some theorems on the axioms of order


Let B be a betweenness relation on a set A.

Theorem 1.
  • If (a,b,c)B and (a,c,d)B, then (a,b,d)B.

  • Theorem 2.

    For each pair of elements p,qA, we can define five sets:

    1. 1.

      B*pq:={rA(r,p,q)B},

    2. 2.

      Bp*q:={rA(p,r,q)B},

    3. 3.

      Bpq*:={rA(p,q,r)B},

    4. 4.

      Bpq:=Bp*q{q}Bpq*, and

    5. 5.

      B(p,q):=B*pq{p}Bpq.

    Then

    • (1)

      B*pq=Bqp*.

    • (2)

      Bp*q=Bq*p.

    • (3)

      The intersection of any pair of the first three sets contains at most one element, either p or q.

    • (4)

      Each of the sets can be partially ordered.

    • (5)

      The partial orderMathworldPlanetmath on Bpq and B(p,q) extends that of the subsets.

    Title some theorems on the axioms of order
    Canonical name SomeTheoremsOnTheAxiomsOfOrder
    Date of creation 2013-03-22 17:18:47
    Last modified on 2013-03-22 17:18:47
    Owner Mathprof (13753)
    Last modified by Mathprof (13753)
    Numerical id 6
    Author Mathprof (13753)
    Entry type Theorem
    Classification msc 51G05
    Related topic BetweennessRelation