Separation relation

In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. It is defined as a quaternary relation S(a, b, c, d) satisfying certain axioms, which is interpreted as asserting that a and c separate b from d.[1]

Whereas a linear order endows a set with a positive end and a negative end, a separation 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.[2]

## Application

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The separation may be used in showing the real projective plane is a complete space. The separation relation was described with axioms in 1898 by Giovanni Vailati.[3]

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

The relation of separation of points was written AC//BD by H. S. M. Coxeter in his textbook The Real Projective Plane.[4] The axiom of continuity used is "Every monotonic sequence of points has a limit." The separation relation is used to provide definitions:

* {An} is monotonic ≡ ∀ n > 1  A 0 A n / / A 1 A n + 1 . {\displaystyle A_{0}A_{n}//A_{1}A_{n+1}.}
* M is a limit ≡ (∀ n > 2  A 1 A n / / A 2 M {\displaystyle A_{1}A_{n}//A_{2}M} ) ∧ (∀ P  A 1 P / / A 2 M {\displaystyle A_{1}P//A_{2}M} ⇒ ∃ n A 1 A n / / P M {\displaystyle A_{1}A_{n}//PM} ).

## References

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1. ^ Huntington, Edward V. (July 1935), "Inter-Relations Among the Four Principal Types of Order" (PDF), Transactions of the American Mathematical Society, 38 (1): 1–9, doi:10.1090/S0002-9947-1935-1501800-1, retrieved 8 May 2011
2. ^ Macpherson, H. Dugald (2011), "A survey of homogeneous structures" (PDF), Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024, retrieved 28 April 2011
3. ^ Bertrand Russell (1903) Principles of Mathematics, page 214
4. ^ H. S. M. Coxeter (1949) The Real Projective Plane, Chapter 10: Continuity, McGraw Hill