Let $L$ be a lattice. A filter (of $L$ is the dual concept of an ideal. Specifically, a filter $F$ of $L$ is a non-empty subset of $L$ such that

1. $F$ is a sublattice of $L$ and
2. for any $a\in F$ and $b\in L$ $a\vee b\in F$

The first condition can be replaced by a weaker one: for any $a,b\in F$ $a\wedge b\in F$

An equivalent characterization of a filter $I$ in a lattice $L$ is

1. for any $a,b\in F$ $a\wedge b\in F$ and
2. for any $a\in F$ if $a\le b$ then $b\in F$

Note that the dualization switches the meet and join operations, as well as reversing the ordering relationship.

Special Filters. Let $F$ be a filter of a lattice $L$ Some of the common types of filters are defined below.

• $F$ is a proper filter if $F\ne L$ and, if $L$ contains $0$ $F\ne 0$
• $F$ is a prime filter if it is proper, and $a\vee b\in F$ implies that either $a\in F$ or $b\in F$
• $F$ is an ultrafilter (or maximal filter) of $L$ if $F$ is proper and the only filter properly contains $F$ is $L$
• filter generated by a set. Let $X$ be a subset of a lattice $L$ Let $T$ be the set of all filters of $L$ containing $X$ Since $T\ne\varnothing$ ($L\in T$ , the intersection $N$ of all elements in $T$ is also a filter of $L$ that contains $X$ $N$ is called the filter generated by $X$ written $[X)$ If $X$ is a singleton $\lbrace x\rbrace$ then $N$ is said to be a principal filter generated by $x$ written $[x)$

Examples.

1. Consider the positive integers, with meet and join defined by the greatest common divisor and the least common multiple operations. Then the positive even numbers form a filter, generated by $2$ If we toss in $3$ as an additional element, then $1=2\wedge 3\in[\lbrace 2,3\rbrace)$ and consequently any positive integer $i\in[\lbrace 2,3\rbrace)$ since $1\le i$ In general, if $p,q$ are relatively prime, then $[\lbrace p,q\rbrace)=\mathbb{Z}^{+}$ In fact, any proper filter in $\mathbb{Z}^{+}$ is principal. When the generator is prime, the filter is prime, which is also maximal. So prime filters and ultrafilters coincide in $\mathbb{Z}^{+}$
2. Let $A$ be a set and $2^A$ the power set of $A$ If the set inclusion is the ordering defined on $2^A$ then the definition of a filter here coincides with the ususal definition of a filter on a set in general.

Remark. If $F$ is both a filter and an ideal of a lattice $L$ then $F=L$