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