| Version 7 |
Version 6 |
| Let $P$ be a poset and $A$ a subset of $P$. The \emph{upper set} of $A$ is defined to be the set |
Let $P$ be a poset and $A$ a subset of $P$. The \emph{upper set} of $A$ is defined to be the set |
| $$\lbrace b\in P\mid a\le b \mbox{ for some } a\in A\rbrace,$$ |
$$\lbrace b\in P\mid a\le b \mbox{ for some } a\in A\rbrace,$$ |
| and is denoted by $\up A$. In other words, $\up A$ is the set of all upper bounds of elements of $A$. |
and is denoted by $\up A$. In other words, $\up A$ is the set of all upper bounds of elements of $A$. |
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| $\uparrow$ can be viewed as a unary operator on $2^P$ sending $A\in 2^P$ to $\up A \in 2^P$. $\uparrow$ has the following properties |
$\uparrow$ can be viewed as a unary operator on $2^P$ sending $A\in 2^P$ to $\up A \in 2^P$. $\uparrow$ has the following properties |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\up P=P$, |
\item $\up P=P$, |
| \item $A\subseteq \up A$, |
\item $A\subseteq \up A$, |
| \item $\uparrow \up A=\up A$, and |
\item $\uparrow \up A=\up A$, and |
| \item if $A\subseteq B$, $\up A\subseteq \up B$. |
\item if $A\subseteq B$, $\up A\subseteq \up B$. |
| \end{enumerate} |
\end{enumerate} |
| So $\uparrow$ is a closure operator. |
So $\uparrow$ is a closure operator. |
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| An \emph{order filter} $F$ of $P$ is a subset of $P$ such that $F$ is closed under $\uparrow$, that is, $\up F=F$. In other words, an order filter is the upper set of itself. From this, we see that an order filter is synonymous with an upper set generated by some set (at most itself). So an upper set (without mentions of the implied generating set) is also called an order filter. |
An \emph{order filter} $F$ of $P$ is a subset of $P$ such that $F$ is closed under $\uparrow$, that is, $\up F=F$. In other words, an order filter is the upper set of itself. From this, we see that an order filter is synonymous with an upper set generated by some set (at most itself). So an upper set (without mentions of the implied generating set) is also called an order filter. |
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Dually, the \emph{lower set} of $A$ is the set of all lower bounds of elements of $A$. The lower set of $A$ is denoted by $\down A$. A lower set is the lower set of some set, and is also called an \emph{order ideal}. Like $\uparrow$, $\downarrow$ is also a closure operator on $P$.
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Dually, the \emph{lower set} of $A$ is the set of all lower bounds of elements of $A$. The lower set of $A$ is denoted by $\down A$. A lower set is the lower set of some set, and is also called an \emph{order ideal}..
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| \textbf{Remark}. If $P$ is a lattice and $A=\lbrace x\rbrace$, then $\up A$ is the principal filter generated by $x$, and $\down A$ is the principal ideal generated by $x$. |
\textbf{Remark}. If $P$ is a lattice and $A=\lbrace x\rbrace$, then $\up A$ is the principal filter generated by $x$, and $\down A$ is the principal ideal generated by $x$. |
