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:

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 $(a,b,c)\in B$;

O2
if $(a,b,a)\in B$, then $a=b$;

O3
for each $a,b\in A$, there is a $c\in A$ such that $(a,b,c)\in B$;

O4
for each $a,b\in A$, there is a $c\in A$ such that $(a,c,b)\in B$;

O5
if $(a,b,c)\in B$ and $(b,a,c)\in B$, then $a=b$;

O6
if $(a,b,c)\in B$ and $(b,c,d)\in B$, then $(a,b,d)\in B$;

O7
if $(a,b,d)\in B$ and $(b,c,d)\in B$, then $(a,b,c)\in B$.
Title  betweenness relation 

Canonical name  BetweennessRelation 
Date of creation  20130322 17:18:44 
Last modified on  20130322 17:18:44 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  6 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 51G05 
Synonym  axioms of order 
Related topic  SomeTheoremsOnTheAxiomsOfOrder 