In this, the final chapter, we consider a generalization of the concept of generalized quadrangle. Generalized quadrangles were first introduced by Tits (1959). The generalization, partial geometry, first appeared independently of Tits' work in a paper by Bose (1963).

The definition

A partial geometry is a finite near-linear space S = (P, L) such that

PG1 for each point p and line ℓ, p∉ℓ implies c(p, ℓ) = α,

PG2 each line has s + 1 points,

PG3 each point is on t + 1 lines,

where α, s and t are fixed positive integers.

Note that Ø is trivially a partial geometry. If S is a partial geometry

which is not Ø, we say S has parameters α, s and t.

We note also that, as for generalized quadrangles, the dual of a partial geometry is again a partial geometry.

Clearly, if α = 1 and GQ2 is satisfied, then S is a generalized quadrangle.

It is not difficult to show that if S ≠ Ø, and α = 1, then GQ2 is satisfied, and we leave this as an exercise.

From the definition of α, s and t, we see that α < s + 1 and α < t + 1. So α ≤ min {s, t} + 1.