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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: |
2007-02-12 10:22:57 |
| Classification: |
msc:06A06 |
| Defines: |
lower set, order filter, order ideal |
| Synonyms: |
upper set=up set
upper set=down set
upper set=filter
upper set=ideal |
Preamble:
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% define commands here
\newcommand{\up}{\uparrow\!\!}
\newcommand{\down}{\downarrow\!\!} |
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 $\up A$. In other words, $\up A$ is the set of all upper bounds of elements of $A$.
$\uparrow$ can be viewed as a unary operator on the power set $2^P$ sending $A\in 2^P$ to $\up A \in 2^P$. $\uparrow$ has the following properties
\begin{enumerate}
\item $\up P=P$,
\item $A\subseteq \up A$,
\item $\uparrow \up A=\up A$, and
\item if $A\subseteq B$, $\up A\subseteq \up B$.
\end{enumerate}
So $\uparrow$ is a closure operator.
An \emph{upper set} in $P$ is a subset $A$ such that its upper set is itself: $\up A=A$.
An \emph{order filter} (or simply a \emph{filter}) in $P$ is a subset $F$ which is both an upper set and a filtered set (every pair of elements in $F$ has a lower bound in $F$). This is a generalization of the notion of a \PMlinkname{filter}{Filter} in a 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 $\down A$. If the lower set of $A$ is $A$ itself, then $A$ is a called a \emph{lower set}. If a lower set $A$ is also a directed set, then $A$ is said to be an \emph{order ideal}, or simply an \emph{ideal} (of the poset $P$). Like $\uparrow$, $\downarrow$ is also a closure operator on $P$.
\textbf{Remarks}.
\begin{itemize}
\item $\up A$ is \emph{not} the same as the set of upper bounds of $A$, commonly denoted by $A^u$, which is defined as the set $\lbrace b\in P\mid a\le b\mbox{ for all }a\in A\rbrace$. Similarly, in general, $\down A\neq A^{\ell}$ where $A^{\ell}$ is the set of lower bounds of $A$.
\item When $A=\lbrace x\rbrace$, we write $\up x$ for $\up A$ and $\down x$ for $\down A$. $\up x = \lbrace x\rbrace ^u$ and $\down x=\lbrace x\rbrace ^d$.
\item If $P$ is a lattice and $x\in P$, then $\up x$ is the principal filter generated by $x$, and $\down x$ is the principal ideal generated by $x$.
\end{itemize} |
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