Ultrafilter - Types and Existence of Ultrafilters

Types and Existence of Ultrafilters

There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form F_a = {x | a ≤ x} for some (but not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. For the case of filters on sets, the elements that qualify as principals are exactly the one-element sets. Thus, a principal ultrafilter on a set S consists of all sets containing a particular point of S. An ultrafilter on a finite set is principal. Any ultrafilter which is not principal is called a free (or non-principal) ultrafilter.

Note that an ultrafilter on an infinite set S is non-principal if and only if it contains the Fréchet filter of cofinite subsets of S. This is obvious, since a non-principal ultrafilter contains no finite set, it means that, by taking complements, it contains all cofinite subsets of S, which is exactly the Fréchet filter.

One can show that every filter of a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice in the form of Zorn's Lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter of a finite poset (or on a finite set) is principal, since any finite filter has a least element.