Separation relation

In

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

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

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