betweenness relationBetweennessRelation2007-06-24 15:56:562007-06-24 17:00:57Definition% this is the default PlanetMath preamble. as your knowledge
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\section{Definition}
Let $A$ be a set. A ternary relation $B$ on
$A$ is said to be a \emph{betweenness relation} if it has the following properties:
\begin{enumerate}
\item[O1] if $(a,b,c)\in B$, then $(c,b,a)\in B$; in other words, the set $$B(b)=
\lbrace (a,c)\mid (a,b,c)\in B\rbrace$$ is a \PMlinkname{symmetric relation}{Symmetric} for \emph{each}
$b$; thus, from now on, we may say, without any ambiguity, that $b$
is \emph{between} $a$ and $c$ if $(a,b,c)\in B$;
\item[O2] if $(a,b,a)\in B$, then $a=b$;
\item[O3] for each $a,b\in A$, there is a $c\in A$ such that $(a,b,c)\in B$;
\item[O4]for each $a,b\in A$, there is a $c\in A$ such that $(a,c,b)\in B$;
\item[O5] if $(a,b,c)\in B$ and $(b,a,c)\in B$, then $a=b$;
\item[O6] if $(a,b,c)\in B$ and $(b,c,d)\in B$, then $(a,b,d)\in B$;
\item[O7] if $(a,b,d)\in B$ and $(b,c,d)\in B$, then $(a,b,c)\in B$.
\end{enumerate}