Difference between revisions of "Filter"

Revision as of 19:59, 13 October 2019

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 $(P,\leq)$ : 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.

@@ Line 4: / Line 4: @@
 Let <math>\mathcal{F}</math> be a set of subsets of <math>X</math>.  We say that <math>\mathcal{F}</math> is a filter on <math>X</math> if and only if each of the following conditions hold:
-* The [[empty set]] is not an element of <math>\mathcal{F}</math>
+* The [[empty set]] is not an element of <math>\mathcal{F}</math>.
 * If <math>A</math> and <math>B</math> are subsets of <math>X</math>, <math>A</math> is a subset of <math>B</math>, and <math>A</math> is an element of <math>\mathcal{F}</math>, then <math>B</math> is an element of <math>\mathcal{F}</math>.
 * The intersection of two elements of <math>\mathcal{F}</math> is an element of <math>\mathcal{F}</math>.
 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.
+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
+* <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>.
+* If <math>x\in F</math> and <math>x\leq y\in P</math>, then <math>y\in F</math>.
+A filter on a set <math>S</math> is a filter on the poset <math>(\mathcal{P}(S),\subseteq)</math>.
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Difference between revisions of "Filter"

Revision as of 19:59, 13 October 2019

Definition

Examples

See also