Journal of Symbolic Logic 73 (1):238 - 260 (2008)

Abstract
We characterize the (κ, Λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games $\scr{G}_{<\mu}^{\kappa}(\lambda)$ , 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 Pκ+Λ in the ground model is preserved by B. In this vein, we develop the theory of κ-club and κ-stationary subsets of Pκ+Λ. We also construct Boolean algebras in which Player I wins $\scr{G}_{\kappa}^{\kappa}(\kappa ^{+})$ but the (κ, ∞, κ)-d.1, holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined
Keywords Boolean algebra   distributive law   game   κ-stationary set
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DOI 10.2178/jsl/1208358752
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References found in this work BETA

Perfect-Set Forcing for Uncountable Cardinals.Akihiro Kanamori - 1980 - Annals of Mathematical Logic 19 (1-2):97-114.
More Game-Theoretic Properties of Boolean Algebras.Thomas J. Jech - 1984 - Annals of Pure and Applied Logic 26 (1):11-29.
Stationary Logic and its Friends. II.Alan H. Mekler & Saharon Shelah - 1986 - Notre Dame Journal of Formal Logic 27 (1):39-50.

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When P(Λ) (Vaguely) Resembles Κ.Pierre Matet - 2021 - Annals of Pure and Applied Logic 172 (2):102874.

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