### Definitions of Ultrafilter

#### Posted by Tom Leinster

One of these days I want to explain a precise sense in which the notion of ultrafilter is inescapable. But first I want to do a bit of historical digging.

If you’re subscribed to Bob Rosebrugh’s categories mailing list, you might have seen one of my historical questions. Here’s another: have you ever seen the following definition of ultrafilter?

Definition 1Anultrafilteron a set $X$ is a set $U$ of subsets with the following property: for all partitions $X = X_1 \amalg \cdots \amalg X_n$ of $X$ into a finite number $n \geq 0$ of subsets, there is precisely one $i$ such that $X_i \in U$.

This is equivalent to any of the usual definitions. It’s got to be in the literature somewhere, but I haven’t been able to find it. Can anyone help?

Just for fun, here’s a list of other equivalent definitions of ultrafilter. I wouldn’t be at all surprised if there’s some text where someone has compiled a similar list; but again, I haven’t found one.

Throughout, let $X$ be a set. I’ll write $P(X)$ for its power set.

Definition 2Anultrafilteron $X$ is a set $U$ of subsets with the following property: for all partitions $X = X_1 \amalg X_2 \amalg X_3$ of $X$ into three subsets, there is precisely one $i \in \{1, 2, 3\}$ such that $X_i \in U$.

This is the same as the first definition except that $n$ is constrained to be equal to $3$. You can do the same with $4, 5, \ldots$, but not $2$.

Another way of defining ultrafilter is in the spirit of my recent post on Hadwiger’s theorem. The idea is that an ultrafilter is a way of measuring the “size” of subsets of $X$.

Definition 3Anultrafilteron $X$ is a function $\phi: P(X) \to \{0, 1\}$ such that (i) $\phi$ is a valuation: $\phi(\emptyset) = 0$ and $\phi(Y \cup Z) = \phi(Y) + \phi(Z) - \phi(Y \cap Z)$ for all $Y, Z \subseteq X$, and (ii) $\phi(X) = 1$.

So an ultrafilter is almost the same thing as a $\{0, 1\}$-valued valuation. The only difference is the extra condition that $\phi(X) = 1$, which could equivalently be replaced with “$\phi$ is not identically zero”.

A valuation is something like a measure. Measures are closely related to integrals. So, we can try to come up with a way of defining ultrafilters so that they look like integrals. I had a go at that a year ago, but I think the following feels more authentically integral-esque.

Choose your favourite rig $k$. I’ll assume that your favourite rig has the property that there are no natural numbers $n \neq 1$ satisfying $n.1 = 1$ in $k$. For example, $k$ might be a field of characteristic zero, or $\mathbb{N}$, or $\mathbb{Z}$.

Definition 4Anultrafilteron $X$ is a $k$-linear function $\int: \{functions X \to k with finite image\} \to k$ such that $\int \lambda = \lambda$ for all $\lambda \in k$ (where the integrand is a constant function) and $\int f \in image(f)$ for all $f$.

Let’s look now at the classic way of defining “ultrafilter”. In fact there are two classic ways, closely related to each other. Before stating either, we need a preliminary definition. A **filter** on $X$ is a collection $F$ of subsets that
is upwards closed ($Y \supseteq Z \in F$ implies $Y \in F$) and closed under
finite intersections (or equivalently (i) $Y, Z \in F$ implies $Y \cap Z \in
F$, and (ii) $X \in F$).

Definition 5Anultrafilteron $X$ is a filter $U$ such that the only filters containing $U$ are $P(X)$ and $U$ itself.

In other words, an ultrafilter is a maximal proper filter. There is an alternative way of framing the maximality, which gives the other classic definition:

Definition 6Anultrafilteron $X$ is a filter $U$ such that for all $Y \subseteq X$, either $Y \in U$ or $X \setminus Y \in U$, but not both.

This is ripe for restating order-theoretically. We’ll use the inclusion ordering on $P(X)$, and we’ll use the two-element totally ordered set $2$. A filter on $X$ is nothing but a map $P(X) \to 2$ of meet-semilattices—that is, a map preserving finite meets ($=$ infs $=$ greatest lower bounds). An ultrafilter is a filter that, viewed as a map, also preserves complements.

Definition 7Anultrafilteron $X$ is a map $P(X) \to 2$ of Boolean algebras (or equivalently, of lattices).

We’re now getting into the realm of Stone duality (the equivalence between the category of Boolean algebras and the opposite of the category of totally disconnected compact Hausdorff spaces). So it’s no surprise that accompanying the Boolean algebra definition, there’s a topological definition:

Definition 8Anultrafilteron $X$ is a point of the Stone–Čech compactification of $X$.

Finally, here’s a definition I learned from the $n$Lab page on ultrafilters, but haven’t digested yet:

Definition 9Anultrafilteron $X$ is a set $U$ of subsets such that for all $Y \subseteq X$, $Y \in U \Leftrightarrow \forall n \geq 0, \forall Z_1, \ldots, Z_n \in U, Y \cap Z_1 \cap \cdots \cap Z_n \neq \emptyset.$

Anyway, the last eight of those nine were mostly for entertainment. What I’d most like is if someone can give me a reference for the first one. Thanks!

## Re: Definitions of Ultrafilter

Can you give me an example of an ultrafilter $U$ on a set $X$ that isn’t of the form $\exists x \in X$ such that $S \in U \Leftrightarrow x \in S$?