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 ${\displaystyle X}$.^{[note 1]} 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.^[1]: §4

Ultrafilters have many applications in set theory, model theory, topology^[2]: 186 and combinatorics.^[3]

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