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

Abstract

AbstractWe characterize the (κ, λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games , 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 in the ground model is preserved by . In this vein, we develop the theory of κ-club and κ-stationary subsets of . We also construct Boolean algebras in which Player I wins but the (κ, ∞, κ)-d.1. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.