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

• $B_{*pq}=B_{qp*}.$

• $B_{p*q}=B_{q*p}.$

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

• Each of the sets can be partially ordered.

• The partial order on $B_{pq}$ and $B(p,q)$ extends that of the subsets.