There are probably dozens of ways of defining "ultrafilter". 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

ultrafilteron $X$ 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.

everywherethat ultrafilters are discussed. $\endgroup$5more comments