HOME

TheInfoList



OR:

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 In mathematics, a finitary relation over sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples consisting of elements ''x'i'' in ''X'i''. Typically, the relation describes a possible connection between the elemen ...
' satisfying certain axioms, which is interpreted as asserting that ''a'' and ''c'' separate ''b'' from ''d''. Whereas a
linear order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
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 In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. In ...
. There is nothing else that can be forgotten: up to the relevant sense of interdefinability, these three relations are the only nontrivial
reduct In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion." Definition Let ''A'' be an algebraic structu ...
s of the ordered set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s.


Application

The separation may be used in showing the
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
is a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bo ...
. The separation relation was described with axioms in 1898 by
Giovanni Vailati Giovanni Vailati (24 April 1863 – 14 May 1909) was an Italian proto-analytic philosopher, historian of science, and mathematician. Life Vailati was born in Crema, Lombardy, and studied engineering at the University of Turin. He went on to le ...
. * ' = ' * ' = ' * ' ⇒ ¬ ' * ' ∨ ' ∨ ' * ' ∧ ' ⇒ '. The relation of separation of points was written AC//BD by
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...
in his textbook ''The Real Projective Plane''.
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...
(1949) ''The Real Projective Plane'', Chapter 10: Continuity,
McGraw Hill McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes referen ...
The axiom of continuity used is "Every monotonic sequence of points has a limit." The separation relation is used to provide definitions: * is monotonic ≡ ∀ ''n'' > 1 A_0 A_n // A_1 A_. * ''M'' is a limit ≡ (∀ ''n'' > 2 A_1 A_n // A_2 M) ∧ (∀ P A_1P // A_2 M ⇒ ∃ ''n'' A_1 A_n // P M ).


References

{{Reflist Order theory