lattice filterLatticeFilter2006-03-27 18:30:312007-07-25 00:02:42Definitionfilterprime filterultrafilterfilter generated byprincipal filter\usepackage{amssymb,amscd}
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% define commands hereLet $L$ be a lattice. A \emph{filter} (of $L$) is the dual concept of an \PMlinkname{ideal}{LatticeIdeal}. Specifically, a filter $F$ of $L$ is a non-empty subset of $L$ such that
\begin{enumerate}
\item $F$ is a sublattice of $L$, and
\item for any $a\in F$ and $b\in L$, $a\vee b\in F$.
\end{enumerate}
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
\begin{enumerate}
\item for any $a,b\in F$, $a\wedge b\in F$, and
\item for any $a\in F$, if $a\le b$, then $b\in F$.
\end{enumerate}
Note that the dualization switches the meet and join operations, as well as reversing the ordering relationship.
\textbf{Special Filters}. Let $F$ be a filter of a lattice $L$. Some of the common types of filters are defined below.
\begin{itemize}
\item $F$ is a \emph{proper filter} if $F\ne L$, and, if $L$ contains $0$, $F\ne 0$.
\item $F$ is a \emph{prime filter} if it is proper, and $a\vee b\in F$ implies that either $a\in F$ or $b\in F$.
\item $F$ is an \emph{ultrafilter} (or \emph{maximal filter}) of $L$ if $F$ is proper and the only filter properly contains $F$ is $L$.
\item \textbf{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 \emph{filter generated by} $X$, written $[X)$. If $X$ is a singleton $\lbrace x\rbrace$, then $N$ is said to be a \emph{principal filter} generated by $x$, written $[x)$.
\end{itemize}
\textbf{Examples}.
\begin{enumerate}
\item 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 $\mathbb{Z}^{+}$.
\item 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 \PMlinkname{filter}{Filter} on a set in general.
\end{enumerate}
\textbf{Remark}. If $F$ is both a filter and an ideal of a lattice $L$, then $F=L$.