There are probably dozens of [ways of defining "ultrafilter"](http://golem.ph.utexas.edu/category/2011/07/definitions_of_ultrafilter.html). The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. 


But there's another, shorter way to state the definition:



> 
> Let $X$ be a set. An **ultrafilter** on $X$ is a set $\mathcal{U}$ of subsets such that for all partitions
>  $$
> X = X\_1 \amalg \cdots \amalg X\_n
> $$
>  of $X$ into a finite number $n \geq 0$ of subsets, there is exactly one $i$ for which $X\_i \in \mathcal{U}$.
> 
> 
> 


I'd be amazed if this wasn't in the literature somewhere, but I haven't been able to track it down. Can anyone help?


Actually, there's an even more economical definition: instead of allowing $n$ to be any natural number, you take it to be 3. Thus, the condition is that whenever $X = A \amalg B \amalg C$, exactly one of $A$, $B$ and $C$ is in $\mathcal{U}$. (The same thing works with 4, or 5, etc., though not with 2.) I'm mostly interested in the version with arbitrary $n$, which seems more natural, but if you've seen the $n = 3$ version in the literature then I'd like to hear about that, too. 


**Edit** To be clear, when I use the word "partition" I don't mean to imply that the sets $X\_i$ are nonempty. I just mean a family of pairwise disjoint sets $X\_i$ whose union is $X$. They can be empty.



An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request
You can find that characterization (even with n = 3), as well as the generalization to $\kappa$-complete ultrafilters, in: Fred Galvin and Alfred Horn, Operations preserving all equivalence relations, Proc. Amer. Math. Soc. 24 (1970), 521-523.


