Ultrafilter

This article deals with the mathematical term ultrafilter , for the technical filtration process see ultrafiltration

An ultrafilter is in the mathematics an amount filters on a quantity , so that for each subset of either itself or its complement element is the amount of filter. Ultra filters are therefore precisely those volume filters for which there is no real refinement. This definition of ultrafilters can be transferred from bulk filters to general filters in the sense of association theory . ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X\setminus A}$

Ultra filters with the property that the intersection of all their elements is not empty are called fixed ultrafilters , point filters or elementary filters : They consist of all subsets that contain a certain point. All ultrafilters on finite quantities are fixed ultrafilters . Fixed filters are the only ultrafilter that can be explicitly constructed. The second type of ultrafilter are the free ultrafilters , for which the intersection of all their elements is the empty set .

Ultra filters are used, for example, in topology and model theory .

The dual concept of the ultrafilter is that of the prime ideal .

Formal definition and basic properties

It is a lot. A filter is a family of subsets with the following properties: ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$

1. ${\displaystyle \emptyset \notin {\mathcal {F}},X\in {\mathcal {F}}}$
2. ${\displaystyle F\in {\mathcal {F}},G\supseteq F\Rightarrow G\in {\mathcal {F}}}$
3. ${\displaystyle F_{1},\dotsc ,F_{n}\in {\mathcal {F}}\Rightarrow \left(\bigcap \limits _{i=1}^{n}F_{i}\right)\in {\mathcal {F}}}$

An ultrafilter is a filter with the property: ${\displaystyle {\mathcal {F}}}$

4. Is filter with , the following applies . ${\displaystyle {\mathcal {G}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {G}}\supseteq {\mathcal {F}}}$ ${\displaystyle {\mathcal {G}}={\mathcal {F}}}$

This point can also be expressed in such a way that in the set of all filters on is maximal , whereby the inclusion on , i.e. on the power set of the power set of , is used as the order. (Note: A filter is a subset of and therefore an element of .) ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}({\mathcal {P}}(X))}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}(X)}$ ${\displaystyle {\mathcal {P}}({\mathcal {P}}(X))}$

The following sentence applies: If there is a filter on the set , then there is an ultrafilter that includes the filter . Since there is a filter on the set , there is an ultrafilter on every non-empty set. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle \{X\}}$ ${\displaystyle X}$

Ultra filters can be characterized by the following theorem:

Let there be a filter . Then the following statements are equivalent (L1): ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$

1. For all filter on with follows . ${\displaystyle {\mathcal {G}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {G}}\supseteq {\mathcal {F}}}$ ${\displaystyle {\mathcal {G}}={\mathcal {F}}}$
2. The following applies to all subsets : or . ${\displaystyle A,B\subset X}$ ${\displaystyle A\cup B\in {\mathcal {F}}\Rightarrow A\in {\mathcal {F}}}$ ${\displaystyle B\in {\mathcal {F}}}$
3. ${\displaystyle \forall A\subseteq X}$holds that either or . ${\displaystyle A\in {\mathcal {F}}}$ ${\displaystyle X\setminus A\in {\mathcal {F}}}$

Furthermore, if ultrafilters are on a set , then they are of equal power. This can be seen from the following figures: ${\displaystyle {\mathcal {F}}_{1},{\mathcal {F}}_{2}}$ ${\displaystyle X}$

${\displaystyle f_{1}\colon {\mathcal {F}}_{1}\rightarrow {\mathcal {F}}_{2},A\mapsto {\begin{cases}A,&{\text{wenn }}A\in {\mathcal {F}}_{2},\\X\setminus A,&{\text{wenn }}X\setminus A\in {\mathcal {F}}_{2}\end{cases}}}$ such as ${\displaystyle f_{2}\colon {\mathcal {F}}_{2}\rightarrow {\mathcal {F}}_{1},A\mapsto {\begin{cases}A,&{\text{wenn }}A\in {\mathcal {F}}_{1},\\X\setminus A,&{\text{wenn }}X\setminus A\in {\mathcal {F}}_{1}\end{cases}}}$

First you can see that the maps are well-defined because of (L1). One sees immediately and . Thus it is a question of bijections. ${\displaystyle f_{1}\circ f_{2}=\operatorname {id} _{{\mathcal {F}}_{1}}}$ ${\displaystyle f_{2}\circ f_{1}=\operatorname {id} _{{\mathcal {F}}_{2}}}$

completeness

The completeness of an ultrafilter is the smallest cardinal number so that there are elements of the filter whose average is not an element of the filter. This does not contradict the definition of an ultrafilter, since according to this only the average of a finite number of elements has to be contained in the filter again. From this prerequisite, however, it follows that the completeness of an ultrafilter is at least . An ultrafilter, the completeness of which is greater than , i.e. uncountable , is called countable complete or -complete , since every intersection of countable (also countable infinite) many elements of the filter is again an element of the filter. ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{0}}$ ${\displaystyle \sigma }$

