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A [[Filter|filter]] which is maximal, in the sense that every filter containing it coincides with it. An ultrafilter may be defined as a system of subsets satisfying three conditions: 1) the empty set is not included; 2) the intersection of two subsets in the system again belongs to it; and 3) for any subset, either it or its complement belongs to the system.
 
A [[Filter|filter]] which is maximal, in the sense that every filter containing it coincides with it. An ultrafilter may be defined as a system of subsets satisfying three conditions: 1) the empty set is not included; 2) the intersection of two subsets in the system again belongs to it; and 3) for any subset, either it or its complement belongs to the system.
  
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "General topology" , ''Elements of mathematics'' , Springer  (1989)  pp. Chapts. 1–2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "General topology" , ''Elements of mathematics'' , Springer  (1989)  pp. Chapts. 1–2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Ultrafilters support a considerable body of theory both in general topology and in mathematical logic. For a topologist, they are primarily the elements of free compact spaces — that is, of the Stone–Čech compactifications (cf. [[Stone–Čech compactification|Stone–Čech compactification]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950201.png" /> of discrete spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950202.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950203.png" /> is a free compact Hausdorff space on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950204.png" /> of generators, just like a free group on a set of generators; the defining characteristic is that every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950205.png" /> from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950206.png" /> to a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950207.png" /> extends uniquely to a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950208.png" />.
+
Ultrafilters support a considerable body of theory both in general topology and in mathematical logic. For a topologist, they are primarily the elements of free compact spaces — that is, of the Stone–Čech compactifications (cf. [[Stone–Čech compactification|Stone–Čech compactification]]) $  \beta D $
 +
of discrete spaces $  D $.  
 +
$  \beta D $
 +
is a free compact Hausdorff space on the set $  D $
 +
of generators, just like a free group on a set of generators; the defining characteristic is that every mapping $  f $
 +
from the set $  D $
 +
to a compact Hausdorff space $  X $
 +
extends uniquely to a continuous mapping $  \beta f : \beta D \rightarrow X $.
  
To see that free ultrafilters are hard to describe, consider the mapping that assigns to each subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u0950209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502010.png" /> the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502011.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502013.png" /> is a free ultrafilter on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502014.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502015.png" /> is non-measurable.
+
To see that free ultrafilters are hard to describe, consider the mapping that assigns to each subset $  A $
 +
of $  \mathbf N $
 +
the number $  x _ {A} = \sum _ {n \in A }  2  ^ {-} n $
 +
in the interval $  [ 0, 1 ] $.  
 +
If u $
 +
is a free ultrafilter on $  \mathbf N $,  
 +
then the set $  \{ {x _ {A} } : {A \in u } \} $
 +
is non-measurable.
  
For a logician, ultrafilters are primarily the indexing structures over which ultraproducts are formed. A number of simple but important existential results in model theory are proved in a rather uniform way: To get a model for an infinite set of sentences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502016.png" />, form models for arbitrarily large finite subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502017.png" /> (which is often easy to prove possible) and take any ultraproduct of them. For more control of the construction, one uses restricted ultrafilters, for instance good ultrafilters, or uniform ultrafilters; see [[#References|[a1]]].
+
For a logician, ultrafilters are primarily the indexing structures over which ultraproducts are formed. A number of simple but important existential results in model theory are proved in a rather uniform way: To get a model for an infinite set of sentences $  S $,  
 +
form models for arbitrarily large finite subsets of $  S $(
 +
which is often easy to prove possible) and take any ultraproduct of them. For more control of the construction, one uses restricted ultrafilters, for instance good ultrafilters, or uniform ultrafilters; see [[#References|[a1]]].
  
 
For a discussion of models of set theory without free ultrafilters see [[#References|[a8]]].
 
For a discussion of models of set theory without free ultrafilters see [[#References|[a8]]].
  
