Ultrafilter

An ultrafilter is in mathematics, a lot of filters on a set, such that for every subset of either itself or its complement (difference amount ) is the amount of element filter. Ultrafilters are thus precisely those filters amount to which there is no real refinement. This definition of ultrafilters can be transferred from quantity filters to general filter in the sense of lattice theory.

Ultrafilter with the property that the intersection of all its elements is not empty, hot fixed ultrafilter point filter or elementary filters: They consist of all subsets that contain a given point. All ultrafilter on finite sets are fixed ultrafilter. Fixed filters are the only explicitly constructible ultrafilter. The second type of the ultrafilter, the free ultrafilter to the intersection of all of its elements is the empty set.

Ultrafilter find applications for instance in the topology and the model theory.

Formal definition and basic properties

It was a lot. A filter is a family of subsets with the following properties:

An ultra- filter is a filter having the feature:

This point can also be expressed that in the set of all filters is a maximum, is being used as an ordering on the inclusion, ie on the power set of the power set of. (Note: A filter is a subset of, and therefore is a member of. )

We have the following theorem: If a filter on the set. Then there exists an ultrafilter which includes the filter. Because a filter on the set is found on every non-empty set an ultrafilter.

Ultrafilter can be characterized by the following sentence:

Either on a filter. Then the following statements are equivalent (L1 ):

Furthermore: Are ultrafilter on a set, then they are equally powerful. This can be seen by the following figures a:

As well as

First we see that the images are well defined because of ( L1). One sees immediately. Thus, there are bijections.

Completeness

Under the completeness of an ultrafilter, understood to be the smallest cardinal number, so that the average of elements of the filter is not an element of the filter. This does not contradict the definition of ultrafilters, since after this just the average must be an element of the ultrafilter of finitely many elements of an ultrafilter. From the definition follows that the integrity of at least one ultrafilter. An ultrafilter whose completeness is greater than, that is uncountable, ie, countably complete, or - Complete, since each intersection of a countable (even countably ) many elements of the filter, again is an element of the filter.

Generalization of ultrafilters on posets

In the context of the more general definition of filters as a subset of a partially ordered set (for example, the power set with inclusion ) is called a filter ultrafilter if it as there is no finer filter, which already is not quite - expressed formally: if a filter is, then applies or. This general definition is consistent in the special case that the power set of a set, with the first given match. Using Zorn's lemma, it can be shown that each filter is contained in an ultrafilter.

Ultrafilter on associations

As a special case of the definition of partial orders gives a definition of associations. An ultrafilter on a bandage can be alternatively defined as the two-element Boolean algebra in Verbandshomomorphismus. A countably complete ultrafilter can be regarded as 0,1 -valued measure.

Species and existence of ultrafilters

There are two types of filters. To distinguish the following definition is used:

A filter is called free if it is, otherwise it is called fixed.

Easily one sees that ultrafilter are fixed on a finite set; on finite partially ordered sets have ultrafilter a smallest element, they can be as present for an item. More generally, on any amount: A filter is a fixed ultrafilter if and only if it satisfies one of the following equivalent conditions:

• There is a with.
• The filter has a finite element.

In this case, is the main element of the ultrafilter.

Free ultrafilters can only exist on infinite sets. It can be shown ( Ultrafilterlemma ) that each filter a lot (more generally, any subset for which the intersection of finitely many subsets of again in lies ) is contained in an ultrafilter of what secures the existence of free ultrafilters ( form about the kofiniten subsets an infinite set a filter, the free ultrafilters are precisely the ultrafilter, the upper filter this filter are ). However, the evidence this are not constructive and result, eg from the axiom of choice. Therefore can not be explicitly specified free ultrafilter, although most are free ultrafilters on infinite sets.

An example of fixed filters are around filter.

Examples

• On the empty set, there is only the empty filter, which is the empty set. This is thus an ultrafilter.
• Is a finite set, then every ultrafilter is fixed at exactly through a point. Would not that be so, and the filter would be fixed by the amount, so you could really refine it by adding. Thus, the ultrafilter are on a lot of just the point filter.
• Near a point in the filter topology is then just a ultrafilter if the point is isolated.

Applications

• In the model theory and universal algebra ultrafilters are used to define products and Ultra Ultra powers of algebraic structures. These constructions inherit this certain properties of the underlying structures.
• The for nonstandardanalysis basic hyper real numbers can be constructed as such ultra potency.
• In topology ultrafilter allow a characterization of compactness: A topological space is compact if every ultrafilter converges on it. This characterization can be used to prove the theorem of Tychonoff, which is fundamental for the set-theoretic topology.
• In metric geometry using ultrafilter to construct the asymptotic cone, an important tool to study the "large-scale geometry" of non-compact spaces.