Playful Boolean Algebras

We show that for an atomless complete Boolean algebra 8 of density < 2N°, the Banach-Mazur, the split and choose, and the Ulam game on S are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff B adds a real; Nonempty wins iff B has a (r-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáá. 0. Introduction. In [Jel and Je3] Jech initiated the study of game-theoretic properties of Boolean algebras. There are mainly two sources of ideas for defining these games. One is to consider classical games of Banach, Mazur, Mycielski, Ulam and others and to translate them into the Boolean algebraic context. For the origin of these games see the Scottish book [Ma, problems 43 and 67]. The other is to look at some well-known properties of Boolean algebras, such as various distributivity conditions, the existence of dense sets with certain closedness properties, Axiom A, properness and others (these can usually be characterized by properties of the corresponding forcing extensions) and to try to devise games that would reflect them. We are mainly interested in characterizing those Boolean algebras in which one of the players has a winning strategy in a certain game. The hope is that by doing this some new and interesting concepts and problems would emerge that would improve our understanding of the structure of Boolean algebras. Most of the results of this paper were motivated by a list of problems from [Je3]. Some of them we solve, and to others we give partial solutions. For example, we prove that if the Nonempty player has a winning strategy in the cut ¿c choose game on a complete Boolean algebra 8 which has a dense set of size < 2N°, then S has a (r-closed dense set. By this we improve a previous result of Foreman [Fo], and Vojtáá [Vo2], though their ideas are essential ingredients of our proof. We also give a consistent example of a poset P such that P x P is equivalent to CWl (the usual Cohen poset for adding a subset of u>i) but P is not. By a result from [Fo] this cannot happen under CH. As an application we give a proof of the following result of Gregory announced in [Gr]. If "ZFC +3 weakly compact cardinal" is consistent, then so is "ZFC+GCH+ Every N2-Suslin tree is essentially <r-closed". We show that Nonempty may or may not have a winning strategy in the both players cut k, choose game on Prikry forcing, depending on the model of set theory. Since, as is easily seen, Nonempty wins the ordinary cut &; choose game on Prikry forcing; this shows that it sometimes makes a difference if we require Nonempty to cut, too. Received by the editors May 15, 1985. 1980 Mathematics Subject Classification. Primary 03G05, 06E05, 06E10; Secondary 03E55.