betweenness relation

1 Definition

Let A be a set. A ternary relationPlanetmathPlanetmath B on A is said to be a betweenness relation if it has the following properties:

  1. O1

    if (a,b,c)B, then (c,b,a)B; in other words, the set


    is a symmetric relationMathworldPlanetmath ( for each b; thus, from now on, we may say, without any ambiguity, that b is between a and c if (a,b,c)B;

  2. O2

    if (a,b,a)B, then a=b;

  3. O3

    for each a,bA, there is a cA such that (a,b,c)B;

  4. O4

    for each a,bA, there is a cA such that (a,c,b)B;

  5. O5

    if (a,b,c)B and (b,a,c)B, then a=b;

  6. O6

    if (a,b,c)B and (b,c,d)B, then (a,b,d)B;

  7. O7

    if (a,b,d)B and (b,c,d)B, then (a,b,c)B.

Title betweenness relation
Canonical name BetweennessRelation
Date of creation 2013-03-22 17:18:44
Last modified on 2013-03-22 17:18:44
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 6
Author Mathprof (13753)
Entry type Definition
Classification msc 51G05
Synonym axioms of order
Related topic SomeTheoremsOnTheAxiomsOfOrder