some theorems on strict betweenness relations


Let B be a strict betweenness relation. In the following the sets B*pq,Bp*q,Bpq*,Bpq,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, Bp*q and Bpq* are pairwise disjoint. Furthermore, if p=q then all three sets are empty.

Theorem 3.

If B is strict, then BpqBqp=Bp*q and BpqBqp=B(p,q).

Theorem 4.

If B is strict, then for any p,qA, pq, B*pq, Bp*q and Bpq* are infiniteMathworldPlanetmath.

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