Filter

dual ideal

A non-empty subset $F$ of a partially ordered set $(P,{\le})$ satisfying the conditions: a) if $a,b \in F$ and if the infimum $\inf\{a,b\}$ exists, then $\inf\{a,b\} \in F$; and b) if $a \in F$ and $a \le b$, then $b \in F$. The concept of a filter is dual to that of an ideal of a partially ordered set.

A filter over a non-empty set $E$ (or in a set $E$) is a proper filter of the set of subsets of $E$, ordered by inclusion i.e. any non-empty collection $F$ of subsets of $E$ satisfying the conditions: If $A,B \in F$ then $A \cap B \in F$; if $A \in F$ and $A \subset B$, then $B \in F$; the empty set does not belong to $F$.

A filter base is a system of subsets of $E$ satisfying the two conditions: 1) the empty set does not belong to it; and 2) the intersection of two subsets belonging to it contains some third subset belonging to it. Every filter is completely determined by any of its filter bases. The system of all subsets of $E$ that contain some element of a given filter base is a filter. It is said to be spanned by this base.

The set of all filters over a given set is partially ordered by inclusion. A maximal element of it is called an ultrafilter (a maximal proper filter in any Boolean algebra is also called an ultrafilter).

Examples of filters. 1) Let $N_k$ be the subset of the natural numbers consisting of those numbers that are multiples of $k$; the system $\{N_k : k=1,2,\ldots \}$ is a filter base; the filter spanned by this base consists of those subsets that contain some $N_k$. 2) The collection of all subsets of $E$ containing a certain fixed non-empty subset $A \subseteq E$ is a filter over $E$, called a principal filter. All filters over a finite set are principal. 3) If $E$ is an infinite set of cardinality $\mathfrak{a}$ and $F$ is the collection of all subsets of $E$ whose complements have cardinality less than $\mathfrak{a}$, then $F$ is a filter (called a Fréchet filter). A Fréchet filter is an example of a non-principal filter. 4) The system of subsets containing some fixed point of a set is also a filter; moreover, it is an ultrafilter. 5) Suppose a topology is given on $E$; then the neighbourhoods of any point $x \in E$ (the subsets of $E$ containing $x$ in the interior) form a filter.

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There is some disagreement about the definition of a filter in a partially ordered set where finite infima do not always exist. Most authors would replace condition a) in the first paragraph above by "if $a,b \in F$, then there exists $c \in F$ with $c \le a$ and $c \le b$" , cf. [a1]. This avoids some rather pathological seeming cases.

A further meaning of the word "filter" occurs in the theory of (partially observed) stochastic processes, cf. Stochastic processes, filtering of.

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