Games and General Distributive Laws in Boolean Algebras

The games G 1 (κ) and G η <λ(κ) are played by two players in η +complete and max(η+ , λ)-complete Boolean algebras, respectively. For cardinals η, κ such that κ<η = η or κ<η = κ, the (η, κ)-distributive law holds in a Boolean algebra B iff Player 1 does not have a winning strategy in G 1 (κ). Furthermore, for all cardinals κ, the (η,∞)-distributive law holds in B iff Player 1 does not have a winning strategy in G 1 (∞). More generally, for cardinals η, λ, κ such that (κ<λ)<η = η, the (η, < λ, κ)-distributive law holds in B iff Player 1 does not have a winning strategy in G <λ(κ). For η regular and λ ≤ min(η, κ), ♦η+ implies the existence of a Suslin algebra in which G η <λ(κ) is undetermined.