| Version 5 |
Version 4 |
| 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. |
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}. |
\textbf{Remarks}. |
| Let $L$ be a bounded lattice with $0$ and $1$ as described above. |
Let $L$ be a bounded lattice with $0$ and $1$ as described above. |
| \begin{itemize} |
\begin{itemize} |
| \item $0\land x=0$ and $0\lor x=x$ for all $x\in L$. |
\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 $1\land x=x$ and $1\lor x=1$ for all $x\in L$. |
| \item As a result, $0$ and $1$, if they exist, are necessarily unique. For |
\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 |
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 |
$0=0\land 0^{\prime}=0^{\prime}\land 0=0^{\prime}$. Similarly |
| $1=1^{\prime}$. |
$1=1^{\prime}$. |
| \item $0$ is called the \emph{bottom} of $L$ and $1$ is called the \emph{top} of $L$. |
\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 $[0,1]$. |
| \item $L$ is a lattice interval and can be written as $[0,1]$. |
|
| \end{itemize} |
\end{itemize} |
