strict betweenness relationStrictBetweennessRelation2007-06-24 17:21:402007-06-24 20:47:22Definition% this is the default PlanetMath preamble. as your knowledge
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\section{Definition} A \emph{strict betweenness relation} is a betweenness relation that satisfies
the following axioms:
\begin{itemize}
\item[$O2^{\prime}$] $(p,q,p)\notin B)$ for \emph{each} pair of points $p$ and $q$.
\item[$O3^{\prime}$] for each $p,q\in A$ such that $p\ne q$, there is an $r\in A$ such that $(p,q,r)\in B$.
\item[$O4^{\prime}$] for each $p,q\in A$ such that $p\ne q$, there is an $r\in A$ such that $(p,r,q)\in B$.
\item[$O5^{\prime}$] if $(p,q,r)\in B$, then $(q,p,r)\notin B$.
\end{itemize}
\section{Remarks}
\begin{itemize}
\item A very simple example of a strict betweenness relation is the empty set.
In $\varnothing$, all the conditions are vacuously satisfied.
The empty set, in this context, is called the trivial strict betweenness relation.
\item Any strict betweenness relation can be enlarged to a betweenness
relation by including all triples of the forms $(p,p,q),(p,q,p),$ or
$(p,q,q)$.
\item Conversely, any betweenness relation can be reduced
to a strict betweenness relation by removing all triples of the
forms just listed. However, it is possible that the ``derived''
strict betweenness relation is trivial.
\item From axiom $O2^{\prime}$ we have $(p,p,p) \notin B.$
\end{itemize}