Generalized Quadrangle

# 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 [itex]n=4[/itex]. They are also precisely the partial geometries [itex]pg(s,t,alpha)[/itex] with [itex]alpha = 1 [/itex].

## Definition

A generalized quadrangle is an incidence structure [itex](P,B,I)[/itex], with [itex]Isubseteq Ptimes B[/itex] an incidence relation, satisfying certain axioms. Elements of [itex]P[/itex] are by definition the points of the generalized quadrangle, elements of [itex]B[/itex] the lines. The axioms are the following:
• There is a [itex]s[/itex] ([itex]sgeq 1[/itex]) such that on every line there are exactly [itex]s+1[/itex] points. There is at most one point on two distinct lines.
• There is a [itex]t[/itex] ([itex]tgeq 1[/itex]) such that through every point there are exactly [itex]t+1[/itex] lines. There is at most one line through two distinct points.
• For every point [itex]p[/itex] not on a line [itex]L[/itex], there is a unique line [itex]M[/itex] and a unique point [itex]q[/itex], such that [itex]p[/itex] is on [itex]M[/itex], and [itex]q[/itex] on [itex]M[/itex] and [itex]L[/itex].

[itex](s,t)[/itex]...
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