 ##  [Ultrafilter](/ultrafilter) 

  ##  [Ultrafilter](https://natural.quantumdictionary.io/ultrafilter-0) 

  
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**Natural &amp; Formal Sciences Dictionary**

 
Definition

A maximal proper filter on a set: a collection of subsets closed under finite intersection and supersets, not containing the empty set, and such that for every subset A either A or its complement belongs to the ultrafilter.

 
 ##  [Ultrafilter](https://mathlogic.quantumdictionary.io/ultrafilter-1) 

  
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- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A maximal proper filter on a Boolean algebra or on the power set of a set: a collection of subsets closed under finite intersection and supersets that contains exactly one of each complementary pair; principal ultrafilters concentrate on a single point, nonprincipal ones do not.

 
 ##  [Ultrafilter](https://puremath.quantumdictionary.io/ultrafilter-2) 

  
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- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

A filter U on a set X that is maximal with respect to inclusion: no strictly larger filter on X contains U. Equivalently, for every subset A of X, either A ∈ U or X\A ∈ U, giving a two-valued decision of membership for each subset.