## Thursday, January 9, 2014

### Partial orders and betweenness

Given any partial order we can produce a ternary betweenness relation defined by the condition that b is between a and c if either a <= b <= c or c <= b <= a. Here is an example of a betweenness relation corresponding to a partial order relation:
(= (betweenness-relation (weak-order [#{0} #{1} #{2}]))
#user.relation{
:vertices #{0 1 2}
:edges #{[2 1 0] [1 0 0] [2 2 1] [1 1 1]
[0 0 1] [1 2 2] [0 1 2] [2 1 1]
[2 2 2] [1 1 2] [0 0 2] [2 0 0]
[2 2 0] [1 1 0] [0 0 0]
[0 1 1] [0 2 2]})

Like with the ternary relations of infima and suprema the ternary relation of betweenness preserves the connectivity of the underlying partial order.
(= (connected-components order)
(connected-components (betweenness-relation order)))

A convex set of a betweenness relation is a set that includes all elements between its members. Ordered geometry uses betweenness relations as part of its foundation as we can define the elements that are between any points in a geometric space.