- Ultrafilter
In the mathematical field of

set theory , an**ultrafilter**on a set "X" is a collection ofsubset s of "X" that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as afinitely additive measure. Then every subset of "X" is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0). If "A" is a subset of "X", then either "A" or "X""A" is an element of the ultrafilter (here "X""A" is the relative complement of "A" in "X"; that is, the set of all elements of "X" that are not in "A"). The concept can be generalized to Boolean algebras or even to generalpartial order s, and has many applications in set theory,model theory , andtopology .**Formal definition**Given a set "X", an ultrafilter on "X" is a set "U" consisting of subsets of "X" such that

#The empty set is not an element of "U"

#If "A" and "B" are subsets of "X", "A" is a subset of "B", and "A" is an element of "U", then "B" is also an element of "U".

#If "A" and "B" are elements of "U", then so is the intersection of "A" and "B".

#If "A" is a subset of "X", then either "A" or $X\; setminus\; A$ is an element of "U". (Note: axioms 1 and 3 imply that "A" and $X\; setminus\; A$ cannot "both" be elements of "U".)A characterization is given by the following theorem.A filter "U" on a set "X" is an ultrafilter if one of the following conditions is true.

#There is no filter "F" finer than "U", $Usubset\; F$ implies $U=F$.

#$Acup\; Bin\; U$ implies $Ain\; U$ or $Bin\; U$.

#$forall\; Asubset\; X:\; Ain\; U$ or $Xsetminus\; A\; in\; U$. Another way of looking at ultrafilters on a set "X" is to define a function "m" on thepower set of "X" by setting "m"("A") = 1 if "A" is an element of "U" and "m"("A") = 0 otherwise. Then "m" is a finitely additive measure on "X", and every property of elements of "X" is either truealmost everywhere or false almost everywhere. Note that this does not define a measure in the usual sense, which is required to be "countably additive".For a filter "F" which is not an ultrafilter, one would say "m"("A") = 1 if "A" ∈ "F" and "m"("A") = 0 if "X""A" ∈ "F", leaving "m" undefined elsewhere.

**Completeness**The

**completeness**of an ultrafilter "U" on a set is the smallest cardinal κ such that there are κ elements of "U" whose intersection is not in "U". The definition implies that the completeness of any ultrafilter is at least $aleph\_0$. An ultrafilter whose completeness is "greater" than $aleph\_0$ — that is, the intersection of any countable collection of elements of "U" is still in "U" — is called**countably complete**or $sigma$**-complete**.The completeness of a countably complete nonprincipal ultrafilter on a set is always a

measurable cardinal .**Generalization to partial orders**In

order theory , an**ultrafilter**is asubset of apartially ordered set (a "poset") which is maximal among allproper filter s. 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, as in the case of an ultrafilter on a set (defined as a filter of the correspondingpowerset ). 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").Ultrafilters on a Boolean algebra can be identified with

prime ideal s,maximal ideal s, and homomorphisms to the 2-element Boolean algebra {true, false}, as follows:

*Maximal ideals of a Boolean algebra are the same as prime ideals.

*Given a homomorphism of a Boolean algebra onto {true, false}, the inverse image of "true" is an ultrafilter, and the inverse image of "false" is a maximal ideal.

*Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".

*Given an ultrafilter of a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".Let us see another theorem which could be used for definition of concept “ultrafilter”. Let

**B**denote a Bool-algebra and "F" a proper filter [*i.e. a filter "F" with the surplus restriction $0\; otin\; F$, i.e. being a filter that does not “degenerate” to coincide with the whole (universe of) the Boolean algebra*] in it. "F" is an ultrafilter iff::for all $a,b\; in\; mathbf\; B$, if $a\; vee\; b\; in\; F$, then $a\; in\; f$ or $b\; in\; f$(To avoid confusion: signs $0$, $vee$ are used here to denote operations of the Boolean algebra, and logical connectives are rendered by English circumlocutions.) See details (and proof) in [*[*] .*http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html A Course in Universal Algebra*] (written by [*http://www.math.uwaterloo.ca/~snburris/index.html Stanley N. Burris*] and H.P. Sankappanavar), Corrolary 3.13 on p. 149.**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 aleast 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.One can show that every filter (or more generally, any subset with the

finite intersection property ) is contained in an ultrafilter (seeUltrafilter lemma ) and that free ultrafilters therefore exist, but the proofs involve theaxiom of choice in the form ofZorn'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, and inmodel theory in the construction of ultraproducts and ultrapowers. 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 theprime filter s. Ultrafilters in this form play a central role inStone'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 "D"

