# 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. 1.

$F$ is a sublattice of $L$, and

2. 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 $a,b\in F$, $a\wedge b\in F$.

1. 1.

for any $a,b\in F$, $a\wedge b\in F$, and

2. 2.

for any $a\in F$, if $a\leq b$, then $b\in F$.

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

Examples.

1. 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 $1\leq i$. 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. 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