In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 eleme ...
' 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 ( reflexive ...
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 Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affin ...
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 operation (mathematics), operations and relation (mathematics), relations of that structure. The opposite of "reduct" is "expansion. ...
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 ration ...
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 bas ...
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 boun ...
. 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 t ...
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 t ...
(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 referenc ...
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
* ''M'' is a limit ≡ (∀ ''n'' > 2
) ∧ (∀ P
⇒ ∃ ''n''
).
References
{{Reflist
Order theory