some theorems on the axioms of order
Let $B$ be a betweenness relation on a set $A$.
Theorem 1.
If $(a,b,c)\in B$ and $(a,c,d)\in B$, then $(a,b,d)\in B$.
Theorem 2.
For each pair of elements $p\mathrm{,}q\mathrm{\in}A$, we can define five sets:

1.
${B}_{*pq}:=\{r\in A\mid (r,p,q)\in B\}$,

2.
${B}_{p*q}:=\{r\in A\mid (p,r,q)\in B\}$,

3.
${B}_{pq*}:=\{r\in A\mid (p,q,r)\in B\}$,

4.
${B}_{pq}:={B}_{p*q}\cup \{q\}\cup {B}_{pq*}$, and

5.
$B(p,q):={B}_{*pq}\cup \{p\}\cup {B}_{pq}$.
Then

(1)
${B}_{*pq}={B}_{qp*}.$

(2)
${B}_{p*q}={B}_{q*p}.$

(3)
The intersection of any pair of the first three sets contains at most one element, either $p$ or $q$.

(4)
Each of the sets can be partially ordered.

(5)
The partial order^{} on ${B}_{pq}$ and $B(p,q)$ extends that of the subsets.
Title  some theorems on the axioms of order 

Canonical name  SomeTheoremsOnTheAxiomsOfOrder 
Date of creation  20130322 17:18:47 
Last modified on  20130322 17:18:47 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  6 
Author  Mathprof (13753) 
Entry type  Theorem 
Classification  msc 51G05 
Related topic  BetweennessRelation 