betweenness relation

1 Definition

Let $A$ be a set. A ternary relation $B$ on$A$ is said to be a betweenness relation if it has the following properties:

1. if $(a,b,c)\in B$, then $(c,b,a)\in B$; in other words, the set

   $$B(b)=\{(a,c)\mid (a,b,c)\in B\}$$

   is a symmetric relation (http://planetmath.org/Symmetric) for each$b$; thus, from now on, we may say, without any ambiguity, that $b$is between $a$ and $c$ if $(a,b,c)\in B$;

2. if $(a,b,a)\in B$, then $a=b$;

3. for each $a,b\in A$, there is a $c\in A$ such that $(a,b,c)\in B$;

4. for each $a,b\in A$, there is a $c\in A$ such that $(a,c,b)\in B$;

5. if $(a,b,c)\in B$ and $(b,a,c)\in B$, then $a=b$;

6. if $(a,b,c)\in B$ and $(b,c,d)\in B$, then $(a,b,d)\in B$;

7. if $(a,b,d)\in B$ and $(b,c,d)\in B$, then $(a,b,c)\in B$.