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 $B_{\mathrm{*}p\mathit{}q}$, $B_{p\mathrm{*}q}$ and $B_{p\mathit{}q\mathit{}\mathrm{*}}$ are pairwise disjoint. Furthermore, if $p\mathrm{=}q$ then all three sets are empty.

Theorem 3.

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

Theorem 4.

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

Title	some theorems on strict betweenness relations
Canonical name	SomeTheoremsOnStrictBetweennessRelations
Date of creation	2013-03-22 17:18:59
Last modified on	2013-03-22 17:18:59
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	6
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 51G05
Related topic	StrictBetweennessRelation