Four Games on Boolean Algebras

Abstract

The games ${\cal G}_2$ and ${\cal G}_3$ are played on a complete Boolean algebra $\Bbb B$ in
$\omega$-many moves. At the beginning White picks a non-zero element $p$ of $\Bbb B$ and, in the $n$-th move,
White picks a positive $p_n<p$ and Black chooses an $i_n\in\{0,1\}$. White wins ${\cal G}_2$ iff $\liminf
p_n^{i_n}=0$ and wins ${\cal G}_3$ iff $\bigvee_{A\in [\omega ]^{\omega }}\bigwedge_{n\in A}p_n^{i_n}=0$. It is
shown that White has a winning strategy in the game ${\cal G}_2$ iff White has a winning strategy in the
cut-and-choose game ${\cal G}_{\rm{c\&c}}$ introduced by Jech. Also, White has a winning strategy in the game
${\cal G}_3$ iff forcing by $\Bbb B$ produces a subset $R$ of the tree ${}^{< \omega }2$ containing either
$\varphi ^{\smallfrown } 0 $ or $\varphi ^{\smallfrown }1$, for each $\varphi\in {}^{<\omega }2$, and having
unsupported intersection with each branch of the tree ${}^{< \omega }2$ belonging to $V$. On the other
hand, if forcing by $\Bbb B$ produces independent (splitting) reals then White has a winning strategy in the game ${\cal
G}_3$ played on $\Bbb B$. It is shown that $\diamondsuit$ implies the existence of an algebra on which these
games are undetermined.