strict betweenness relation

1 Definition

A strict betweenness relation is a betweenness relation that satisfiesthe following axioms:

• $(p,q,p)\notin B)$ for each pair of points $p$ and $q$.

• for each $p,q\in A$ such that $p\ne q$, there is an $r\in A$ such that $(p,q,r)\in B$.

• for each $p,q\in A$ such that $p\ne q$, there is an $r\in A$ such that $(p,r,q)\in B$.

• 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 $\mathrm{\varnothing }$, 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 betweennessrelation by including all triples of the forms $(p,p,q),(p,q,p),$ or$(p,q,q)$.

• Conversely, any betweenness relation can be reducedto a strict betweenness relation by removing all triples of theforms just listed. However, it is possible that the “derived”strict betweenness relation is trivial.

• From axiom $O{2}^{\prime }$ we have $(p,p,p)\notin B.$