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Revision difference : ordered group |
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| \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 |
\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} |
\begin{itemize} |
| \item $a^{-1}Sa \subseteq S \quad \forall a\in G,$ |
\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. |
\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} |
\end{itemize} |
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| The order ``$<$'' of the group $G$ is explicitely given by setting in $G$: |
The order ``$<$'' of the group $G$ is explicitely given by setting in $G$: |
| $$a < b \,\, \Leftrightarrow \,\,ab^{-1}\in S$$ |
$$a < b \,\, \Leftrightarrow \,\,ab^{-1}\in S$$ |
| Then we speak of the {\em ordered group} $(G,\,<)$ or simply $G$. |
Then we speak of the {\em ordered group} $(G,\,<)$ or simply $G$. |
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| \begin{thmplain} |
\textbf{Theorem.} \,The order ``$<$'' defined by the subsemigroup $S$ of the group $G$ has the following properties. |
| \,\,The order ``$<$'' defined by the subsemigroup $S$ of the group $G$ has the following properties. |
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| \begin{enumerate} |
\begin{enumerate} |
| \item For all \,$a,\,b\in G$, exactly one of the conditions \,\,$a < b,\,\,a = b,\,\,b < a$\,\, holds. |
\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 $a < b \,\land\, b < c \,\,\Rightarrow\,\,a < c$ |
| \item $a < b \,\,\Rightarrow\,\, ac < bc \,\land\, ca < cb$ |
\item $a < b \,\,\Rightarrow\,\, ac < bc \,\land\, ca < cb$ |
| \item $a < b \,\land\, c < d \,\,\Rightarrow\,\, ac < bd$ |
\item $a < b \,\land\, c < d \,\,\Rightarrow\,\, ac < bd$ |
| \item $a < b \,\,\Leftrightarrow\,\, b^{-1} < a^{-1}$ |
\item $a < b \,\,\Leftrightarrow\,\, b^{-1} < a^{-1}$ |
| \item $a < 1 \,\,\Leftrightarrow\,\, a\in S$ |
\item $a < 1 \,\,\Leftrightarrow\,\, a\in S$ |
| \end{enumerate} |
\end{enumerate} |
| \end{thmplain} |
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| \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 form an ordered group \,$(G^*,\,<)$\, and if |
\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 form an ordered group \,$(G^*,\,<)$\, and if |
| \begin{itemize} |
\begin{itemize} |
| \item $0a = a0 = 0 \quad\forall a\in G,$ |
\item $0a = a0 = 0 \quad\forall a\in G,$ |
| \item $0 < a \quad\forall a\in G^*.$ |
\item $0 < a \quad\forall a\in G^*.$ |
| \end{itemize} |
\end{itemize} |
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| \begin{thebibliography}{9} |
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| \bibitem{artin} Emil Artin: {\em Theory of Algebraic Numbers}. \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959). |
\subsection*{References} |
| \bibitem{Jaffard} Paul Jaffard: {\em Les syst\`emes d'id\'eaux}. \,Dunod, Paris (1960). |
Emil Artin: {\em Theory of Algebraic Numbers}. \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959). |
| \end{thebibliography} |
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Paul Jaffard: {\em Les syst\`emes d'id\'eaux}. \,Dunod, Paris (1960). |
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