7 - Partial geometries
Published online by Cambridge University Press: 26 March 2010
Summary
‘What's the good of Mercators, North Poles and Equators
Tropics, Zones, and Meridian Lines?’
So the Bellman would cry: and the crew would reply
‘They are merely conventional signs!’
Lewis Carroll The Bellman's SpeechIn 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.
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- Information
- Combinatorics of Finite Geometries , pp. 138 - 157Publisher: Cambridge University PressPrint publication year: 1997