebras in which the games are undetermined. This is a correction to a previous "result" where we only assumed $\kappa$ regular and $\diamond_{\kappa^+}$ to construct such a $\kappa^+$-Suslin tree. Balcar pointed out the error and also that $\kappa^{<\kappa}=\kappa$ and $\diamond_{\kappa^+}(E(\kappa))$ suffice to construct a $\kappa^+$-Suslin tree. 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.