lattice filter
Let $L$ be a lattice^{}. A filter (of $L$) is the dual concept of an ideal (http://planetmath.org/LatticeIdeal). Specifically, a filter $F$ of $L$ is a nonempty 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 $T\ne \mathrm{\varnothing}$ ($L\in T$), 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 $\{x\}$, 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 $1=2\wedge 3\in [\{2,3\})$ and consequently any positive integer $i\in [\{2,3\})$, since $1\le i$. In general, if $p,q$ are relatively prime, then $[\{p,q\})={\mathbb{Z}}^{+}$. 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 (http://planetmath.org/Filter) on a set in general.
Remark. If $F$ is both a filter and an ideal of a lattice $L$, then $F=L$.
Title  lattice filter 
Canonical name  LatticeFilter 
Date of creation  20130322 15:49:01 
Last modified on  20130322 15:49:01 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B10 
Synonym  ultra filter 
Synonym  ultrafilter 
Synonym  maximal filter 
Related topic  Ultrafilter 
Related topic  UpperSet 
Related topic  LatticeIdeal 
Related topic  OrderIdeal 
Defines  filter 
Defines  prime filter 
Defines  ultrafilter 
Defines  filter generated by 
Defines  principal filter 