Point-pair separation

Template:Short description In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order. Thus the ordering a b c d of four points has (a,c) and (b,d) 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: Template:Quote

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:

• {A_n} is monotonic ≡ ∀ n > 1 $A_0{A}_n//A_1{A}_{n+1}.$
• M is a limit ≡ (∀ n > 2 $A_1{A}_n//A_{2}M$) ∧ (∀ P $A_{1}P//A_{2}M$ ⇒ ∃ n $A_1{A}_n//PM$ ).

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 Template:Not a typo 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.

• Template:Not a typo = Template:Not a typo
• Template:Not a typo = Template:Not a typo
• Template:Not a typo ⇒ ¬ Template:Not a typo
• Template:Not a typo ∨ Template:Not a typo ∨ Template:Not a typo
• Template:Not a typo ∧ Template:Not a typo ⇒ Template:Not a typo.