Generalized n-gon

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In combinatorial mathematics, generalized n-gons are geometric structures introduced by Jacques Tits. They are a generalization of the projective planes, which form the most complex kind of axiomatic projective spaces, and generalized quadrangles, which form the most complex kind of polar spaces.

[edit] Definition

Generalized n-gons (n \geq 2), are incidence structures (P,B,I), with I\subseteq P\times B an incidence relation, satisfying certain conditions. These are best expressed by use of the (bipartite) incidence graph :

  • There is a s (s\geq 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\geq 1) such that through every point there are exactly t+1 lines. There is at most one line through two distinct points.
  • The diameter of the graph is n.
  • The girth of the graph is 2n.

[edit] Examples

Every usual-sense polygon is an example of a generalized n-gon, but they are trivial with s = t = 1.

[edit] Properties

Walter Feit and Graham Higman proved that if we assume

s\geq 2,t\geq 2,,

and both of them finite then n can only be

2, 3, 4, 6 or 8.

More specifically,

  • If n = 2 the structure is trivial.
  • If n = 3, the assumption of only s\geq 2, already implies the structure is a projective plane
  • If n = 4, the structure is, without any assumptions on the parameters, a generalized quadrangle.

If s and t are both infinite then generalized n-gons exist for each n greater or equal to 2. Whether or not there exist generalized n-gons with one of the parameters finite and the other infinite is not known (these cases are called semi-finite).

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