There are two important partial orderings of isomorphism types of ultrafilters on a set, both originating in : the Rudin–Keisler order, defined over an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502018.png" />, and the Rudin–Frolik order, defined only over a countable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502019.png" />. Two ultrafilters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502021.png" />, i.e. two points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502022.png" />, are related by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502023.png" /> in the Rudin–Keisler order if there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502029.png" /> are said to be of the same type. The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502030.png" /> induces a partial ordering of the types. The Rudin–Frolik order over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502031.png" /> is defined similarly, but using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502032.png" /> with discrete image.
+
There are two important partial orderings of isomorphism types of ultrafilters on a set, both originating in : the Rudin–Keisler order, defined over an arbitrary set $  D $,  
 +
and the Rudin–Frolik order, defined only over a countable set $  \omega $.  
 +
Two ultrafilters $  p, q $
 +
on $  D $,  
 +
i.e. two points of $  \beta D $,  
 +
are related by $  p \leq  q $
 +
in the Rudin–Keisler order if there is a mapping $  f : D \rightarrow D \subset  \beta D $
 +
such that $  \beta f( q) = p $.  
 +
If $  p \leq  q $
 +
and $  q \leq  p $,  
 +
$  p $
 +
and $  q $
 +
are said to be of the same type. The relation $  p \leq  q $
 +
induces a partial ordering of the types. The Rudin–Frolik order over $  \omega $
 +
is defined similarly, but using $  f: \omega \rightarrow \beta \omega $
 +
with discrete image.
  
[[#References|[a1]]] is a rather full topologically oriented treatment of ultrafilter theory as of 1974, and still the best introduction to the subject. It has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502033.png" />-page chapter on large cardinals, a subject which has had near-revolutionary growth since 1974.
+
[[#References|[a1]]] is a rather full topologically oriented treatment of ultrafilter theory as of 1974, and still the best introduction to the subject. It has a $  40 $-
 +
page chapter on large cardinals, a subject which has had near-revolutionary growth since 1974.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502034.png" /> be an ultrafilter on an index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502035.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502036.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502037.png" /> be a set. Using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502038.png" /> one defines an equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502039.png" /> as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502041.png" />, are equivalent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502042.png" /> (written: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502043.png" />). The quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502045.png" /> by this equivalence relation is called the ultraproduct of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502046.png" /> (with respect to the ultrafilter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502047.png" />).
+
Let $  \Phi $
 +
be an ultrafilter on an index set $  I $.  
 +
For each $  i $,  
 +
let $  A _ {i} $
 +
be a set. Using $  \Phi $
 +
one defines an equivalence relation on $  \prod A _ {i} $
 +
as follows: $  a = ( a ( i) ) _ {i \in I }  $,  
 +
$  b = ( b ( i) ) _ {i \in I }  $,  
 +
are equivalent if and only if $  \{ {i } : {a( i) = b( i) } \} \in \Phi $(
 +
written: $  a \equiv b $).  
 +
The quotient $  ( \prod A _ {i} ) / \Phi $
 +
of $  \prod A _ {i} $
 +
by this equivalence relation is called the ultraproduct of the $  A _ {i} $(
 +
with respect to the ultrafilter $  \Phi $).
  
For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502048.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502049.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502050.png" />-ary relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502051.png" /> (eventually corresponding to one and the same predicate of a language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502052.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502053.png" /> are supposed to be interpretations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502054.png" />). Then a corresponding relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502055.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502056.png" /> is defined by:
+
For each $  i $,  
 +
let $  R _ {i} $
 +
be an $  n $-
 +
ary relation on $  A _ {i} $(
 +
eventually corresponding to one and the same predicate of a language $  L $,  
 +
where the $  ( A _ {i} , \{ R _ {i} \} ) $
 +
are supposed to be interpretations of $  L $).  
 +
Then a corresponding relation $  R $
 +
on $  ( \prod A _ {i} )/ \Phi $
 +
is defined by:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502057.png" /></td> </tr></table>
+
$$
 +
( \overline{a}\; {} _ {1} \dots \overline{a}\; {} _ {n} ) \in R  \iff \
 +
\{ {i } : {( a _ {1} ( i) \dots a _ {n} ( i) ) \in R _ {i} } \}
 +
\in \Phi .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502058.png" /> is the equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502060.png" />. (This is well defined by the properties of ultrafilters.) Functions and individual constants are similarly defined.
+
Here $  \overline{a}\; {} _ {m} $
 +
is the equivalence class of $  a _ {m} $
 +
in $  ( \prod A _ {i} ) / \Phi $.  
 +
(This is well defined by the properties of ultrafilters.) Functions and individual constants are similarly defined.
  
