Quasi-bodies

A quasi-body , also called the Veblen-Wedderburn system after Oswald Veblen and Joseph Wedderburn , is an algebraic structure that is used in synthetic geometry as a coordinate range for certain affine planes , the affine translation planes . Quasi-bodies are always Cartesian groups and every alternative body is a quasi-body.

Definitions

In the geometrical definition, an affine coordinate system is introduced on an affine translation plane by choosing a point base . The points on the first axis of this coordinate system serve as coordinates. On the coordinate area , addition and multiplication are introduced through geometric construction. ${\displaystyle (O,E_{1},E_{2})}$ ${\displaystyle OE_{1}}$ ${\displaystyle K=OE_{1}}$

With the algebraic definition, the quasi-body is characterized by its algebraic properties and an affine translation plane is built up on the set of pairs as point coordinates by algebraic equations that describe the straight lines. ${\displaystyle (K,+,\cdot )}$ ${\displaystyle K^{2}}$

Geometric definition

An affine plane is called an affine translation plane if there is a translation for every pair of points , i.e. a collineation with the properties ${\displaystyle A}$ ${\displaystyle (P,Q)\in A^{2}}$ ${\displaystyle \tau ={\overrightarrow {PQ}}}$ ${\displaystyle \tau :A\rightarrow A}$

• ${\displaystyle \tau (P)=Q}$,
• for each line is the level , ${\displaystyle g}$ ${\displaystyle \tau (g)\parallel g}$
• ${\displaystyle \tau }$ is the identity or free of fixed points.

An affine plane is a translation plane if and only if Desargues' small affine theorem holds in it.

In the affine translation plane, three different points are selected that do not lie on a common straight line. The points on the first coordinate axis serve as coordinates . A pair can be assigned to each point of the plane in a reversible and unambiguous manner using the coordinate construction. ${\displaystyle O,E_{1},E_{2}}$ ${\displaystyle K=OE_{1}}$ ${\displaystyle (x_{1},x_{2})\in K^{2}}$

addition

Addition of two elements . The sum is independent of the position of the auxiliary point ${\displaystyle a,b\in K=OE_{1}}$ ${\displaystyle a+b}$ ${\displaystyle H}$

Let two points be on the first coordinate axis . Their sum is again obtained as a point on this axis by the following construction, compare the figure on the right: ${\displaystyle a,b\in K}$ ${\displaystyle OE_{1}}$ ${\displaystyle a+b}$

1. Pick an auxiliary point outside the first coordinate axis. ${\displaystyle H}$
2. The parallel to through cuts the parallel to through in . ${\displaystyle OE_{1}}$ ${\displaystyle H}$ ${\displaystyle OH}$ ${\displaystyle a}$ ${\displaystyle P}$
3. The parallel to through intersects the first coordinate axis in the point . This point is the sum you are looking for. ${\displaystyle bH}$ ${\displaystyle P}$ ${\displaystyle OE_{1}}$ ${\displaystyle a+b}$

The result of the construction is independent of which auxiliary point outside the first coordinate axis is used. Of the underlying coordinate system, only the origin and the first coordinate axis go into the construction as a straight line. That means: If you choose a different coordinate system with the same origin and the same first coordinate axis, but a different first unit point on this axis and any second unit point outside the first axis, then the addition does not change. ${\displaystyle H}$

The addition so constructed becomes a commutative group . Your neutral element is the origin of the coordinate system. It is isomorphic to the group of parallel displacements in the direction of the first coordinate axis - and thus to every group of parallel displacements of the plane in a fixed direction . ${\displaystyle (K,+)}$ ${\displaystyle O}$

multiplication

Multiplication of two elements . ${\displaystyle a,b\in K}$

Let two points be on the first coordinate axis . Their product is obtained again as a point on this axis through the following construction, compare the figure on the right: ${\displaystyle a,b\in K}$ ${\displaystyle OE_{1}}$ ${\displaystyle a\cdot b}$

1. The parallel to through intersects the second coordinate axis in . ${\displaystyle E_{1}E_{2}}$ ${\displaystyle b}$ ${\displaystyle OE_{2}}$ ${\displaystyle B}$
2. The parallel to through intersects the first coordinate axis in the point . For technical reasons, this point is also labeled in the drawing . ${\displaystyle aE_{2}}$ ${\displaystyle B}$ ${\displaystyle OE_{1}}$ ${\displaystyle a\cdot b}$ ${\displaystyle a*b}$

With the two connections of addition and multiplication, the first coordinate axis fulfills the following algebraic properties of a quasi-body. The neutral element of multiplication is the first unit point . ${\displaystyle K=OE_{1}}$ ${\displaystyle 1=E_{1}}$

