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: ${\displaystyle K}$ ${\displaystyle +,\;\cdot }$ ${\displaystyle 0,1\in K}$

1. ${\displaystyle (K,+)}$ is a group with the neutral element 0.
2. It applies and ${\displaystyle 0\cdot x=x\cdot 0=0}$ ${\displaystyle 1\cdot x=x\cdot 1=x}$
3. Are and holds , then there is exactly one and at least one such that and holds. ${\displaystyle a,b,c\in K}$ ${\displaystyle a\neq b}$ ${\displaystyle x\in K}$ ${\displaystyle y\in K}$ ${\displaystyle xa=c+xb}$ ${\displaystyle by=ay+c}$

Equivalent: is a Cartesian group if and only if ${\displaystyle (K,+,\cdot ,0,1)}$

1. ${\displaystyle (K,T,0,1)}$is a ternary body with the ternary link and ${\displaystyle T(a,b,c)=a\cdot b+c}$
2. in the associative law holds, i.e. is always fulfilled. ${\displaystyle (K,+,0)}$ ${\displaystyle a,b,c\in K}$ ${\displaystyle a+(b+c)=(a+b)+c}$

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 . ${\displaystyle (K,+,\cdot ,0,1)}$
• The affine level above is built up via the ternary link as described in the article Ternary body (for the linear case). ${\displaystyle K}$
• 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 . ${\displaystyle +}$
• 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 ${\displaystyle K=\mathbb {R} }$

${\displaystyle a\ast _{c}b={\begin{cases}a\cdot c\cdot b&a<0\land b<0\\a\cdot b&{\text{sonst.}}\end{cases}}}$

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. ${\displaystyle c\neq 1}$ ${\displaystyle (\mathbb {R} ,+,\ast _{c},0,1)}$

• Obviously, any ordered body can be used instead of in the last example . This leads to an infinite number of non-isomorphic Cartesian groups that all contain an infinite number of elements. The projective planes are arranged projective planes of the Lenz-Barlotti class III.2 and, if the base field is a part of the real numbers, arranged Archimedes and therefore homeomorphic to a sub-plane of the real projective plane. ${\displaystyle \mathbb {R} }$
• 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. ${\displaystyle K_{c}=(K,+,\ast _{c},0,1)}$ ${\displaystyle K_{c}}$

• 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 : ${\displaystyle C=C(k,\rho ,\sigma )=(\mathbb {R} ,+,\circ )}$ ${\displaystyle (\mathbb {R} ,+,\cdot )}$ ${\displaystyle k,\rho ,\sigma >0;\tau =\rho +\sigma -1>0}$ ${\displaystyle a,x\in \mathbb {R} }$

${\displaystyle a\circ x={\begin{cases}a\cdot x&(0\leq a,x),\\a\cdot x^{\rho ^{-1}}&(a<0<x),\\a^{\sigma ^{-1}}\cdot x&(x<0<a),\\k\cdot (-a)^{\sigma \tau ^{-1}}\cdot (-x)^{\sigma \tau ^{-1}}&(a,x<0).\end{cases}}}$

Each such Cartesian group coordinates a parameter- dependent, non-desargue, arranged, flat projective plane that belongs to Lenz-Barlotti class II.1. ${\displaystyle C(k,\rho ,\sigma )}$ ${\displaystyle 0<k,\rho ,\sigma \in \mathbb {R} ,\sigma <1}$ ${\displaystyle (\rho ,\sigma )\neq (1,1)}$

• If you start from the body , the arrangement and addition are retained and a new multiplication is declared through ${\displaystyle (\mathbb {R} ,+,\cdot )}$ ${\displaystyle s,x\in \mathbb {R} }$

${\displaystyle s\circ x=\alpha ^{-1}(\alpha (s)\cdot \alpha (x))\quad \mathrm {mit} \quad \alpha (x)={\begin{cases}x&(|x|<1),\\{\frac {1}{2}}(x+\operatorname {sgn}(x))&(|x|\geq 1),\end{cases}}}$

then a Cartesian group is obtained which coordinates a flat projective plane of the Lenz-Barlotti type II.2. ${\displaystyle (\mathbb {R} ,+,\circ )}$

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

1. ↑ Lenz (1955)
2. ^ ^{A b} Hauke ​​Klein: Cartesian Group. In: Geometry. University of Kiel, November 29, 2002, accessed on December 25, 2011 (English). 
3. ↑ Benz (1990), p. 244
4. ↑ All statements and examples given in this section can be found with evidence of the original literature in the book by Prieß-Crampe (1983)
5. ↑ Pierce (1961)