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'upper set'
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| Title of object: |
upper set |
| Canonical Name: |
UpperSet |
| Type: |
Definition |
| Created on: |
2006-04-03 13:59:31 |
| Modified on: |
2006-04-03 15:16:27 |
| Classification: |
msc:06A06 |
| Defines: |
lower set |
| Synonyms: |
upper set=up set
upper set=order filter
upper set=down set
upper set=order ideal |
Revision comment (for changes between this and next version):
| Changes for correction #7794 ('notation'). |
Preamble:
\usepackage{amssymb,amscd}
\usepackage{amsmath}
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%\usepackage{psfrag}
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Content:
Let $P$ be a poset and $A$ a subset of $P$. The \emph{upper set} of $A$ is defined to be the set
$$\lbrace b\in P\mid a\le b \mbox{ for some } a\in A\rbrace,$$
and is denoted by $\uparrow A$. In other words, $\uparrow A$ is the set of all upper bounds of elements of $A$.
$\uparrow$ can be viewed as a unary operator on $2^P$ sending $A\in 2^P$ to $\uparrow A\in 2^P$. $\uparrow$ has the following properties
\begin{enumerate}
\item $A\subseteq \uparrow A$,
\item $\uparrow\uparrow A=\uparrow A$, and
\item if $A\subseteq B$, $\uparrow A\subseteq \uparrow B$.
\end{enumerate}
So $\uparrow$ is a closure operator.
An \emph{order filter} $F$ of $P$ is a subset of $P$ such that $F$ is closed under $\uparrow$: $\uparrow F=F$. 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. An order filter is also called an upper set (without mentions of an implied generating set).
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 $\downarrow A$. A lower set is the lower set of some set, and is also called an \emph{order ideal}.
\textbf{Remark}. If $P$ is a lattice and $A=\lbrace x\rbrace$, then $\uparrow A$ is the principal filter generated by $x$, and $\downarrow A$ is the principal ideal generated by $x$. |
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