Ultrafilter on a Boolean algebra

Summary

An ultrafilter on a Boolean algebra $B$ is a proper filter of $B$ that is maximal among proper filters. Equivalently, it is a filter that contains exactly one of $a$ and $\neg a$ for every $a \in B$. Ultrafilters are the same as homomorphisms from $B$ onto the two-element Boolean algebra, and they are the points of the Stone space of $B$. For the power set of a set $X$ one gets the ultrafilters on $X$.