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

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:

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

References

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