# some theorems on strict betweenness relations

Let $B$ be a strict betweenness relation. In the following the sets $B_{*pq},B_{p*q},B_{pq*},B_{pq},B(p,q)$ are defined in the entry about some theorems on the axioms of order.

###### Theorem 1.

Three elements are in a strict betweenness relation only if they are pairwise distinct.

###### Theorem 2.

If $B$ is strict, then $B_{*pq}$, $B_{p*q}$ and $B_{pq*}$ are pairwise disjoint. Furthermore, if $p=q$ then all three sets are empty.

###### Theorem 3.

If $B$ is strict, then $B_{pq}\cap B_{qp}=B_{p*q}$ and $B_{pq}\cup B_{qp}=B(p,q)$.

###### Theorem 4.

If $B$ is strict, then for any $p,q\in A$, $p\neq q$, $B_{*pq}$, $B_{p*q}$ and $B_{pq*}$ are infinite.

Title some theorems on strict betweenness relations SomeTheoremsOnStrictBetweennessRelations 2013-03-22 17:18:59 2013-03-22 17:18:59 Mathprof (13753) Mathprof (13753) 6 Mathprof (13753) Theorem msc 51G05 StrictBetweennessRelation