If all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502061.png" /> are the same, one speaks of ultrapowers instead of ultraproducts.
+
If all the $  A _ {i} $
 +
are the same, one speaks of ultrapowers instead of ultraproducts.
  
Ultraproducts have important applications in the theory of [[Diophantine equations|Diophantine equations]] and [[Algebraic number theory|algebraic number theory]]. For instance, for each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502062.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502063.png" /> be the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502064.png" />-adic numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502065.png" /> be the field of Laurent series over the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502066.png" />. Then the Ax–Kochen theorem says that for each non-principal ultrafilter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502067.png" />,
+
Ultraproducts have important applications in the theory of [[Diophantine equations|Diophantine equations]] and [[Algebraic number theory|algebraic number theory]]. For instance, for each prime number $  p $,  
 +
let $  \mathbf Q _ {p} $
 +
be the field of $  p $-
 +
adic numbers and let $  \mathbf F _ {p} (( t)) $
 +
be the field of Laurent series over the finite field $  \mathbf F _ {p} = \{ 0 \dots p - 1 \} $.  
 +
Then the Ax–Kochen theorem says that for each non-principal ultrafilter $  \Phi $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502068.png" /></td> </tr></table>
+
$$
 +
\prod _ { p } \mathbf Q _ {p} / \Phi  \simeq  \prod _ { p }
 +
\mathbf F _ {p} (( t)) / \Phi .
 +
$$
  
This gives an immediate positive partial solution to Artin's conjecture on Diophantine equations in the form of the theorem: For each positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502069.png" /> there exists a finite set of primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502070.png" /> such that every homogeneous polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502071.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502072.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502073.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502074.png" /> has a non-trivial zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502075.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502076.png" />. This result can also be deduced from results of Yu.L. Ershov ([[#References|[a6]]]), which also use ultraproducts in their proof. Artin's conjecture in full generality says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502077.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502079.png" />-field, which means that the conclusion just formulated must hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502080.png" /> (the  "2"  in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502081.png" />-field refers to the  "2"  in  "n&gt;d2" ). However, G. Terjanian gave in 1966 a counterexample to the full Artin conjecture by providing a quartic form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502082.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502083.png" /> with only non-trivial zeros.
+
This gives an immediate positive partial solution to Artin's conjecture on Diophantine equations in the form of the theorem: For each positive integer $  d $
 +
there exists a finite set of primes $  P( d) $
 +
such that every homogeneous polynomial $  f ( X _ {1} \dots X _ {n} ) $
 +
of degree $  d $
 +
over $  \mathbf Q _ {p} $
 +
with $  n > d  ^ {2} $
 +
has a non-trivial zero in $  \mathbf Q _ {p} $
 +
for all $  p \notin P( d) $.  
 +
This result can also be deduced from results of Yu.L. Ershov ([[#References|[a6]]]), which also use ultraproducts in their proof. Artin's conjecture in full generality says that $  \mathbf Q _ {p} $
 +
is a $  C _ {2} $-
 +
field, which means that the conclusion just formulated must hold for all $  p $(
 +
the  "2"  in $  C _ {2} $-
 +
field refers to the  "2"  in  "n&gt;d2" ). However, G. Terjanian gave in 1966 a counterexample to the full Artin conjecture by providing a quartic form in $  18 $
 +
variables over $  \mathbf Q _ {2} $
 +
with only non-trivial zeros.
  