Algebraic definition

A set with two-digit connections and two different structural constants is called a (left) quasi-body if the following axioms hold: ${\displaystyle K}$ ${\displaystyle +,\;\cdot }$ ${\displaystyle 0,1\in K}$

1. ${\displaystyle (K,+)}$is an Abelian group with neutral element 0.
2. ${\displaystyle (K\setminus \lbrace 0\rbrace ,\cdot )}$is a loop with neutral element 1, i.e. a quasi-group with both left and right neutral element 1.
3. ${\displaystyle a\cdot 0=0\cdot a=0}$applies to everyone . ${\displaystyle a\in K}$
4. It applies the Linksdistributivgesetz : for all . ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle a,b,c\in K}$
5. To with there is exactly one with . ${\displaystyle a,b,c\in K}$ ${\displaystyle a\neq b}$ ${\displaystyle y\in K}$ ${\displaystyle a\cdot y-b\cdot y=c}$

If the structure fulfills these properties of a quasi-body, then straight lines can be defined by coordinate equations on the set of points given by the set of pairs . The structure of points and straight lines then forms an affine translation plane . → The straight line equations are described in the article Ternary bodies in the section Geometry of the plane . ${\displaystyle (K,+,\cdot )}$ ${\displaystyle A=K^{2}}$

Core of a quasi-body

The amount

${\displaystyle \operatorname {Kern} (K)=\lbrace x\in K:\;\forall a,b\in K\left((a+b)x=ax+bx\land (ab)x=a(bx)\right)\rbrace }$

is called the core of the quasi-body. This core is an oblique body . The quasi-body is a module above its core.

Features and remarks

• The quasi-body defined by the axioms is more precisely a left-hand quasi-body , because the law of left distribution applies in it. Also right quasifield - with Rechtsdistributivgesetz instead 4th and accordingly formulated with reverse multiplication 5. Axiom - are drawn in the literature simply as quasifield, but here come the eligible terms before.
• A quasi-body in which both distributive laws apply is called a half-body in geometry . Note, however, that this term is not used uniformly in mathematics and compare half-bodies .
• Obviously, by reversing the multiplication, a left quasi-body becomes a right quasi-body and vice versa.
• If the alternative , a weakened form of the associative law of multiplication, applies in a half body in the sense of synthetic geometry in addition to both distributive laws, then this half body is even an alternative field .
• Through the definition , a ternary link can be introduced on each quasi-body , with which the quasi-body becomes a linear ternary body . ${\displaystyle T(a,b,c)=a\cdot b+c}$
• Regarding the 5th axiom of the quasi-body in the algebraic definition, it should be noted:

1. If the legal distributive law also applies , then the 5th axiom follows from the first three axioms, it is a real weakening of the legal distributive law. ${\displaystyle K}$
2. It is dispensable, that is, it follows from the other axioms without further assumptions if is finite. ${\displaystyle K}$

• Quasi-bodies were referred to in the literature as the Veblen-Wedderburn system until 1975 .
• Each quasi-body is a Cartesian group .
• Every almost body is a quasi-body. A quasi-body is an almost-body if and only if its multiplication is associative.

Quasi-bodies as coordinate areas of projective planes

• Quasi-bodies also appear as coordinate areas of special projective planes . In the classification of projective levels according to Hanfried Lenz, these are the levels of classes IV, V and VII.
  • More precisely, the following applies: A projective level of class IVa or IVb can be coordinated by choosing a suitable point base using a left quasi-body or a right-hand quasi-body. Every ternary body that is assigned to the plane when any point base is chosen is isotopic to a left or right quasi-body.
  • All coordinate areas of a projective plane of class V are half-bodies isotopic to one another, i.e. right and left quasi-bodies at the same time. In general, however, these half-bodies are not isomorphic to one another.
  • All coordinate areas of a projective plane of class VII are alternative bodies isomorphic to one another.

Examples

• Every stronger algebraic structure, i.e. every almost body , alternative body , inclined body or body provides an example of a quasi-body.
• The article Ternary Body contains some examples of quasi-bodies, in particular a detailed example of a finite, non-commutative quasi-body (→ in the subsection Examples of order 9 ).

