Four Games on Boolean Algebras

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.