More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502085.png" /> be integers. Then a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502086.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502088.png" />-field if every homogeneous polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502089.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502090.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502091.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502092.png" /> variables has a non-trivial zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502093.png" />. A field that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502094.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502095.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502097.png" />-field. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502098.png" />-fields are the algebraically closed fields. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u09502099.png" />-fields are also called quasi-algebraically closed. The rational functions in one variable over an algebraically closed field form a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u095020100.png" />-field (Tsen's theorem). The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u095020101.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u095020102.png" />-field (H. Hasse, 1923) and also a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u095020103.png" />-field (D.J. Lewis, 1952).
+
More precisely, let $  i \geq  0 $,  
 +
$  d \geq  1 $
 +
be integers. Then a field $  F $
 +
is called a $  C _ {i} ( d) $-
 +
field if every homogeneous polynomial $  f( X _ {1} \dots X _ {n} ) $
 +
of degree $  d $
 +
over $  F $
 +
in $  n > d  ^ {i} $
 +
variables has a non-trivial zero in $  F $.  
 +
A field that is $  C _ {i} ( d) $
 +
for all $  d \geq  1 $
 +
is called a $  C _ {i} $-
 +
field. The $  C _ {0} $-
 +
fields are the algebraically closed fields. The $  C _ {1} $-
 +
fields are also called quasi-algebraically closed. The rational functions in one variable over an algebraically closed field form a $  C _ {1} $-
 +
field ([[Tsen's theorem]]). The field $  \mathbf Q _ {p} $
 +
is a $  C _ {2} ( 2) $-
 +
field (H. Hasse, 1923) and also a $  C _ {2} ( 3) $-
 +
field (D.J. Lewis, 1952).
  
Other important applications of ultraproducts are in [[Non-standard analysis|non-standard analysis]]; in particular, non-standard models of the reals, integers, etc. can be obtained as ultrapowers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u095020104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095020/u095020105.png" />, etc.
+
Other important applications of ultraproducts are in [[Non-standard analysis|non-standard analysis]]; in particular, non-standard models of the reals, integers, etc. can be obtained as ultrapowers of $  \mathbf R $,  
 +
$  \mathbf Z $,  
 +
etc.
  
 
Cf. [[Model theory|Model theory]] and [[#References|[a2]]] for results in logic involving ultrafilters and ultraproducts.
 
Cf. [[Model theory|Model theory]] and [[#References|[a2]]] for results in logic involving ultrafilters and ultraproducts.

Latest revision as of 08:27, 6 June 2020


A filter which is maximal, in the sense that every filter containing it coincides with it. An ultrafilter may be defined as a system of subsets satisfying three conditions: 1) the empty set is not included; 2) the intersection of two subsets in the system again belongs to it; and 3) for any subset, either it or its complement belongs to the system.

All ultrafilters are divided into two classes: trivial (or fixed or principal) and free ultrafilters. An ultrafilter is called trivial or principal if it is the system of all subsets containing a given point; such an ultrafilter is also called fixed in that point. An ultrafilter is called free if the intersection of all its elements is the empty set, in other words, if it is not fixed in any point. The existence of free ultrafilters is unprovable without the axiom of choice.

For every filter there is an ultrafilter containing it; moreover, every filter is precisely the intersection of all the ultrafilters containing it.

References

[1] N. Bourbaki, "General topology" , Elements of mathematics , Springer (1989) pp. Chapts. 1–2 (Translated from French)
[2] K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)

Comments

Ultrafilters support a considerable body of theory both in general topology and in mathematical logic. For a topologist, they are primarily the elements of free compact spaces — that is, of the Stone–Čech compactifications (cf. Stone–Čech compactification) $ \beta D $ of discrete spaces $ D $. $ \beta D $ is a free compact Hausdorff space on the set $ D $ of generators, just like a free group on a set of generators; the defining characteristic is that every mapping $ f $ from the set $ D $ to a compact Hausdorff space $ X $ extends uniquely to a continuous mapping $ \beta f : \beta D \rightarrow X $.

