strict betweenness relation

1 Definition

$O2^{\prime}$

$(p,q,p)\notin B)$ 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 $(q,p,r)\notin B$ .

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

Title	strict betweenness relation
Canonical name	StrictBetweennessRelation
Date of creation	2013-03-22 17:18:56
Last modified on	2013-03-22 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