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Homelattice filter
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lattice filter
Let $L$ be a lattice. A filter (of $L$) is the dual concept of an ideal. 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 $a\leq b$, 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 $F\neq L$, and, if $L$ contains $0$, $F\neq 0$.

$F$ 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 $T\neq\varnothing$ ($L\in T$), 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 $1\leq i$. 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. 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 $F=L$.
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Comments
Least Element
I am not sure, but I think to define filter of lattices. One does assume that the Lattice has a least element. Although, I do not see why this is necessary.
Re: Least Element
What do you mean filter of lattices? Or do you mean filter of a lattice? Given a lattice or more generally a poset, a filter can be defined without the lattice (or poset) having a least element.
Re: Least Element
This is how Lambek has defined it in his book "Lectures on Rings and Modules" .. he had a special reason for doing so, but Im rather shaggy today .. Im sure though that the definition with least element became necessary later on in his book.