To see that free ultrafilters are hard to describe, consider the mapping that assigns to each subset $ A $ of $ \mathbf N $ the number $ x _ {A} = \sum _ {n \in A } 2 ^ {-} n $ in the interval $ [ 0, 1 ] $. If $ u $ is a free ultrafilter on $ \mathbf N $, then the set $ \{ {x _ {A} } : {A \in u } \} $ is non-measurable.

For a logician, ultrafilters are primarily the indexing structures over which ultraproducts are formed. A number of simple but important existential results in model theory are proved in a rather uniform way: To get a model for an infinite set of sentences $ S $, form models for arbitrarily large finite subsets of $ S $( which is often easy to prove possible) and take any ultraproduct of them. For more control of the construction, one uses restricted ultrafilters, for instance good ultrafilters, or uniform ultrafilters; see [a1].

For a discussion of models of set theory without free ultrafilters see [a8].

There are two important partial orderings of isomorphism types of ultrafilters on a set, both originating in : the Rudin–Keisler order, defined over an arbitrary set $ D $, and the Rudin–Frolik order, defined only over a countable set $ \omega $. Two ultrafilters $ p, q $ on $ D $, i.e. two points of $ \beta D $, are related by $ p \leq q $ in the Rudin–Keisler order if there is a mapping $ f : D \rightarrow D \subset \beta D $ such that $ \beta f( q) = p $. If $ p \leq q $ and $ q \leq p $, $ p $ and $ q $ are said to be of the same type. The relation $ p \leq q $ induces a partial ordering of the types. The Rudin–Frolik order over $ \omega $ is defined similarly, but using $ f: \omega \rightarrow \beta \omega $ with discrete image.

[a1] is a rather full topologically oriented treatment of ultrafilter theory as of 1974, and still the best introduction to the subject. It has a $ 40 $- page chapter on large cardinals, a subject which has had near-revolutionary growth since 1974.

Let $ \Phi $ be an ultrafilter on an index set $ I $. For each $ i $, let $ A _ {i} $ be a set. Using $ \Phi $ one defines an equivalence relation on $ \prod A _ {i} $ as follows: $ a = ( a ( i) ) _ {i \in I } $, $ b = ( b ( i) ) _ {i \in I } $, are equivalent if and only if $ \{ {i } : {a( i) = b( i) } \} \in \Phi $( written: $ a \equiv b $). The quotient $ ( \prod A _ {i} ) / \Phi $ of $ \prod A _ {i} $ by this equivalence relation is called the ultraproduct of the $ A _ {i} $( with respect to the ultrafilter $ \Phi $).

For each $ i $, let $ R _ {i} $ be an $ n $- ary relation on $ A _ {i} $( eventually corresponding to one and the same predicate of a language $ L $, where the $ ( A _ {i} , \{ R _ {i} \} ) $ are supposed to be interpretations of $ L $). Then a corresponding relation $ R $ on $ ( \prod A _ {i} )/ \Phi $ is defined by:

$$ ( \overline{a}\; {} _ {1} \dots \overline{a}\; {} _ {n} ) \in R \iff \ \{ {i } : {( a _ {1} ( i) \dots a _ {n} ( i) ) \in R _ {i} } \} \in \Phi . $$

Here $ \overline{a}\; {} _ {m} $ is the equivalence class of $ a _ {m} $ in $ ( \prod A _ {i} ) / \Phi $. (This is well defined by the properties of ultrafilters.) Functions and individual constants are similarly defined.

If all the $ A _ {i} $ are the same, one speaks of ultrapowers instead of ultraproducts.

