Linear space (geometry)

A linear space, sometimes referred to as an incidence area, has a basic structure in finite geometry. As an independent concept he introduced in 1964 by Paul Libois. Except in trivial cases ( at most one-dimensional spaces ) can be linear spaces view as a generalization of weak affine spaces, which are in turn a generalization of affine spaces. At the same time provide linear spaces and a generalization of the two-dimensional projective spaces at least dar. finite linear spaces can turn as a generalization of 2 - are considered block plans, where you omitted the fact that on each straight line ( = block ) is the same number of points.

Definitions

Linear space

Is an incidence structure in which is defined as the elements of the points and the elements of the straight lines ( or blocks). Furthermore, one used for the incidence relation and the ways of speaking a point p lies on a line g ( with ) and a line g goes through a point p ( with ). The incidence structure is called a linear space if the following three axioms are satisfied:

  • (L1 ) through two points is exactly one line.
  • (L2 ) are at least two points on each line.
  • (L3 ) L has at least two lines.

Occasionally the axiom L3 is not required in the literature, in such a case applies to those linear spaces which still meet as a non-trivial linear spaces. Does the linear space has a finite number of points, so we also speak briefly of a finite linear space.

There are almost always only finite linear spaces studied. As indicated in block diagrams, the number of dots in the rule is V, the number of lines designated by b.

Partial linear space

Apply for an incidence structure of the axioms (L2) and (L3 ), but instead of ( L1) only the weaker axiom

  • ( L1p ) " by 2 points is at most a straight line. "

It is called a partial linear space ( eng. partial linear space).

Degenerate linear space, near- pencil

  • A linear space with n points, which has a straight line with n-1 points are called degenerate linear space ( degenerate linear space or near- pencil ). All linear spaces, linear spaces other is called non-degenerate linear spaces.
  • The class of projective planes is grouped with the class of the degeneracy of the linear space to a class of generalized projective plane.

Properties

  • Each linear space is a simple incidence structure, ie, a straight line is uniquely determined by the inzidierenden with her points, so can be considered as a set of its points.
  • Finite linear spaces, the number B of straight lines is never less than the number v of the points, so it is always. This is the statement of a set of de Brujin and Erdos.
  • Equality holds if and only if the linear space is a generalized projective plane, ie either a projective plane or a near- pencil.
  • The maximum number of lines for a given score is v. The linear space is then the complete graph on v nodes.

Examples

  • The normal Euclidean plane an infinite linear space.
  • More generally all affine and projective areas, whose dimension is greater than or equal to 2, and thus in particular projective planes, such as the Fano - level ( non-trivial ) linear spaces.
  • A dotted projective plane arises from a projective plane by omitting exactly one point: such a level always forms a linear space.
  • An affine plane with a remote point arises from an affine plane by adding exactly one far point as the intersection of a fixed chosen exactly parallel class level. These levels always form a linear space.

In the following, all four linear spaces with five points () are listed. In this case, it is common to only two points not to draw all the lines in the graph for clarity.

Affine geometry Projective geometry Fano plane Albrecht Beutelspacher
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