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Homestrict betweenness relation
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strict betweenness relation
1 Definition
A strict betweenness relation is a betweenness relation that satisfies the following axioms:
 $O2^{{\prime}}$
$(p,q,p)\notin B)$ for each pair of points $p$ and $q$.
 $O3^{{\prime}}$
for each $p,q\in A$ such that $p\neq q$, there is an $r\in A$ such that $(p,q,r)\in B$.
 $O4^{{\prime}}$
for each $p,q\in A$ such that $p\neq q$, there is an $r\in A$ such that $(p,r,q)\in B$.
 $O5^{{\prime}}$
if $(p,q,r)\in B$, then $(q,p,r)\notin B$.
2 Remarks

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 $(p,q,q)$.

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.

From axiom $O2^{{\prime}}$ we have $(p,p,p)\notin B.$
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