Point-pair separation


In mathematics, two pairs of points in a cyclic order such as the real projective line separate each other when they occur alternately in the order. Thus the ordering a b c d of four points has and as separating pairs. This point-pair separation is an invariant of projectivities of the line.

Concept

The concept was described by G. B. Halsted at the outset of his Synthetic Projective Geometry:
Given any pair of points on a projective line, they separate a third point from its harmonic conjugate.
A pair of lines in a pencil separates another pair when a transversal crosses the pairs in separated points.
The point-pair separation of points was written AC//BD by H. S. M. Coxeter in his textbook The Real Projective Plane.

Application

The relation may be used in showing the real projective plane is a complete space.
The axiom of continuity used is "Every monotonic sequence of points has a limit." The point-pair separation is used to provide definitions:
  • is monotonic ≡ ∀ n > 1
  • M is a limit ≡ ∧.

    Unoriented circle

Whereas a linear order endows a set with a positive end and a negative end, an other relation forgets not only which end is which, but also where the ends are located. In this way it is a final, further weakening of the concepts of a betweenness relation and a cyclic order. There is nothing else that can be forgotten: up to the relevant sense of interdefinability, these three relations are the only nontrivial reducts of the ordered set of rational numbers.
A quaternary relation is defined satisfying certain axioms, which is interpreted as asserting that a and c separate b from d.

Axioms

The separation relation was described with axioms in 1898 by Giovanni Vailati.
  • ' = '
  • ' = '
  • ' ⇒ ¬ '
  • ' ∨ ' ∨ '
  • ' ∧ ' ⇒ '.