Κ-stationary Subsets of Pκ+λ, Infinitary Games, and Distributive Laws in Boolean Algebras

We characterize the (κ, λ,< μ)-distributive law in Boolean algebras in terms of cut and choose games G <μ(λ), when μ ≤ κ ≤ λ and κ <κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, λ, μ with μ ≤ λ, under GCH. In the case when μ ≤ κ ≤ λ and κ = κ, we show that it is necessary to consider whether the κ-stationarity of Pκ+λ in the ground model is preserved by B. In this vein, we develop the theory of κ-club and κ-stationary subsets of Pκ+λ. We also construct Boolean algebras in which Player I wins G κ(κ +) but the (κ,∞, κ)-d.l. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined. §