We characterize the (κ,λ, <μ)-distributive law in Boolean algebras in terms of cut and choose games 𝔖κ<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=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 𝔖κκ(κ+) but the (κ,∞,κ)-d.l. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.
"κ-stationary subsets of 𝒫κ +λ, infinitary games, and distributive laws in Boolean algebras." J. Symbolic Logic 73 (1) 238 - 260, March 2008. https://doi.org/10.2178/jsl/1208358752