strict betweenness relation

1 Definition

A strict betweenness relation is a betweenness relation that satisfies the following axioms:

  • O2

    (p,q,p)B) for each pair of points p and q.

  • O3

    for each p,qA such that pq, there is an rA such that (p,q,r)B.

  • O4

    for each p,qA such that pq, there is an rA such that (p,r,q)B.

  • O5

    if (p,q,r)B, then (q,p,r)B.

2 Remarks

  • A very simple example of a strict betweenness relation is the empty setMathworldPlanetmath. In , all the conditions are vacuously satisfied. The empty set, in this context, is called the trivial strict betweenness relation.

  • Any strict betweenness relation can be enlarged to a betweenness relation by including all triples of the forms (p,p,q),(p,q,p), or (p,q,q).

  • Conversely, any betweenness relation can be reduced to a strict betweenness relation by removing all triples of the forms just listed. However, it is possible that the “derived” strict betweenness relation is trivial.

  • From axiom O2 we have (p,p,p)B.

Title strict betweenness relation
Canonical name StrictBetweennessRelation
Date of creation 2013-03-22 17:18:56
Last modified on 2013-03-22 17:18:56
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Definition
Classification msc 51G05
Related topic SomeTheoremsOnStrictBetweennessRelations