Generalized Quadrangle

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A **generalized quadrangle** is an incidence structure. A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with <math>n=4</math>. They are also precisely the partial geometries <math>pg(s,t,alpha)</math> with <math>alpha = 1 </math>.

## Definition

A generalized quadrangle is an incidence structure <math>(P,B,I)</math>, with <math>Isubseteq Ptimes B</math> an incidence relation, satisfying certain axioms. Elements of <math>P</math> are by definition the points of the generalized quadrangle, elements of <math>B</math> the lines. The axioms are the following:

<math>(s,t)</math>...

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- There is a <math>s</math> (<math>sgeq 1</math>) such that on every line there are exactly <math>s+1</math> points. There is at most one point on two distinct lines.
- There is a <math>t</math> (<math>tgeq 1</math>) such that through every point there are exactly <math>t+1</math> lines. There is at most one line through two distinct points.
- For every point <math>p</math> not on a line <math>L</math>, there is a unique line <math>M</math> and a unique point <math>q</math>, such that <math>p</math> is on <math>M</math>, and <math>q</math> on <math>M</math> and <math>L</math>.

<math>(s,t)</math>...

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