κ-stationary subsets of 𝒫κ +λ, infinitary games, and distributive laws in Boolean algebras

Abstract

We characterize the (κ,λ, <μ)-distributive law in Boolean algebras in terms of cut and choose games 𝔖^κ_<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 𝒫_κ⁺λ in the ground model is preserved by 𝔹. In this vein, we develop the theory of κ-club and κ-stationary subsets of 𝒫_κ⁺λ. We also construct Boolean algebras in which Player I wins 𝔖^κ_κ(κ⁺) but the (κ,∞,κ)-d.l. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.