Ultraproducts have important applications in the theory of Diophantine equations and algebraic number theory. For instance, for each prime number $ p $, let $ \mathbf Q _ {p} $ be the field of $ p $- adic numbers and let $ \mathbf F _ {p} (( t)) $ be the field of Laurent series over the finite field $ \mathbf F _ {p} = \{ 0 \dots p - 1 \} $. Then the Ax–Kochen theorem says that for each non-principal ultrafilter $ \Phi $,

$$ \prod _ { p } \mathbf Q _ {p} / \Phi \simeq \prod _ { p } \mathbf F _ {p} (( t)) / \Phi . $$

This gives an immediate positive partial solution to Artin's conjecture on Diophantine equations in the form of the theorem: For each positive integer $ d $ there exists a finite set of primes $ P( d) $ such that every homogeneous polynomial $ f ( X _ {1} \dots X _ {n} ) $ of degree $ d $ over $ \mathbf Q _ {p} $ with $ n > d ^ {2} $ has a non-trivial zero in $ \mathbf Q _ {p} $ for all $ p \notin P( d) $. This result can also be deduced from results of Yu.L. Ershov ([a6]), which also use ultraproducts in their proof. Artin's conjecture in full generality says that $ \mathbf Q _ {p} $ is a $ C _ {2} $- field, which means that the conclusion just formulated must hold for all $ p $( the "2" in $ C _ {2} $- field refers to the "2" in "n>d2" ). However, G. Terjanian gave in 1966 a counterexample to the full Artin conjecture by providing a quartic form in $ 18 $ variables over $ \mathbf Q _ {2} $ with only non-trivial zeros.

More precisely, let $ i \geq 0 $, $ d \geq 1 $ be integers. Then a field $ F $ is called a $ C _ {i} ( d) $- field if every homogeneous polynomial $ f( X _ {1} \dots X _ {n} ) $ of degree $ d $ over $ F $ in $ n > d ^ {i} $ variables has a non-trivial zero in $ F $. A field that is $ C _ {i} ( d) $ for all $ d \geq 1 $ is called a $ C _ {i} $- field. The $ C _ {0} $- fields are the algebraically closed fields. The $ C _ {1} $- fields are also called quasi-algebraically closed. The rational functions in one variable over an algebraically closed field form a $ C _ {1} $- field (Tsen's theorem). The field $ \mathbf Q _ {p} $ is a $ C _ {2} ( 2) $- field (H. Hasse, 1923) and also a $ C _ {2} ( 3) $- field (D.J. Lewis, 1952).

Other important applications of ultraproducts are in non-standard analysis; in particular, non-standard models of the reals, integers, etc. can be obtained as ultrapowers of $ \mathbf R $, $ \mathbf Z $, etc.

Cf. Model theory and [a2] for results in logic involving ultrafilters and ultraproducts.

There are further important applications of ultrafilters to topology, cf. [a1], [a7].

References

[a1] W.W. Comfort, S. Negrepontis, "The theory of ultrafilters" , Springer (1974)
[a2] J.L. Bell, A.B. Slomson, "Models and ultraproducts" , North-Holland (1969)
[a3] J. Ax, S. Kochen, "Diophantine problems over local fields I" Amer. J. Math. , 87 (1965) pp. 605–630
[a4] J. Ax, S. Kochen, "Diophantine problems over local fields II. A complete set of axioms for -adic number theory" Amer. J. Math. , 87 (1965) pp. 631–648
[a5] J. Ax, S. Kochen, "Diophantine problems over local fields III. Decidable fields" Ann. of Math. , 83 (1966) pp. 437–456
[a6] Yu.L. Ershov, "On the elementary theory of maximal normal fields" Soviet Math. Dokl. , 6 (1965) pp. 1390–1393 Dokl. Akad. Nauk SSSR , 165 (1965) pp. 21–23
[a7] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[a8] T.J. Jech, "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)
[a9] J. van Mill, "An introduction to " K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984)
[a10] W. Rudin, "Homogeneity problems in the theory of Čech compactification" Duke Math. J. , 23 (1956) pp. 409–419
How to Cite This Entry:
Ultrafilter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultrafilter&oldid=18223
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article