A game on Boolean algebras describing the collapse of the continuum

Abstract

The game ${\mathcal{G}}_{\mathrm{ls}}$ is played on a complete Boolean algebra $\mathbb{B}$ in $\omega$-many moves. At the beginning White chooses a non-zero element $p$ of $\mathbb{B}$ and, in the $n$th move, White chooses a positive $p_n<p$ and Black responds by choosing an $i_n\in \{0,1\}$. White wins the play iff $lim\; sup{p}_n^{i_n}=0$. It is shown that White has a winning strategy in this game iff forcing by $\mathbb{B}$ collapses the continuum to $\omega$ in some generic extension. On the other hand, if a complete Boolean algebra $\mathbb{B}$ carries a strictly positive Maharam submeasure or contains a countable dense subset, then Black has a winning strategy in the game ${\mathcal{G}}_{\mathrm{ls}}$ played on $\mathbb{B}$. A Suslin algebra on which the game is undetermined is constructed and the game ${\mathcal{G}}_{\mathrm{ls}}$ is compared with the well-known cut-and-choose games ${\mathcal{G}}_{\mathrm{c\&c}}$, ${\mathcal{G}}_{\mathrm{fin}}\left(\lambda \right)$ and ${\mathcal{G}}_{\omega }\left(\lambda \right)$ introduced by Jech.