alternative characterization of ultrafilter
Let $X$ be a set. A filter $\mathcal{F}$ over $X$ is an ultrafilter^{} if and only if it satisfies the following condition: if $A\coprod B=X$ (see disjoint union^{}), then either $A\in \mathcal{F}$ or $B\in \mathcal{F}$.
This result can be generalized somewhat: a filter $\mathcal{F}$ over $X$ is an ultrafilter if and only if it satisfies the following condition: if $A\cup B=X$ (see union), then either $A\in \mathcal{F}$ or $B\in \mathcal{F}$.
This theorem can be extended to the following two propositions^{} about finite unions:

1.
A filter $\mathcal{F}$ over $X$ is an ultrafilter if and only if, whenever ${A}_{1},\mathrm{\dots},{A}_{n}$ are subsets of $X$ such that ${\coprod}_{i=1}^{n}{A}_{i}=X$ then there exists exactly one $i$ such that ${A}_{i}\in \mathcal{F}$.

2.
A filter $\mathcal{F}$ over $X$ is an ultrafilter if and only if, whenever ${A}_{1},\mathrm{\dots},{A}_{n}$ are subsets of $X$ such that ${\bigcup}_{i=1}^{n}{A}_{i}=X$ then there exists an $i$ such that ${A}_{i}\in \mathcal{F}$.
Title  alternative characterization of ultrafilter 

Canonical name  AlternativeCharacterizationOfUltrafilter 
Date of creation  20130322 14:42:20 
Last modified on  20130322 14:42:20 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  13 
Author  yark (2760) 
Entry type  Theorem 
Classification  msc 54A20 