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

gollark: The internet connection here is *apiohazardously* bad and I'm forced to `pacman`-update over it.

gollark: I mean, JSON has problems, but it is at least simple and coherent.

gollark: Maybe? Less so for some.

gollark: I was looking at the WHATWG's documentation on application/x-www-form-urlencoded and it says this:> The application/x-www-form-urlencoded format is in many ways an aberrant monstrosity, the result of many years of implementation accidents and compromises leading to a set of requirements necessary for interoperability, but in no way representing good design practices.

gollark: I feel like they could probably just, if it's for array literals, get away with limiting it to 16 or so with no major issues.

References

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
Macpherson, H. Dugald (2011), "A survey of homogeneous structures" (PDF), Discrete Mathematics, doi:10.1016/j.disc.2011.01.024, retrieved 28 April 2011
Bertrand Russell (1903) Principles of Mathematics, page 214
H. S. M. Coxeter (1949) The Real Projective Plane, Chapter 10: Continuity, McGraw Hill

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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, 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