Cartesian group
A Cartesian group (also: Cartesian group , engl. Cartesian Group ) is an algebraic structure , in the synthetic geometry as coordinate range for certain affine and projective serves levels. The term goes back to Reinhold Baer . Every Cartesian group can be made into a ternary body , every quasi-body is a Cartesian group. The projective level above a Cartesian group belongs to Lenz class II or a higher class (III, IVa, IVb, V or VII).
definition
A set with the two-digit connections and two different structural constants is called a Cartesian group if the following axioms hold:
- is a group with the neutral element 0.
- It applies and
- Are and holds , then there is exactly one and at least one such that and holds.
Equivalent: is a Cartesian group if and only if
- is a ternary body with the ternary link and
- in the associative law holds, i.e. is always fulfilled.
Features and remarks
- In the 3rd axiom of the first system, the existence of at least one “left solution” x and exactly one “right solution” y can be demanded equally . The uniqueness of the solution, which is not specifically required in the axioms, can then be derived from the other axioms.
- The ternary field named in the second system of axioms and clearly determined by it is always linear .
- The affine level above is built up via the ternary link as described in the article Ternary body (for the linear case).
- The projective conclusion of the affine level mentioned belongs at least to Lenz class II.
- The "addition" in a Cartesian group does not have to be commutative .
- A Cartesian group is a special linear ternary field, i.e. an algebraic structure with two different links, in contrast to the usual concept of a group . With the addition alone, each Cartesian group forms a group in the otherwise usual sense of algebra.
Examples
- Every quasi-body and every stronger algebraic structure, i.e. every near-body , half-body , alternative body , oblique body or body provides an example for a Cartesian group.
Coordinate areas of arranged and planar projective planes
If an affine plane allows a (“strong”) arrangement , then its projective closure is also an arranged projective plane. Then its coordinate ternary body is also arranged and thus infinite. In the 1960s, some examples of arranged , true Cartesian groups were found that coordinate such arranged planes.
An arranged projective plane is referred to as a plane projective plane if it is homeomorphic to the usual real projective plane in its “natural” topology , which is induced here by the arrangement of one (and thus each) of its coordinate ternary bodies . The coordinate ternary body of a planar projective plane then always allows an Archimedean arrangement .
- The coordinate area of the Moulton plane is a Cartesian group that is not a quasi-body. One uses the usual addition in the field of real numbers and defines a new multiplication through
- with a positive constant . Then there is a Cartesian group with commutative addition and commutative, non-associative multiplication. None of the distributive laws are fulfilled, so it is not a quasi-body.
- Obviously, any
- With the above modified multiplication for a Moulton plane, one can also start from a non-commutative, arranged inclined body K instead of a commutative body. Even then, a Cartesian group always forms . A projective plane that can be coordinated by is an arranged, non-desargue projective plane and has the Lenz-Barlotti type III.2, if
c is in the center of K and otherwise the Lenz-Barlotti type III.1. Since inclined bodies arranged non-commutatively cannot be ordered in an Archimedean way, these planes are not in an Archimedean order either.
- An uncountable set of examples for a Cartesian group can be obtained from the field by choosing three real parameters . One chooses the usual real addition as addition and replaces the multiplication with the combination for :
- Each such Cartesian group coordinates a parameter- dependent, non-desargue, arranged, flat projective plane that belongs to Lenz-Barlotti class II.1.
- If you start from the body , the arrangement and addition are retained and a new multiplication is declared through
- then a Cartesian group is obtained which coordinates a flat projective plane of the Lenz-Barlotti type II.2.
literature
- Walter Benz : Fundamentals of Geometry . In: A Century of Mathematics, 1890–1990. Festschrift for the anniversary of the DMV . Vieweg, Braunschweig 1990, ISBN 3-528-06326-2 .
- Hanfried Lenz : Small Desarguessian sentence and duality in projective levels . In: Annual report of the German Mathematicians Association . tape 57 . Teubner, 1955, p. 20–31 ( Permalink to the digitized full text [accessed December 25, 2011]).
- WA Pierce: Moulton Planes . In: Canadian J. Math . tape 13 , 1961, pp. 427-436 .
- Sibylla Prieß-Crampe : Arranged structures . Groups, bodies, projective levels (= results of mathematics and its border areas . Volume 98 ). Springer, Berlin / Heidelberg / New York 1983, ISBN 3-540-11646-X ( permalink to a review of the book [accessed on June 16, 2012]).
- Charles Weibel : Survey of Non-Desarguesian Planes . In: Notices of the American Mathematical Society . tape 54 . American Mathematical Society, November 2007, pp. 1294–1303 ( ams.org [PDF; 702 kB ; accessed on July 30, 2013]).
Individual evidence
- ↑ Lenz (1955)
- ^ A b Hauke Klein: Cartesian Group. In: Geometry. University of Kiel, November 29, 2002, accessed on December 25, 2011 (English).
- ↑ Benz (1990), p. 244
- ↑ All statements and examples given in this section can be found with evidence of the original literature in the book by Prieß-Crampe (1983)
- ↑ Pierce (1961)