Four Games on Boolean Algebras

TL;DR

This paper investigates two infinite games played on complete Boolean algebras, establishing conditions under which White has winning strategies and connecting these to forcing and combinatorial principles.

Contribution

It characterizes winning strategies in the games G_2 and G_3 on Boolean algebras and links them to forcing properties and combinatorial set theory principles.

Findings

White has a winning strategy in G_2 iff in the cut-and-choose game G_c&c

White has a winning strategy in G_3 iff forcing produces a special subset of the binary tree

Under , there exist algebras where these games are undetermined

Abstract

The games G_2 and G_3 are played on a complete Boolean algebra B in \omega-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive p_n < p and Black chooses an i_n belonging to {0,1}. White wins G_2 iff liminf p_n^{i_n}=0 and wins 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 G_2 iff White has a winning strategy in the cut-and-choose game G_c&c introduced by Jech. Also, White has a winning strategy in the game G_3 iff forcing by B produces a subset R of the binary tree 2^{<\omega} containing either f^0 or f^1, for each f in 2^{<\omega}, and having unsupported intersection with each branch of the tree 2^{<\omega} belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G_3 played on B. It is shown that $\diamondsuit$ implies the existence of an algebra on which these games are undetermined.