Games and general distributive laws in Boolean algebras

Abstract:

The games $\mathcal {G}_{1}^{\eta }(\kappa )$ and $\mathcal {G}_{<\lambda }^{\eta }(\kappa )$ are played by two players in $\eta ^{+}$-complete and max$(\eta ^{+},\lambda )$-complete Boolean algebras, respectively. For cardinals $\eta ,\kappa$ such that $\kappa ^{<\eta }=\eta$ or $\kappa ^{<\eta }=\kappa$, the $(\eta ,\kappa )$-distributive law holds in a Boolean algebra $\mathbf {B}$ iff Player 1 does not have a winning strategy in $\mathcal {G}_{1}^{\eta }(\kappa )$. Furthermore, for all cardinals $\kappa$, the $(\eta ,\infty )$-distributive law holds in $\mathbf {B}$ iff Player 1 does not have a winning strategy in $\mathcal {G}_{1}^{\eta }(\infty )$. More generally, for cardinals $\eta ,\lambda ,\kappa$ such that $(\kappa ^{<\lambda })^{<\eta }=\eta$, the $(\eta ,<\lambda ,\kappa )$-distributive law holds in $\mathbf {B}$ iff Player 1 does not have a winning strategy in $\mathcal {G}_{<\lambda }^{\eta }(\kappa )$. For $\eta$ regular and $\lambda \le \text {min}(\eta ,\kappa )$, $\lozenge _{\eta ^{+}}$ implies the existence of a Suslin algebra in which $\mathcal {G}_{<\lambda }^{\eta }(\kappa )$ is undetermined.