BoundedLattice 2005-02-16 01:45:13 2007-04-29 23:48:29 Definition top bottom bounded poset % 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 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. For example, any finite lattice $L$ is bounded, as $\bigvee L$ and $\bigwedge L$, being join and meet of finitely many elements, exist. $\bigvee L=1$ and $\bigwedge L=0$. \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 $L$. \item $L$ is a lattice interval and can be written as $[0,1]$. \end{itemize} \textbf{Remark}. More generally, a poset $P$ is said to be \emph{bounded} if it has both a greatest element $1$ and a least element $0$.