there is no proper filter on that properly extends (that is, such that is a proper subset of ).
Types and existence of ultrafilters
Every ultrafilter falls into exactly one of two categories: principal or free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form for some (but not all) elements of the given poset. In this case is called the principal element of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter.
For ultrafilters on a powerset a principal ultrafilter consists of all subsets of that contain a given element Each ultrafilter on that is also a principal filter is of this form.: 187 Therefore, an ultrafilter on is principal if and only if it contains a finite set.[note 2] If is infinite, an ultrafilter on is hence non-principal if and only if it contains the Fréchet filter of cofinite subsets of [note 3] If is finite, every ultrafilter is principal.: 187
If is infinite then the Fréchet filter is not an ultrafilter on the power set of but it is an ultrafilter on the finite–cofinite algebra of
Every filter on 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 (AC) 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). In general, proofs involving the axiom of choice do not produce explicit examples of free ultrafilters, though it is possible to find explicit examples in some models of ZFC; for example, Gödel showed that this can be done in the constructible universe where one can write down an explicit global choice function. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.
Ultrafilter on a Boolean algebra
An important special case of the concept occurs if the considered poset is a Boolean algebra. In this case, ultrafilters are characterized by containing, for each element of the Boolean algebra, exactly one of the elements and (the latter being the Boolean complement of ):
If is a Boolean algebra and is a proper filter on then the following statements are equivalent:
Given an arbitrary set its power set ordered by set inclusion, is always a Boolean algebra; hence the results of the above section apply. An (ultra)filter on is often called just an "(ultra)filter on ".[note 1] Given an arbitrary set an ultrafilter on is a set consisting of subsets of such that:
If is a subset of then either[note 4] or its complement is an element of .
An equivalent form of a given is a 2-valued morphism, a function on defined as if is an element of and otherwise. Then is finitely additive, and hence a content on and every property of elements of is either true almost everywhere or false almost everywhere. However, is usually not countably additive, and hence does not define a measure in the usual sense.
For a filter that is not an ultrafilter, one can define if and if leaving undefined elsewhere.
The set of all ultrafilters of a poset can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element of , let This is most useful when is again a Boolean algebra, since in this situation the set of all is a base for a compact Hausdorff topology on . Especially, when considering the ultrafilters on a powerset the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality
The ultraproduct construction in model theory uses ultrafilters to produce a new model starting from a sequence of -indexed models; for example, the compactness theorem can be proved this way.
In the special case of ultrapowers, one gets elementary extensions of structures. For example, in nonstandard analysis, the hyperreal numbers can be constructed as an ultraproduct of the real numbers, extending the domain of discourse from real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead the functions and relations are defined "pointwise modulo" , where is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If is nonprincipal, then the extension thereby obtained is nontrivial.
In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This construction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces.
In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for aggregating the preferences of infinitely many individuals. Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (for example, Kirman and Sondermann, 1972). Mihara (1997, 1999) shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.
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^ abIf happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on or an (ultra)filter just on is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter of a partial ordered set" vs. "on an arbitrary set"; i.e. they write "(ultra)filter on " to abbreviate "(ultra)filter of ".
^ is non-principal if and only if it contains no finite set, that is, (by Nr.3 of the above characterization theorem) if and only if it contains every cofinite set, that is, every member of the Fréchet filter.
^Properties 1 and 3 imply that and cannot both be elements of