maximal element
MaximalElement
2002-03-02 13:28:20
2006-10-28 23:16:56
Definition
greatest element
least element
minimal element
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{psfrag}
%\usepackage{graphicx}
%\usepackage{xypic}
Let $\le$ be an ordering on a set $S$, and let $A \subseteq S$. Then, with respect to the ordering $\le$,
\begin{itemize}
\item $a \in A$ is the \emph{least} element of $A$ if $a \le x$, for all $x \in A$.
\item $a \in A$ is a \emph{minimal} element of $A$ if there exists no $x \in A$ such that $x \le a$ and $x \ne a$.
\item $a \in A$ is the \emph{greatest} element of $A$ if $x \le a$ for all $x \in A$.
\item $a \in A$ is a \emph{maximal} element of $A$ if there exists no $x \in A$ such that $a \le x$ and $x \ne a$.
\end{itemize}
\paragraph{Examples.}
\begin{itemize}
\item The natural numbers $\mathbb{N}$ ordered by divisibility ($\mid$) have a least element, $1$. The natural numbers greater than 1 ($\mathbb{N} \setminus \{1\}$) have no least element, but infinitely many minimal elements (the primes.) In neither case is there a greatest or maximal element.
\item The negative integers ordered by the standard definition of $\le$ have a maximal element which is also the greatest element, $-1$. They have no minimal or least element.
\item The natural numbers $\mathbb{N}$ ordered by the standard $\le$ have a least element, $1$, which is also a minimal element. They have no greatest or maximal element.
\item The rationals greater than zero with the standard ordering $\le$ have no least element or minimal element, and no maximal or greatest element.
\end{itemize}