Journal of Symbolic Logic

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

Natasha Dobrinen

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We characterize the (κ,λ, <μ)-distributive law in Boolean algebras in terms of cut and choose games 𝔖κ(λ), when μ≤κ≤λ and 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, 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 𝔖κκ+) but the (κ,∞,κ)-d.l. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.

Article information

J. Symbolic Logic, Volume 73, Issue 1 (2008), 238-260.

First available in Project Euclid: 16 April 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G05: Boolean algebras [See also 06Exx] 03E40: Other aspects of forcing and Boolean-valued models 03E05: Other combinatorial set theory 06E10: Chain conditions, complete algebras

Boolean algebra distributive law game κ-stationary set


Dobrinen, Natasha. κ-stationary subsets of 𝒫 κ + λ, infinitary games, and distributive laws in Boolean algebras. J. Symbolic Logic 73 (2008), no. 1, 238--260. doi:10.2178/jsl/1208358752.

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