|
|
|
Viewing Version
4
of
'bounded lattice'
|
[ view 'bounded lattice'
|
back to history
]
| Title of object: |
bounded lattice |
| Canonical Name: |
BoundedLattice |
| Type: |
Definition |
| Created on: |
2005-02-16 01:45:13 |
| Modified on: |
2006-03-18 11:21:43 |
| Classification: |
msc:06B05 |
| Defines: |
bounded from below, bounded from above, top, bottom |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\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}
% there are many more packages, add them here as you need them
% define commands here |
Content:
A lattice $L$ is said to be \emph{\PMlinkescapetext{bounded from below}} if there is an element $0\in L$ such that $0\leq x$ for all $x\in L$. Dually, $L$ is \emph{\PMlinkescapetext{bounded from above}} if there exists an element $1\in L$ such that $x\leq1$ for all $x\in L$. A \emph{bounded lattice} is one that is \PMlinkescapetext{bounded} both from above and below.
\textbf{Remarks}.
Let $L$ be a bounded lattice with $0$ and $1$ as described above.
\begin{itemize}
\item $0\land x=0$ and $0\lor x=x$ for all $x\in L$.
\item $1\land x=x$ and $1\lor x=1$ for all $x\in L$.
\item As a result, $0$ and $1$, if they exist, are necessarily unique. For
if there is another such a pair $0^{\prime}$ and $1^{\prime}$, then
$0=0\land 0^{\prime}=0^{\prime}\land 0=0^{\prime}$. Similarly
$1=1^{\prime}$.
\item $0$ is called the \emph{bottom} of $L$ and $1$ is called the \emph{top} of $L$. $L$ is a lattice interval and can be written as $[0,1]$.
\end{itemize} |
|
|
|
|
|
