ultrafilter
Sign in to savethumb|Hasse diagram of the [[divisors of 210, ordered by the relation is divisor of, with the upper set ↑14 colored dark green. It is a , but not an , as it can be extended to the larger nontrivial filter ↑2, by including also the light green elements. Since ↑2 cannot be extended any further, it is an ultrafilter.]] In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P.
Article
10 sectionsContents
- Ultrafilters on partial orders
- {{vanchor|Types and existence of ultrafilters|Types}}
- Ultrafilter on a Boolean algebra
- Ultrafilter on the power set of a set
- Applications
- See also
- Notes
- References
- Bibliography
- Further reading
thumb|Hasse diagram of the [[divisors of 210, ordered by the relation is divisor of, with the upper set ↑14 colored dark green. It is a , but not an , as it can be extended to the larger nontrivial filter ↑2, by including also the light green elements. Since ↑2 cannot be extended any further, it is an ultrafilter.]] In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P.
If X is an arbitrary set, its power set {\mathcal P}(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on {\mathcal P}(X) are usually called X. An ultrafilter on a set X may be considered as a finitely additive 0-1-valued measure on {\mathcal P}(X). In this view, every subset of 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.