Ultrafilter

In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") ${\textstyle P}$ is a certain subset of ${\displaystyle P,}$ namely a maximal filter on ${\displaystyle P;}$ that is, a proper filter on ${\textstyle P}$ that cannot be enlarged to a bigger proper filter on ${\displaystyle P.}$

If ${\displaystyle X}$ is an arbitrary set, its power set ${\displaystyle {\mathcal {P}}(X),}$ ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on ${\displaystyle {\mathcal {P}}(X)}$ are usually called ultrafilters on the set ${\displaystyle X}$. An ultrafilter on a set ${\displaystyle X}$ may be considered as a finitely additive 0-1-valued measure on ${\displaystyle {\mathcal {P}}(X)}$. In this view, every subset of ${\displaystyle X}$ is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.

Ultrafilters have many applications in set theory, model theory, topology and combinatorics.