# 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. O1

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. O2

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

3. O3

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

4. O4

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

5. O5

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

6. O6

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

7. O7

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

Title betweenness relation BetweennessRelation 2013-03-22 17:18:44 2013-03-22 17:18:44 Mathprof (13753) Mathprof (13753) 6 Mathprof (13753) Definition msc 51G05 axioms of order SomeTheoremsOnTheAxiomsOfOrder