Generalization of ultrafilters to half orders

In the context of the more general definition of a filter as a subset of a semi-ordered set (e.g. power set with inclusion) , a filter is called an ultrafilter if there is no finer filter than that which is not already whole - formally expressed: If a filter is open with , then the following applies or . This more general definition agrees with the first given in the special case that the power set is a set . With the help of Zorn's lemma it can be shown that every filter is contained in an ultrafilter. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {F'}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {F'}}}$ ${\displaystyle {\mathcal {F'}}={\mathcal {F}}}$ ${\displaystyle {\mathcal {F'}}={\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle X}$

Ultrafilter on bandages

As a special case of the definition of partial orders, there is a definition of associations . An ultrafilter on a association can be alternatively as Verbandshomomorphismus in the two-element Boolean algebra define. A countable, complete ultrafilter can be understood as a 0.1-valued measure . ${\displaystyle \{\bot ,\top \}}$

Types and existence of ultrafilters

There are two types of filters. The following definition is used to differentiate:

A filter is called free if it is, otherwise it is called fixed. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle \bigcap \limits _{F\in {\mathcal {F}}}F=\emptyset }$

It is easy to see that ultrafilters are fixed on a finite set; on finite, semi-ordered sets, ultrafilters have a smallest element ; they can be represented as for one element . More generally, in any amount: An ultrafilter on a fixed ultrafilter if and only if it meets one of the following equivalent conditions: ${\displaystyle {\mathcal {F}}_{a}=\{x:a\leq x\}}$ ${\displaystyle a}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$

• There is one with . ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {F}}={\mathcal {F}}_{x}:=\{F:x\in F\subseteq X\}}$
• The filter has a finite element.

In this case it is called the main element of the ultrafilter. ${\displaystyle x}$

Free ultrafilters can only exist on infinite quantities. It can be shown ( Tarski's ultrafilter set , English Tarski's ultrafilter theorem ) that every filter of a set (more generally: every subset for which the intersection of finitely many subsets of is again in ) is contained in an ultrafilter of . The proof of the ultrafilter set is not constructive and results from the application of Zorn's lemma, i.e. it assumes the assumption that the axiom of choice is valid . ${\displaystyle X}$ ${\displaystyle {\mathcal {Y}}\subseteq {\mathcal {P}}(X)}$ ${\displaystyle {\mathcal {Y}}}$ ${\displaystyle {\mathcal {Y}}}$ ${\displaystyle X}$

Ambient filters are an example of pinned filters .

Examples

• On the empty set there is only the empty filter, which is the empty set. This is therefore an ultrafilter. ${\displaystyle \emptyset }$
• If it is a finite set, then every ultrafilter is fixed at exactly one point. If this were not the case and the filter was fixed by the amount , it could be refined by adding real. Thus the ultrafilters are precisely the point filters on a finite set. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \{x_{1},...,x_{n}\}}$ ${\displaystyle {x_{n}}}$
• The environment filter of a point in the topology is an ultrafilter if and only if the point is isolated .

Applications

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

literature

• Boto von Querenburg : Set theoretical topology. 3rd revised and expanded edition. Springer-Verlag, Berlin et al. 2001, ISBN 3-540-67790-9 ( Springer textbook ).
• Paul Moritz Cohn : Universal Algebra (=  Mathematics and Its Applications . Volume 6 ). Revised edition. D. Reidel Publishing Company, Dordrecht, Boston 1981, ISBN 90-277-1213-1 .
• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set theoretical and algebraic topology. Heldermann, Lemgo 2006, ISBN 3-88538-115-X ( Berlin study series on mathematics 15), pp. 203ff. Chapter 13.
• Thomas Jech : Set Theory . The Third Millennium edition, revised and expanded (=  Springer Monographs in Mathematics ). Springer Verlag, Berlin, Heidelberg, New York 2003, ISBN 3-540-44085-2 .
• Horst Schubert : Topology (=  mathematical guidelines ). 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .

Individual evidence

1. ↑ Thomas Jech: Set Theory 2003, pp. 74 ff.
2. ↑ Jech, op.cit., P. 75.
3. ↑ In this way the existence of free ultrafilters is ensured. For example, the cofinite subsets of an infinite amount form a filter, the free ultrafilters are precisely the ultrafilters that are the upper filters of this filter.