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Partial geometry

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An incidence structure C = ( P , L , I ) {\displaystyle C=(P,L,I)} consists of a set ⁠ P {\displaystyle P} ⁠ of points, a set ⁠ L {\displaystyle L} ⁠ of lines, and an incidence relation, or set of flags, I ⊆ P × L {\displaystyle I\subseteq P\times L} ; a point p {\displaystyle p} is said to be incident with a line l {\displaystyle l} if ⁠ ( p , l ) ∈ I {\displaystyle (p,l)\in I} ⁠. It is a (finite) partial geometry if there are integers s , t , α ≥ 1 {\displaystyle s,t,\alpha \geq 1} such that:

  • For any pair of distinct points p {\displaystyle p} and ⁠ q {\displaystyle q} ⁠, there is at most one line incident with both of them.
  • Each line is incident with s + 1 {\displaystyle s+1} points.
  • Each point is incident with t + 1 {\displaystyle t+1} lines.
  • If a point p {\displaystyle p} and a line l {\displaystyle l} are not incident, there are exactly α {\displaystyle \alpha } pairs ⁠ ( q , m ) ∈ I {\displaystyle (q,m)\in I} ⁠, such that p {\displaystyle p} is incident with m {\displaystyle m} and q {\displaystyle q} is incident with ⁠ l {\displaystyle l} ⁠.

A partial geometry with these parameters is denoted by ⁠ p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} ⁠.

Properties

  • The number of points is given by ( s + 1 ) ( s t + α ) α {\displaystyle {\frac {(s+1)(st+\alpha )}{\alpha }}} and the number of lines by ⁠ ( t + 1 ) ( s t + α ) α {\displaystyle {\frac {(t+1)(st+\alpha )}{\alpha }}} ⁠.
  • The point graph (also known as the collinearity graph) of a p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} is a strongly regular graph: ⁠ s r g ( ( s + 1 ) ( s t + α ) α , s ( t + 1 ) , s − 1 + t ( α − 1 ) , α ( t + 1 ) ) {\displaystyle \mathrm {srg} {\Big (}(s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1){\Big )}} ⁠.
  • Partial geometries are dualizable structures: the dual of a p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} is simply a ⁠ p g ( t , s , α ) {\displaystyle \mathrm {pg} (t,s,\alpha )} ⁠.

Special cases

  • The generalized quadrangles are exactly those partial geometries p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} with ⁠ α = 1 {\displaystyle \alpha =1} ⁠.
  • The Steiner systems S ( 2 , s + 1 , t s + 1 ) {\displaystyle S(2,s+1,ts+1)} are precisely those partial geometries p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} with ⁠ α = s + 1 {\displaystyle \alpha =s+1} ⁠.

Generalisations

A partial linear space S = ( P , L , I ) {\displaystyle S=(P,L,I)} of order s , t {\displaystyle s,t} is called a semipartial geometry if there are integers α ≥ 1 , μ {\displaystyle \alpha \geq 1,\mu } such that:

  • If a point p {\displaystyle p} and a line l {\displaystyle l} are not incident, there are either 0 {\displaystyle 0} or exactly α {\displaystyle \alpha } pairs ⁠ ( q , m ) ∈ I {\displaystyle (q,m)\in I} ⁠, such that p {\displaystyle p} is incident with m {\displaystyle m} and q {\displaystyle q} is incident with ⁠ l {\displaystyle l} ⁠.
  • Every pair of non-collinear points have exactly μ {\displaystyle \mu } common neighbours.

A semipartial geometry is a partial geometry if and only if ⁠ μ = α ( t + 1 ) {\displaystyle \mu =\alpha (t+1)} ⁠.

It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters ⁠ ( 1 + s ( t + 1 ) + s ( t + 1 ) t ( s − α + 1 ) / μ , s ( t + 1 ) , s − 1 + t ( α − 1 ) , μ ) {\displaystyle (1+s(t+1)+s(t+1)t(s-\alpha +1)/\mu ,s(t+1),s-1+t(\alpha -1),\mu )} ⁠.

A nice example of such a geometry is obtained by taking the affine points of P G ( 3 , q 2 ) {\displaystyle \mathrm {PG} (3,q^{2})} and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters ⁠ ( s , t , α , μ ) = ( q 2 − 1 , q 2 + q , q , q ( q + 1 ) ) {\displaystyle (s,t,\alpha ,\mu )=(q^{2}-1,q^{2}+q,q,q(q+1))} ⁠.

See also

References

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