partially ordered ring
PartiallyOrderedRing
2007-04-12 02:27:08
2009-03-20 08:07:01
Definition
lattice ordered ring
positive cone
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A ring $R$ that is a poset at the same time is called a \emph{partially ordered ring}, or a \emph{po-ring}, if, for $a,b,c\in R$,
\begin{itemize}
\item $a\le b$ implies $a+c\le b+c$, and
\item $0\le a$ and $0\le b$ implies $0\le ab$.
\end{itemize}
Note that $R$ does not have to be associative.
If the underlying poset of a po-ring $R$ is in fact a lattice, then $R$ is called a \emph{lattice-ordered ring}, or an \emph{l-ring} for short.
\textbf{Remark}. The underlying abelian group of a po-ring (with addition being the binary operation) is a po-group. The same is true for l-rings.
Below are some examples of po-rings:
\begin{itemize}
\item Clearly, any (totally) ordered ring is a po-ring.
\item The ring of continuous functions over a topological space is an l-ring.
\item Any matrix ring over an ordered field is an l-ring if we define $(a_{ij})\le (b_{ij})$ whenever $a_{ij}\le b_{ij}$ for all $i,j$.
\end{itemize}
\textbf{Remark}. Let $R$ be a po-ring. The set $R^+:=\lbrace r\in R\mid 0\le r\rbrace$ is called the \emph{positive cone} of $R$.
\begin{thebibliography}{8}
\bibitem{gb} G. Birkhoff {\em Lattice Theory}, 3rd Edition, AMS Volume XXV, (1967).
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