_{"a"}= { "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 "D"_{"a"}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 resultingtopological space is theStone-ech compactification of a discrete space of cardinality |"S"|.The

ultraproduct construction in model theory uses ultrafilters to produceelementary extension s of structures. For example, in constructinghyperreal number s as an ultraproduct of thereal numbers , we first extend thedomain of discourse from the 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 we define the functions and relations "pointwise modulo "U", where "U" is an ultrafilter on the index set of the sequences; byŁoś' theorem , this preserves all properties of the reals that can be stated infirst-order logic . If "U" 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. These 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 ofultralimit s ofmetric space s.Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.**Ordering on ultrafilters****Rudin–Keisler ordering**is apreorder on the class of ultrafilters defined as follows: if "U" is an ultrafilter on "X", and "V" an ultrafilter on "Y", then $Vle\_\{RK\}U$ if and only if there exists a function "f": "X" → "Y" such that:$Cin\; Viff\; f^\{-1\}\; [C]\; in\; U$for every subset "C" of "Y".Ultrafilters "U" and "V" are

**Rudin–Keisler equivalent**, $Uequiv\_\{RK\}V$, if there exist sets $Ain\; U$, $Bin\; V$, and abijection "f": "A" → "B" which satisfies the condition above. (If "X" and "Y" have the same cardinality, the definition can be simplified by fixing "A" = "X", "B" = "Y".)It is known that $equiv\_\{RK\}$ is the kernel of $le\_\{RK\}$, i.e., $Uequiv\_\{RK\}V$ if and only if $Ule\_\{RK\}V$ and $Vle\_\{RK\}U$.

**Notes****See also***

Hyperreal number

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Ultrafilter**— Ein Ultrafilter ist in der Mathematik ein Mengenfilter auf einer Menge X, so dass für jede Teilmenge A von X entweder A selbst oder ihr Komplement X − A (Differenzmenge) Element des Mengenfilters ist. Ultrafilter sind somit genau diejenigen… … Deutsch Wikipedia**ultrafilter**— I. |əltrə+ noun Etymology: International Scientific Vocabulary ultra + filter : a dense filter used for the filtration of a colloidal solution that holds back the dispersed particles but not the liquid II. transitive verb : to cause to pass… … Useful english dictionary**Ultrafilter**— ypač tankus filtras statusas T sritis radioelektronika atitikmenys: angl. ultrafilter vok. Ultrafilter, n rus. фильтр сверхтонкой очистки, m pranc. ultrafiltre, m … Radioelektronikos terminų žodynas**ultrafilter**— ypač tankus filtras statusas T sritis radioelektronika atitikmenys: angl. ultrafilter vok. Ultrafilter, n rus. фильтр сверхтонкой очистки, m pranc. ultrafiltre, m … Radioelektronikos terminų žodynas**Ultrafilter**— ultrafiltras statusas T sritis Standartizacija ir metrologija apibrėžtis Įtaisas didesnėms kaip 0,1 μm dalelėms filtruoti. atitikmenys: angl. ultrafilter vok. Ultrafilter, n rus. ультрафильтр, m pranc. ultrafiltre, m … Penkiakalbis aiškinamasis metrologijos terminų žodynas**ultrafilter**— ultrafiltras statusas T sritis Standartizacija ir metrologija apibrėžtis Įtaisas didesnėms kaip 0,1 μm dalelėms filtruoti. atitikmenys: angl. ultrafilter vok. Ultrafilter, n rus. ультрафильтр, m pranc. ultrafiltre, m … Penkiakalbis aiškinamasis metrologijos terminų žodynas**ultrafilter**— ultrafiltration /ul treuh fil tray sheuhn/, n. /ul treuh fil teuhr/, n. 1. Physical Chem. a filter for purifying sols, having a membrane with pores sufficiently small to prevent the passage of the suspended particles. 2. Math. a filter in a… … Universalium**ultrafilter**— ul•tra•fil•ter [[t]ˌʌl trəˈfɪl tər[/t]] n. 1) chem. a filter for purifying sols, having a membrane with pores sufficiently small to prevent the passage of the suspended particles 2) chem. to purify by means of an ultrafilter • Etymology: 1905–10… … From formal English to slang**ultrafilter**— /ʌltrəˈfɪltə/ (say ultruh filtuh) noun 1. a filter for the separation of solute molecules of different sizes, using a semipermeable membrane filter through which the solute cannot pass. –verb (t) 2. to separate in an ultrafilter … Australian English dictionary**últrafílter**— tra m (ȗ í) teh. naprava za filtracijo, navadno pod zvišanim tlakom, ki se uporablja za ločevanje koloidnih delcev in makromolekulskih snovi od tekočine: ultracentrifuga in ultrafilter … Slovar slovenskega knjižnega jezika