strict betweenness relation (original) (raw)

1 Definition

A strict betweenness relation is a betweenness relation that satisfiesthe following axioms:

• O⁢2′
  (p,q,p)∉B) for each pair of points p and q.
• O⁢3′
  for each p,q∈A such that p≠q, there is an r∈A such that (p,q,r)∈B.
• O⁢4′
  for each p,q∈A such that p≠q, there is an r∈A such that (p,r,q)∈B.
• O⁢5′
  if (p,q,r)∈B, then (q,p,r)∉B.

2 Remarks

• •
  A very simple example of a strict betweenness relation is the empty set. 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 reducedto 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 O⁢2′ we have (p,p,p)∉B.