Quasi-bodies of finite Moulton planes

The finite Moulton planes have “real” quasi-bodies as coordinate areas. The construction is based on a finite field whose characteristic is an odd prime number . In the cyclic , multiplicative group, there is exactly one subgroup of index 2, that is the subgroup of squares . Let be a body automorphism of . Now a new multiplication is introduced: ${\displaystyle F}$ ${\displaystyle F^{\ast }=F\setminus \lbrace 0\rbrace }$ ${\displaystyle Q_{1}}$ ${\displaystyle Q_{1}=(F^{\ast })^{2}=\lbrace x^{2}:x\in F^{\ast }\rbrace }$ ${\displaystyle \varphi }$ ${\displaystyle F}$ ${\displaystyle \odot :F\times F\rightarrow F}$

${\displaystyle a\odot b={\begin{cases}a\cdot b\quad \mathrm {falls} \;a\in Q_{1}\\a\cdot \varphi (b)\quad \mathrm {falls} \;a\not \in Q_{1}.\end{cases}}}$

This becomes a left quasi-body, because the left distribution law is fulfilled. If the chosen body automorphism is not identity, then is ${\displaystyle (F,+,\odot ,0,1)}$ ${\displaystyle \varphi }$

1. the commutative law for the link is not fulfilled, ${\displaystyle \odot }$
2. the associative law for the connection is fulfilled if and only if is involutorial, i.e. if and only then F is a left fast body with the new multiplication ), ${\displaystyle \odot }$ ${\displaystyle \varphi }$ ${\displaystyle \varphi ^{2}=\mathrm {Id_{F}} }$
3. the Rechtsdistributivgesetz not fulfilled as always elements with exist. ${\displaystyle a,b\in Q_{1}}$ ${\displaystyle a+b\not \in Q_{1}}$
4. The 5th axiom for quasi-bodies follows from the other axioms, since is finite. ${\displaystyle F}$
5. ${\displaystyle (F\setminus \{0\},\odot ,1)}$is a loop: The neutrality of the one element of the "ordinary" body multiplication is also evident. The solutions to equations and , are ${\displaystyle \odot }$ ${\displaystyle a\odot x=b}$ ${\displaystyle y\odot a=b}$ ${\displaystyle (a,b\in F\setminus \{0\})}$

   ${\displaystyle x=a\operatorname {\setminus } b={\begin{cases}a^{-1}\cdot b\,&(a\in Q_{1})\\\varphi ^{-1}(a^{-1}\cdot b)\,&(a\not \in Q_{1})\end{cases}}\quad \mathrm {bzw.} \;y=b/a={\begin{cases}b\cdot a^{-1}\,&(a\cdot b\in Q_{1})\\b\cdot \varphi (a^{-1})\,&(a\cdot b\not \in Q_{1})\end{cases}}}$

The core of the quasi body is from Körperautomorphismus fixed finite subfield of . ${\displaystyle (F,+,\odot ,0,1)}$ ${\displaystyle \varphi }$ ${\displaystyle S=\{x\in F:\varphi (x)=x\}}$ ${\displaystyle F}$

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 .
• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . In: Mathematics for teaching at high schools . 1st edition. Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .
• Donald Ervin Knuth : Finite Semifields and Projective Planes . In: Marshall Hall [Research Adviser] (Ed.): Selected Papers on Discrete Mathematics . Dissertation. California Institute of Technology, Stanford, California January 1, 1963 ( full text [accessed April 13, 2012]).
• WA Pierce: Moulton Planes . In: Canadian J. Math. Band 13 , 1961, pp. 427-436 .
• Günter Pickert : Geometric identification of a class of finite Moulton planes . In: Journal for Pure and Applied Mathematics (Crelles Journal) . tape 1964 , no. 214-215 , 1964, ISSN  1435-5345 , pp. 405-411 , doi : 10.1515 / crll.1964.214-215.405 ( full text from DigiZeitschriften [accessed on February 26, 2012]).
• Günter Pickert: Level incidence geometry . 2nd Edition. Frankfurt am Main 1968.
• Oswald Veblen , Joseph Wedderburn : Non-Desarguesian and non-Pascalian geometries . In: Transactions of the American Mathematical Society . tape 8 . American Mathematical Society , 1907, pp. 379-388 .
• 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 February 26, 2012]).

References and comments

1. ↑ Degen (1976), p. 50.
2. ↑ Degen (1976), sentence 2.13
3. ↑ Degen (1976), p. 50.
4. ↑ Degen (1976), sentence 2.17
5. ↑ Weibel (2007) explicitly formulates axioms “(for) a (right) quasi-field ”, otherwise simply calls it “quasi-field”. He mentions (p. 1300) that "(for) a left quasi-field ... is aa right quasi-field". Degen (1976) only knows one type of "quasi-body" that fulfills the law of left distribution, ie "left quasi-field". ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R^{op}}$
6. ↑ Hauke Klein: Semidfields. In: Geometry. University of Kiel, November 29, 2002, accessed on December 13, 2010 (English). 
7. ↑ Weibel (2007) p. 1297.
8. ↑ Weibel (2007), p. 1300.
9. ↑ Knuth (1963)
10. ^ Name after Pierce (1961) and Pickert (1964)
11. ↑ Pickert (1964)