strict betweenness relation
1 Definition
A strict betweenness relation is a betweenness relation that satisfies the following axioms:

$O{2}^{\prime}$
$(p,q,p)\notin B)$ for each pair of points $p$ and $q$.

$O{3}^{\prime}$
for each $p,q\in A$ such that $p\ne q$, there is an $r\in A$ such that $(p,q,r)\in B$.

$O{4}^{\prime}$
for each $p,q\in A$ such that $p\ne q$, there is an $r\in A$ such that $(p,r,q)\in B$.

$O{5}^{\prime}$
if $(p,q,r)\in B$, then $(q,p,r)\notin B$.
2 Remarks
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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 $(p,q,q)$.

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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.

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From axiom $O{2}^{\prime}$ we have $(p,p,p)\notin B.$
Title  strict betweenness relation 

Canonical name  StrictBetweennessRelation 
Date of creation  20130322 17:18:56 
Last modified on  20130322 17:18:56 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  7 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 51G05 
Related topic  SomeTheoremsOnStrictBetweennessRelations 