Moulton plane

The Moulton - level is an often used example of an affine plane in which the set of Desargues does not apply, ie a nichtdesargueschen level. Thus, their coordinates at the same time provide an example of a Ternärkörpers which is not a division ring. It was first described in 1902 by the American astronomer Forest Ray Moulton and later named after him.

The points of the Moulton plane normal points the real plane and the straight line, the normal line of the real level, except that straight line with a negative slope at the y-axis has a bend, that is, when passing through the Y-axis changes its slope: In the right half plane is twice as large as in the left half-plane.

Formal definition

, We define the following as incidence structure, the set of points, the set of lines and the incidence ratio "is located on " refers to:

Which is merely a formal symbol.

The incidence relation is defined for and (see line equation ) by

One can easily prove that this incidence structure satisfies the axioms of an affine plane, ie, in particular, that through two distinct points is exactly one line goes and that there is exactly on a straight line through a given point a parallel.

Invalidity of the set of Desargues

It is based on a Desargues constellation of ten points and ten lines in the ordinary Euclidean plane as in the adjacent figure and placed it in such a way that only one of the ten points and a negative coordinate only one of the three straight lines with a negative slope has (pictured: the straight line ). Going now over to the Moulton plane so all incidences remain active until the case, ie the ( Moulton ) straight, and do not cut all at one point. Thus, the set of Desargues in the Moulton plane has no general validity.

Applications

The Moulton - level is by its very existence is a proof that non- desarguesian affine planes exist and even that affine planes exist that are not affine translation planes. Since one can construct a corresponding projective plane to each affinity level ( the projective completion ), so that the existence of non - desargueschen projective planes is secured, and even the existence of the projective planes that are not Moufangebenen. As is true in the set of Desargues, it follows that: It can not be described any projective planes using the canonical construction of 3- dimensional ( left ) vector spaces over a (skew - ) body.

Generalizations

Moulton - levels of real type

Analogous to the real Moulton level described in this article can be defined on the basis of any parent body affinity levels by modifying the multiplication as the Moulton level. This generalization is described in the article Cartesian group.

Finite Moulton planes

For certain finite fields, one can gain a quasi body by modifying the multiplication. The affine plane over such a quasi body is called by Pierce and Pickert as finite Moulton plane. They are always finite affine translation planes. The algebraic structure of their coordinate ranges is described in the article Quasi body closer.