What is "ultrafilter"

n. 1 (context mathematics of a set, whose subsets are partially ordered by inclusion English) A proper filter which has a law of dichotomy for complements. 2 (context mathematics order theory English) A filter (gloss: subset of a poset) that is maximal as a set with respect to the definition of (term: proper) (term filter).

In the mathematical field of set theory, an ultrafilter on a given partially ordered set (poset) P is a maximal filter on P, that is, a filter on P that cannot be enlarged. Filters and ultrafilters are special subsets of P. If P happens to be a Boolean algebra, each ultrafilter is also a prime filter, and vice versa.

If X is an arbitrary set, its power set ℘(X), ordered by set inclusion, is always a Boolean algebra, and (ultra)filters on ℘(X) are usually called "(ultra)filters on X".Ultrafilters have many applications in set theory, model theory, and topology. An ultrafilter on a set X may be considered as a finitely additive measure. In this view, every subset of X is either considered " almost everything" (has measure 1) or "almost nothing" (has measure 0).