ordered groupOrderedGroup2004-12-27 12:58:112009-04-30 22:08:31Definitiontotal orderordered group equipped with zero% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thmplain}{Theorem}\textbf{Definition 1.}\, We say that the subsemigroup $S$ of the group $G$ (with the operation denoted multiplicatively) defines an \PMlinkescapetext{{\em order for the group}} $G$, if
\begin{itemize}
\item $a^{-1}Sa \subseteq S \quad \forall a\in G,$
\item $G = S\cup \{1\} \cup S^{-1}$\,\, where \,$S^{-1} = \{s^{-1}: \,s\in S\}$\, and the members of the union are pairwise disjoint.
\end{itemize}
The order ``$<$'' of the group $G$ is explicitly given by setting in $G$:
$$a < b \,\, \Leftrightarrow \,\,ab^{-1}\in S$$
Then we speak of the {\em ordered group}\, $(G,\,<)$,\, or simply $G$.\\
\begin{thmplain}
\,\,The order ``$<$'' defined by the subsemigroup $S$ of the group $G$ has the following properties.
\begin{enumerate}
\item For all\, $a,\,b\in G$, exactly one of the conditions\,\, $a < b,\,\,a = b,\,\,b < a$\,\, holds.
\item $a < b \,\land\, b < c \,\,\Rightarrow\,\,a < c$
\item $a < b \,\,\Rightarrow\,\, ac < bc \,\land\, ca < cb$
\item $a < b \,\land\, c < d \,\,\Rightarrow\,\, ac < bd$
\item $a < b \,\,\Leftrightarrow\,\, b^{-1} < a^{-1}$
\item $a < 1 \,\,\Leftrightarrow\,\, a\in S$
\end{enumerate}
\end{thmplain}
\textbf{Definition 2.}\, The set $G$ is an {\em ordered group equipped with zero} 0, if the set $G^*$ of its elements distinct from its element 0 forms an ordered group\, $(G^*,\,<)$\, and if
\begin{itemize}
\item $0a = a0 = 0 \quad\forall a\in G,$
\item $0 < a \quad\forall a\in G^*.$
\end{itemize}
Cf. 7 in examples of semigroups.
\begin{thebibliography}{9}
\bibitem{artin} {\sc Emil Artin}: {\em Theory of Algebraic Numbers}.\, Lecture notes. \,Mathematisches Institut, G\"ottingen (1959).
\bibitem{Jaffard} {\sc Paul Jaffard}: {\em Les syst\`emes d'id\'eaux}.\, Dunod, Paris (1960).
\end{thebibliography}