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 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 $T\ne \mathrm{\varnothing }$ ($L\in T$), 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	2013-03-22 15:49:01
Last modified on	2013-03-22 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	ultra-filter
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