# 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 $p,q\in A$, we can define five sets:

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

$B_{pq}:=B_{p*q}\cup\{q\}\cup B_{pq*}$, and

5. 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 SomeTheoremsOnTheAxiomsOfOrder 2013-03-22 17:18:47 2013-03-22 17:18:47 Mathprof (13753) Mathprof (13753) 6 Mathprof (13753) Theorem msc 51G05 BetweennessRelation