Ultrafilter

A maximal proper filter on a set: a collection of subsets closed under finite intersection and supersets, not containing the empty set, and such that for every subset A either A or its complement belongs to the ultrafilter.

A maximal proper filter on a Boolean algebra or on the power set of a set: a collection of subsets closed under finite intersection and supersets that contains exactly one of each complementary pair; principal ultrafilters concentrate on a single point, nonprincipal ones do not.

A filter U on a set X that is maximal with respect to inclusion: no strictly larger filter on X contains U. Equivalently, for every subset A of X, either A ∈ U or X\A ∈ U, giving a two-valued decision of membership for each subset.