Ultrafilter

Maximal proper filter

In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle P,} namely a maximal filter on P ; {\displaystyle P;} that is, a proper filter on P {\textstyle P} that cannot be enlarged to a bigger proper filter on P . {\displaystyle P.} If X {\displaystyle X} is an arbitrary set, its power set P ( X) , {\displaystyle {\mathcal {P}}(X),} ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on P ( X) {\displaystyle {\mathcal {P}}(X)} are usually called ultrafilters on the set X {\displaystyle X} . An ultrafilter on a set X {\displaystyle X} may be considered as a finitely additive 0-1-valued measure on P ( X) {\displaystyle {\mathcal {P}}(X)} .

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