# ultrafilter

Let $X$ be a set.

## Definitions

A collection^{} $\mathcal{U}$ of subsets of $X$ is an *ultrafilter* if $\mathcal{U}$ is a filter, and whenever $A\subseteq X$ then either $A\in \mathcal{U}$ or $X\setminus A\in \mathcal{U}$.

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

More generally, an ultrafilter of a lattice^{} (http://planetmath.org/LatticeFilter)
is a maximal proper filter of the lattice.
This is indeed a generalization^{}, as an ultrafilter on $X$
can then be defined as an ultrafilter of the power set^{} $\mathcal{P}(X)$.

## Types of ultrafilter

For any $x\in X$ the set $\{A\subseteq X\mid x\in A\}$ 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 set^{} is necessarily fixed.
On any infinite set^{} there are free ultrafilters
(in great abundance (http://planetmath.org/NumberOfUltrafilters)),
but their existence depends on the Axiom of Choice^{},
and so none can be explicitly constructed.

An ultrafilter $\mathcal{U}$ on $X$ is called a *uniform ultrafilter*
if every member of $\mathcal{U}$ 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 |