1. Ultrafilters

A filter is the same thing as a dual ideal in the lattice of all subsets of a set--a power-set Boolean lattice. An ultrafilter is a maximal dual ideal in such a lattice.

Recall that in a Boolean algebra, choosing one prime ideal gets us a little burst of terminology: the prime ideal is also a maximal ideal, and its complement is a maximal dual ideal (an ultrafilter, if in a power-set lattice).

Now let $I = \omega = \{0,1,2,\ldots\}$ and let $B =$   Pow$(I)$. Recall that there are two kinds of prime ideals in $B$:

• Principal prime ideals. Each is generated by $I \setminus \{k\}$ for some $k$.

  The complement of such a prime ideal is the ``principal ultrafilter'' consisting of all subsets containing $\{k\}$.

• Non-principal prime ideals. They exist, since the ideal $I_0$ of all finite subsets of $I$, like any ideal in a distributive lattice, is the intersection of prime ideals, and none of them fit the description of a principal prime ideal. Conversely, any non-principal prime ideal must contain $I_0$; otherwise it would omit some $k$ and be contained in a principal prime ideal, and so, being maximal, would equal that ideal.

  There are $2^{2^{\aleph_0}}$ non-principal prime ideals, if we use the Axiom of Choice, but it is impossible to give even one explicitly!

We shall often treat $I$ as an index set.

Choose a prime ideal in Pow$(I)$ and keep it fixed for the rest of this discussion. We think of its members as ``small'' sets of indices. What is a ``large'' set of indices? There are two possible definitions:

(1) A large set of indices is a set of indices that is not small--a member of the corresponding ultrafilter;

(2) a large set of indices is the complement in $I$ of a small set of indices.

But these two definitions are equivalent! Recall that for a prime ideal in a Boolean lattice, for each $x$ exactly one of $x$ or $x'$ is in the ideal.

Question. For the principal prime ideal generated by $I \setminus \{k\}$, which subsets of $I$ are small and which large? (It is as if only $k$ counts for largeness.)

To summarize, if we have chosen a prime ideal, then with respect to it,

1. Every subset of $I$ is either large or small (not both).
2. The complement in $I$ of a large set is small and vice-versa.
3. The small sets form our chosen prime ideal, by definition. In particular,
   • the union of two small subsets is small;
   • a subset of a small subset is small.
   • The empty set is small.
4. If the prime ideal is nonprincipal, then any finite subset of $I$ is small.
5. The large sets form an ultrafilter. In particular,
   • the intersection of two large subsets is large;
   • a superset of a large subset is large.
   • $I$ itself is large.