Note di Matematica 27, suppl. n. 1, 2007, 65–83. More ubiquitous undetermined games and other results on uncountable length games in Boolean algebras Natasha Dobrinen i The Kurt G¨odel Research Center for Mathematical Logic, University of Vienna, W¨ahringerstrasse 25, 1090 Wien, Austria http://www.logic.univie.ac.at/ ~ dobrinen/ dobrinen@logic.univie.ac.at Current address: Department of Mathematics, University of Denver, Denver, CO 80113, USA ndobrine@du.edu Received: 03/06/2006; accepted: 27/11/2006. 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 uncount- able length games related to distributive laws in Boolean algebras. When |κ <κ | = κ, there is a Boolean algebra in which G κ 1 (2) is undetermined. G κ 1 (∞) is equivalent to G II κ , the strategically closed forcing game. Under certain weak assumptions on cardinal arithmetic, Player II having a winning strategy for G I κ implies B has a dense subtree which is <κ + -closed. Keywords: Boolean algebra, distributive law, game, κ-stationary set MSC 2000 classification: primary 03E05, 03E20, 03E35, 03E40, 03G05, 06E05, 06E10 1 Introduction Distributive laws are a useful tool for classifying and characterising Boolean algebras as well as giving information about their forcing extensions. They pro- vide a means of measuring how close to an algebra of sets a given Boolean al- gebra is: Every κ + -algebra of sets is (κ, κ)-distributive, and a complete Boolean algebra is completely distributive if and only if it is isomorphic to a power set algebra [15]. Measure algebras (those Boolean algebras obtained by taking the σ-algebra of measurable sets of some probability measure space and modding out by the null sets) are weakly (ω,ω)-distributive but not (ω, 2)-distributive. This provides one way of testing whether a Boolean algebra does not carry a strictly positive countably additive measure. By their forcing-equivalent state- i This research was supported by FWF grant P16334-N05. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Università del Salento: ESE - Salento University Publishing