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\begin{document}

\title{Introduction to algebra --- handout 11}
\maketitle

\section{Ordered algebraic structures}
\begin{df}
$\langle G,+,\le\rangle$ is a  (partially) {\bf ordered group} if $\langle G,+\rangle$ is a group, $\langle G, \le\rangle$ is a partially ordered set and, for all $x,y,z\in G$, if $x\le y$, then $x+z\le y+z$ and $z+x\le z+y$.
\end{df}

\begin{ex}
$\langle \Z,+,\le\rangle$, $\langle \Q,+,\le \rangle$, $\langle
  \R,+,\le \rangle$, $\langle C([0,1]), +, \le \rangle$. 
\end{ex}

\begin{e}
Show that every ordered group is infinite or trivial (i.e., contains
only one element).
\end{e}

\begin{df}
An element $x$ of an ordered group $\langle G,+\rangle$ is called {\bf positive} if $e\le x$. The set of positive elements $P_G$ is called the {\bf positive cone} of $\langle G,+\rangle$.   
\end{df}

Note that $y-x\in P_G$ iff $x\le y$. 

\begin{df}
Let $\langle G,+ \rangle$ be a group. Binary relation $\le$ is called {\bf compatible order} on $G$ if $\langle G, +, \le \rangle$ is an ordered group. 
\end{df}

\begin{thm} $P\subseteq G$ is the positive cone of compatible order on  group $\langle G,+ \rangle $ iff
\begin{enumerate}
\item $P\cap (-P)={e}$,
\item $P+P=P$,
\item $x+P-x=P$ for all $x\in G$.
\end{enumerate}
Furthermore,  the order is a total order iff $P\cup (-P)= G$ also holds. 
\end{thm}

\begin{e}
Show that $2\N$ and $\N\setminus\{1\}$ are the positive cones of
compatible orders on $\langle \Z, +\rangle$ and draw the Hesse diagram
of the corresponding orders.
\end{e}

\begin{df} $\langle R,+,\cdot, \le\rangle$ is an ordered ring if $\langle R,+,\cdot\rangle$ is a ring such that $\langle R,+\rangle$ is an ordered (abelian) group and, if $0\le x,y$, then $0\le xy$.
\end{df}

\begin{thm}
A non-empty subset $P$ of an ordered ring $\langle R,+,\cdot\rangle$ is the positive cone relative to some compatible order on
$\langle R,+,\cdot\rangle$ if and only if
\begin{enumerate} 
\item  $P \cap (-P) = \{0\}$;
\item $P + P \subseteq P$;
\item $PP \subseteq P$.
\end{enumerate}
Moreover, the order is a total order iff  $P \cup (-P) = R$ also holds.
\end{thm}

\begin{thm} 
For every ordered integral domain $\langle
R,+,\cdot,\le\rangle$ there is an unique ordering $\preceq$ on the field of fractions of $R$ extending $\le$. 
\end{thm}

\begin{df}
Let $\langle G,\cdot\rangle$ be an ordered group and $x, y\in G$. We
say that $x$ is {\bf infinitely smaller} than $y$, in symbols $x \ll
y$, if $x^n\le y$ for all $n\in\Z$. We say that $\langle
G,\cdot\rangle$ is {\bf archimedean} if $e\le x \ll y$ implies that $x
=e$. An ordered ring (field) is called {\bf archimedean} if its
underlying ordered additive group is archimedean.
\end{df}

\begin{thm}[H\" older]
Let $\langle G,\cdot, \le\rangle$ be a totally ordered group. Then the
following statements are equivalent:
\begin{enumerate}
\item $\langle G,\cdot, \le\rangle$ is archimedean, 
\item $\langle G,\cdot, \le\rangle$ is isomorphic to a subgroup of
  $\langle \R,+\rangle$.
\end{enumerate}
\end{thm}

\end{document}