Voting Rules for Infinite Sets and Boolean Algebras

A voting rule in a Boolean algebra B is an upward closed subset that contains, for each element x ∈ B, exactly one of x and ¬x. We study several aspects of voting rules, with special attention to their relationship with ultrafilters. In particular, we study the set-theoretic hypothesis that all voting rules in the Boolean algebra of subsets of the natural numbers modulo finite sets are nearly ultrafilters. We define the notion of support of a voting rule and use it to describe voting rules that are, in a sense, as different as possible from ultrafilters. Finally, we consider how much of the axiom of choice is needed to guarantee the existence of voting rules.