Definition
A generalized quadrangle is an incidence structure (P,B,I), with I ⊆ P × B an incidence relation, satisfying certain axioms. Elements of P are by definition the points of the generalized quadrangle, elements of B the lines. The axioms are the following:
- There is an s (s ≥ 1) such that on every line there are exactly s + 1 points. There is at most one point on two distinct lines.
- There is a t (t ≥ 1) such that through every point there are exactly t + 1 lines. There is at most one line through two distinct points.
- For every point p not on a line L, there is a unique line M and a unique point q, such that p is on M, and q on M and L.
(s,t) are the parameters of the generalized quadrangle. The parameters are allowed to be infinite. If either s or t is one, the generalized quadrangle is called trivial. A generalized quadrangle with parameters (s,t) is often denoted by GQ(s,t).
The smallest non-trivial generalized quadrangle is GQ(2,2), whose representation has been dubbed "the doily" by Stan Payne in 1973.
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