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More ubiquitous undetermined games and other results on uncountable length games in Boolean algebras


Abstract


This paper surveys some of the known theory for countable length games related to distributive laws in Boolean algebras. The results can be naturally extended to uncountable length games, and detailed proofs are given. In particular, we show the following for uncountable length games related to distributive laws in Boolean algebras.When |k<k|=k there is a Boolean algebra in which 𝓖k1 (2)is undetermined. 𝓖k1(∞) is equivalent to GIIk, the strategically closed forcing game. Under certain weak assumptions on cardinal arithmetic, Player II having a winning strategy for GIk implies 𝔹 has a dense subtree which is < k+-closed.

DOI Code: 10.1285/i15900932v27supn1p65

Keywords: Boolean algebra; Distributive law; Game; k-stationary set

Classification: 03E05; 03E20; 03E35; 03E40; 03G05; 06E05; 06E10

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