For the purposes of classifying “single-bonded” root systems, we would like to consider the following (finite) incidence structure, known as *generalised quadrangle*

A pair with a set (elements of are referred to as

points) and (elements of are referred to aslines) is called ageneralised quadrangleprovided the following holds:

- the lines are of size at least 2
- for every point there is at least one line containing it
- for every point and every line there exists unique line such that

The 3rd axiom implies that two lines intersect in a most one point, and so indeed can be visualised as such. This permits one to talk about *collinearity* of points: two points are collinear iff there exists a line containing them both. Also, note that it implies that in there are no “triangles”, i.e. triples of lines each two of which intersect, so that there are 3 non-collinear intersection points in total.

** Examples**

- a “line” i.e. there is just one line on all the points
- a “claw”, i.e. such that there exists such that for all (and so any is on just one line)
- a complete bipartite graph, i.e. with being the set of vertices of a complete bipartite graph and being the set of its edges (so in particular every line contains just 2 points)
- a “grid”, i.e. with being two sets of parallel lines in and being their pairwise intersections
- for let with being the set of unordered pairs of elements of b and the set of partitions of into three pairs; in particular

Note that the 3rd and the 4th examples are *dual* to each other in the following sense.

In the points are the lines of and the lines are the sets of lines collinear to a point for each

Let be a generalised quadrangle. Then its

dual, the incidence system is also a generalised quadrangle, provided that it is neither a “line” nor a “claw”.

We have to exclude “lines” and “claws” as their duals have lines of size 1.

To emphasise the roles of points and lines as dual to each other, we say that and are *incident* if

It is an interesting exercise to show that the 5th example is *self-dual*, i.e. there exists a bijection preserving the incidence point-line relation.

**Regularity**

The 3rd axiom implies that for any two skew (i.e. non-intersecting) lines there is a bijection between the sets of points incident to them. This means that any two lines without must intersect. One can show that more than two different cardinalities are only possible for a “claw”, as follows:

Three lines with three different cardinalities must have a common intersection point, say A line that is not on can intersect at most one of the three lines, say Therefore there are bijections and and is a bijection between points on and points on contradiction proving the claim.

Similarly, one can show that two different cardinalities are only possible for a “grid” or a “claw”. Indeed, let be two skew lines, and a line of a different cardinality. Then every line that joins a point on to a point on must have the same cardinality as On the other hand, every line of a cardinality different from the one of intersects both and In particular these lines do not intersect. Similarly, the lines with the same cardinality as intersect and It follows that we have a “grid”.

It is also not hard to see that when all the lines are of size 2 we have the complete bipartite graph example.

From now on let us assume that all the lines has the same cardinality and that has the dual. Looking at the dual, we either encounter one of the “easy” cases as above, or derive that all the dual lines must have the same cardinality Such are called *regular*.

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