# 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

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

Title strict betweenness relation StrictBetweennessRelation 2013-03-22 17:18:56 2013-03-22 17:18:56 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 51G05 SomeTheoremsOnStrictBetweennessRelations