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Difference between revisions of "Filter"

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It follows from the definition that the intersection of any finite family of elements of <math>\mathcal{F}</math> is also an element of <math>\mathcal{F}</math>.  Also, if <math>A</math> is an element of <math>\mathcal{F}</math>, then its [[complement]] is not.
 
It follows from the definition that the intersection of any finite family of elements of <math>\mathcal{F}</math> is also an element of <math>\mathcal{F}</math>.  Also, if <math>A</math> is an element of <math>\mathcal{F}</math>, then its [[complement]] is not.
  
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More generally, one can define a filter on any [[Partially ordered set]] <math>(P,\leq)</math>: Let <math>F</math> be a subset of <math>P</math>. We say <math>F</math> is a filter if and only if
+
More generally, one can define a filter on any [[Partially ordered set]] (poset) <math>(P,\leq)</math>: Let <math>F</math> be a subset of <math>P</math>. We say <math>F</math> is a filter if and only if
 
* <math>F\neq\emptyset</math>.
 
* <math>F\neq\emptyset</math>.
 
* For all <math>x,y\in F</math>, there exists <math>z\in F</math> such that <math>z\leq x</math> and <math>z\leq y</math>.
 
* For all <math>x,y\in F</math>, there exists <math>z\in F</math> such that <math>z\leq x</math> and <math>z\leq y</math>.

Revision as of 00:52, 11 November 2024

A filter on a set $X$ is a structure of subsets of $X$.

Definition

Let $\mathcal{F}$ be a set of subsets of $X$. We say that $\mathcal{F}$ is a filter on $X$ if and only if each of the following conditions hold:

  • The empty set is not an element of $\mathcal{F}$.
  • If $A$ and $B$ are subsets of $X$, $A$ is a subset of $B$, and $A$ is an element of $\mathcal{F}$, then $B$ is an element of $\mathcal{F}$.
  • The intersection of two elements of $\mathcal{F}$ is an element of $\mathcal{F}$.

It follows from the definition that the intersection of any finite family of elements of $\mathcal{F}$ is also an element of $\mathcal{F}$. Also, if $A$ is an element of $\mathcal{F}$, then its complement is not.

More generally, one can define a filter on any Partially ordered set (poset) $(P,\leq)$: Let $F$ be a subset of $P$. We say $F$ is a filter if and only if

  • $F\neq\emptyset$.
  • For all $x,y\in F$, there exists $z\in F$ such that $z\leq x$ and $z\leq y$.
  • If $x\in F$ and $x\leq y\in P$, then $y\in F$.

A filter on a set $S$ is a filter on the poset $(\mathcal{P}(S),\subseteq)$.

Examples

Let $Y$ be a subset of $X$. Then the set of subsets of $X$ containing $Y$ constitute a filter on $X$.

If $X$ is an infinite set, then the subsets of $X$ with finite complements constitute a filter on $X$. This is called the cofinite filter, or Fréchet filter.

See also

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