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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:
- $B_{*pq}:=\lbrace r\in A\mid (r,p,q)\in B\rbrace$ ,
- $B_{p*q}:=\lbrace r\in A\mid (p,r,q)\in B\rbrace$ ,
- $B_{pq*}:=\lbrace r\in A\mid (p,q,r)\in B\rbrace$ ,
- $B_{pq}:=B_{p*q}\cup\lbrace q\rbrace\cup B_{pq*}$ , and
- $B(p,q):=B_{*pq}\cup\lbrace p\rbrace\cup B_{pq}$ .
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.
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