Inmathematics, two pairs of points in acyclic order such as thereal projective lineseparate each other when they occur alternately in the order. Thus the orderinga b c d of four points has (a,c) and (b,d) as separating pairs. Thispoint-pair separation is an invariant of projectivities of the line.

The concept was described byG. B. Halsted at the outset of hisSynthetic Projective Geometry:

With regard to a pair of different points of those on a straight, all remaining fall into two classes, such that every point belongs to one and only one. If two points belong to different classes with regard to a pair of points, then also the latter two belong to different classes with regard to the first two. Two such point pairs are said to 'separate each other.' Four different points on a straight can always be partitioned in one and only one way into pairs separating each other.^[1]

Given any pair of points on a projective line, they separate a third point from itsharmonic conjugate.

A pair of lines in apencil separates another pair when atransversal crosses the pairs in separated points.

The point-pair separation of points was written AC//BD byH. S. M. Coxeter in his textbookThe Real Projective Plane.^[2]

The relation may be used in showing thereal projective plane is acomplete 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} ismonotonic ≡ ∀n > 1${\displaystyle A_0{A}_n//A_1{A}_{n+1}.}$
• M is alimit ≡ (∀n > 2${\displaystyle A_1{A}_n//A_{2}M}$) ∧ (∀ P${\displaystyle A_{1}P//A_{2}M}$ ⇒ ∃n${\displaystyle A_1{A}_n//PM}$ ).

Whereas alinear 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 abetweenness relation and acyclic order. There is nothing else that can be forgotten: up to the relevant sense of interdefinability, these three relations are the only nontrivialreducts of the ordered set ofrational numbers.^[3]

Aquaternary relationS(a, b, c, d) is defined satisfying certain axioms, which is interpreted as asserting thata andc separateb fromd.^[4][5]

The separation relation was described with axioms in 1898 byGiovanni Vailati.^[6]

• abcd =badc
• abcd =adcb
• abcd ⇒ ¬acbd
• abcd ∨acdb ∨adbc
• abcd ∧acde ⇒abde.

• Projective geometry
• Order theory

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