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 𝓖^k₁ (2)is undetermined. 𝓖^k₁(∞) is equivalent to G^II_k, the strategically closed forcing game. Under certain weak assumptions on cardinal arithmetic, Player II having a winning strategy for G^I_k implies 𝔹 has a dense subtree which is < k⁺-closed.