Let X be a set.


A collectionMathworldPlanetmath 𝒰 of subsets of X is an ultrafilter if 𝒰 is a filter, and whenever AX then either A𝒰 or XA𝒰.

Equivalently, an ultrafilter on X is a maximal (http://planetmath.org/MaximalElement) filter on X.

More generally, an ultrafilter of a latticeMathworldPlanetmath (http://planetmath.org/LatticeFilter) is a maximal proper filter of the lattice. This is indeed a generalizationPlanetmathPlanetmath, as an ultrafilter on X can then be defined as an ultrafilter of the power setMathworldPlanetmath 𝒫(X).

Types of ultrafilter

For any xX the set {AXxA} is an ultrafilter on X. An ultrafilter formed in this way is called a fixed ultrafilter, or a principal ultrafilter, or a trivial ultrafilter. Any other ultrafilter on X is called a free ultrafilter, or a non-principal ultrafilter. An ultrafilter on a finite setMathworldPlanetmath is necessarily fixed. On any infinite setMathworldPlanetmath there are free ultrafilters (in great abundance (http://planetmath.org/NumberOfUltrafilters)), but their existence depends on the Axiom of ChoiceMathworldPlanetmath, and so none can be explicitly constructed.

An ultrafilter 𝒰 on X is called a uniform ultrafilter if every member of 𝒰 has the same cardinality. (An ultrafilter on a singleton is uniform, but this is a degenerate case and is often excluded. All other uniform ultrafilters are free.)

Title ultrafilter
Canonical name Ultrafilter
Date of creation 2013-03-22 12:13:59
Last modified on 2013-03-22 12:13:59
Owner yark (2760)
Last modified by yark (2760)
Numerical id 11
Author yark (2760)
Entry type Definition
Classification msc 54A20
Related topic Filter
Related topic Ultranet
Related topic EveryBoundedSequenceHasLimitAlongAnUltrafilter
Related topic LatticeFilter
Defines fixed ultrafilter
Defines principal ultrafilter
Defines trivial ultrafilter
Defines free ultrafilter
Defines non-principal ultrafilter
Defines nonprincipal ultrafilter
Defines uniform ultrafilter