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Homesome theorems on strict betweenness relations

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# 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.

Related:

StrictBetweennessRelation

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

51G05*no label found*

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