Generalized quadrangle

Generalized quadrangle is a designation for certain incidence structures are investigated in particular in finite geometry.

• Numbers of 2.1 points and lines

• 3.1 On a hyperboloid

Definition

An incidence structure is called generalized quadrangle if the following axioms hold:

General is also admitted that one of the numbers in the first two axioms is a fixed infinite number.

Order

The number of points on an arbitrary straight line will then be combined with the number of lines through any point and refers to the pair of numbers as the order of the generalized quadrangle. It then writes also, the square is a.

Properties

• If it as a straight line is more than one point and more, the structure is simple. that is, two lines are equal if and only if they contain the same points.
• The dual structure of an incidence that results from exchange of the point with the set of lines and inversion of the relative incidence is one. It holds more generally (as the statement also applies for infinite generalized quadrangles ): The class of all generalized quadrangles is dual to itself.
• Also in case the generalized quadrilateral need not be isomorphic to its dual quad.
• Every finite generalized quadrangle satisfies the regularity conditions and and so is a tactical configuration.
• If the number of points and the number of lines, then there are pairs of points without connecting lines, therefore the generalized quadrangle is then no incidence geometry and no 2- block plan.

Numbers of points and lines

If it is, then:

• A contains exactly points.
• A contains exactly straight.

Examples

• Trivial examples are: Structures with a straight line, which contains all the points
• Dual like previous: Going structures with a point through which all lines,
• The ordinary quadrangle (vertices as points and pages as blocks ) is up to isomorphism only one GQ with just 4 points and isomorphic to its dual structure.
• General is a square grid.
• The " doily " is a. It was named by Payne so, and the diagram shown in the introduction of Doilys was chosen for the cover of the Proceedings.

On a hyperboloid

On a hyperboloid in a three-dimensional affine or projective space is a generalized quadrangle can be explained as follows: The points are the points on the hyperboloid, the lines are the lines contained entirely in the hyperboloid. These lines form two groups, the lines of such a flock are pairwise skew to each other. Through each point go exactly two lines.

In a finite projective space over the finite field contains every straight points. So this is a generalized quadrangle. It is isomorphic to a square lattice.