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 and .
$O3^{\prime}$: for each $p,q\in A$ such that $p\ne q$ , there is an $r\in A$ such that $(p,q,r)\in B$ .
$O4^{\prime}$: for each $p,q\in A$ such that $p\ne q$ , there is an $r\in A$ such that $(p,r,q)\in B$ .
$O5^{\prime}$: if $(p,q,r)\in B$ , then $(q,p,r)\notin B$ .

Remarks

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.
Any strict betweenness relation can be enlarged to a betweenness relation by including all triples of the forms or .
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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Cross-references: reduced, vacuously, empty set, simple, points, axioms, betweenness relation
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This is version 4 of strict betweenness relation, born on 2007-06-24, modified 2007-06-24.
Object id is 9665, canonical name is StrictBetweennessRelation.
Accessed 538 times total.

Classification:

51G05 (Geometry :: Ordered geometries )

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