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Games and general distributive laws in Boolean algebras

Authors:	Natasha Dobrinen

Institution:	Department of Mathematics, The Pennsylvania State University, 218 McAllister Building, University Park, Pennsylvania 16802

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.

Keywords:	Boolean algebra distributivity games

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