Quasifield

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

• 2.1 Quasi body as coordinate ranges projective planes

• 3.1 Quasi- body finite - Moulton planes

Definitions

In the geometric definition an affine coordinate system is introduced to an affine translation plane by choosing a point basis. The points are used on the first axis of this coordinate system as the coordinate. In the coordinate space addition and multiplication are introduced by geometric construction.

In the definition of the algebraic quasi body is characterized by its algebraic properties and the quantity of the pairs of coordinates as an affine translation level, constituted by algebraic equations describing the straight line.

Geometric definition

An affine plane is called affine translation plane if there exists for every pair of points a translation, ie a collineation with the properties

• ,
• Is for every line of the plane
• Is the identity or fixed points.

An affine plane is a translation plane if and only if the following applies to the small affine set of Desargues in it.

In the affine translation plane three points are selected that do not lie on a common straight line. The points of the first coordinate axis as the coordinates. Each point in the plane can be clearly assigned to the reversible coordinate structure includes a pair.

Addition

Are two points on the first coordinate axis. Their sum is again obtained as a point on this axis by the following construction, to compare the picture on the right:

The result of the design does not depend on which auxiliary point outside of the first coordinate axis is used. From the underlying coordinate system go only the origin and the first coordinate axis as a straight line in the construction of a. 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 point outside the second unit of the first axis, then by the addition does not change.

By thus constructed addition is a commutative group. Your neutral element is the origin of the coordinate system. Is the group of the parallel shift in the direction of the first coordinate axis isomorphic - and thus each group of parallel shifts in the plane in a fixed direction.

Multiplication

Are two points on the first coordinate axis. Their product is obtained again as a point on this axis by the following construction, to compare the picture on the right:

With the two shortcuts addition and multiplication, the first coordinate axis, conforms to the following algebraic properties of a quasi body. The neutral element of the multiplication is the first point of unity.

Algebraic definition

A lot with the two-digit links and two different structure constants is called ( left ) quasi- body if the following axioms hold:

Meets the structure, these properties of a quasi body, then can be defined by coordinates equations on the given straight line through the set of pairs of points. The structure of points and lines then forms an affine translation plane. → The line equations are described in the article Ternärkörper section geometry of the plane.

Core of a quasi body

The amount

Is referred to as the core body of the quasi. This core is a skew field. The quasi body is a module over its core.

Properties and observations

• The point defined by the axioms of quasi body is more precisely a left quasi body, because in him the Linksdistributivgesetz applies. Also right quasi body - with Rechtsdistributivgesetz place 4th and accordingly formulated with reversed multiplication fifth axiom - are simply drawn in the literature as quasi body, but here also occur the qualified terms.
• A quasi body in which both distributive laws are valid is called in geometry as a half body. Note, however, that this term in mathematics is not uniformly used and compare to half body.
• Obviously a left quasi body by reversing the multiplication to a right quasi body and vice versa.
• Applies in a half body in the sense of synthetic geometry in addition to the two distributive alternativeness, an attenuated form of the associative law of multiplication, then this half body is even an alternative body.
• By the definition can be implemented on any body of a quasi Ternärverknüpfung, with the quasi- linear body is Ternärkörper.
• For the 5th axiom of quasi body into the algebraic definition is to be noted:

• Quasi bodies were referred to 1975 as Veblen - Wedderburn system in the literature.
• Every body is a quasi- Cartesian group.
• Almost every body is a quasi body. A quasi body is exactly then a fast body if its multiplication is associative.

Quasi body as coordinate ranges projective planes

• Quasi body also occur as coordinate areas of special projective planes. These are in the classification of projective planes by Hanfried Lenz levels of classes IV, V and VII

• More precisely: A projective plane of class IVa and IVb can be koordinatisieren by choosing a suitable base point by a left quasi body or a quasi legal body. Each Ternärkörper, which belongs to level when selecting an arbitrary point base is isotopic to a left or right quasi body.
• All coordinates sections of a projective plane of class V are mutually isotopic half body, so both right and left quasi body. In general, these half body but are not isomorphic to each other.
• All coordinates sections of a projective plane of class VII are mutually isomorphic Alternatively body.

Examples

• Each algebraic structure stronger, so every body fast, Alternative Body, Slanted body or body provides an example of a quasi body.
• The article provides some examples of quasi Ternärkörper body, especially a detail illustrated example of a finite, non-commutative quasi body ( → subsection examples of order 9 ).

Quasi body finite Moulton planes

The finite - Moulton planes have "real" as the coordinate space quasi body. For the construction of the starting material is a finite field whose characteristic is an odd prime number. In the cyclic multiplicative group then there is exactly one subgroup of index 2, which is the subgroup of squares. Be a Körperautomorphismus of. Now a new multiplication is introduced:

This will be a left quasi body because the Linksdistributivgesetz is satisfied. If the selected Körperautomorphismus not the identity, then

The core of the body is the quasi -fixed by Körperautomorphismus finite subfield of.