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In mathematics, especially in order theory, an ultrafilter is a subset of a
partially ordered set which is maximal among all proper filters. Formally, this states that any filter that properly contains an ultrafilter has to be equal to
the whole poset. An important special case of the concept occurs if the considered poset is a Boolean algebra, like in the case of an ultrafilter on a set (defined as a filter of the
corresponding powerset). In this case, ultrafilters are characterized by
containing, for each element a of the Boolean algebra, exactly one of the elements a and ¬a (the
latter being the Boolean complement of a). This can be specialized by stating that an ultrafilter on a
set S is a subset of S that contains for each subset A of S either A or
S \ A. The possibility of including both is eliminated since filters on sets are always assumed to be
proper, i.e. not equal to the whole set. Since the common notions of filters of a poset and on a set differ
only very slightly, this article will always treat both cases in parallel, not without taking care of the fine nuances in
notation.
Another way of looking at ultrafilters on a set S is to define a function m on the power set of S by setting m(A) = 1 if A is contained in F and
m(A) = 0 otherwise. Then m is a finitely additive measure
on S, and every property of elements of S is either true almost everywhere or false almost everywhere.
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
Fa={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. Any ultrafilter which is not principal is called a
free (or non-principal) ultrafilter.
One can show that every filter 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. Consequently explicit examples of free ultrafilters cannot be given. 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.
Applications
Ultrafilters on sets are useful in topology, especially in relation to compact Hausdorff
spaces. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on posets are most
important if the poset is a Boolean algebra, since in this case the ultrafilters coincide with the prime filters. Ultrafilters in this form play a central role in Stone's
representation theorem for Boolean algebras.
The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely
related to the abovementioned representation theorem. For any element a of P, let
Da = { U in G | a in U }. This is most useful when P is
again a Boolean algebra, since in this situation the set of all Da is a base for a compact Hausdorff
topology on G. Especially, when considering the ultrafilters on a set S (i.e. the case that P is the
powerset of S ordered via subset inclusion), the resulting topological space is the Stone-Čech compactification of a discrete space of cardinality |S|.
Ultrafilters on sets are also used in the construction of hyperreal numbers.
Gödel's ontological proof of God's
existence uses as an axiom that the set of all "positive properties" is an ultrafilter.
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