Partial geometry


An incidence structure consists of a set of points, a set of lines, and an incidence relation, or set of flags, ; a point is said to be incident with a line if. It is a partial geometry if there are integers such that:
  • For any pair of distinct points and, there is at most one line incident with both of them.
  • Each line is incident with points.
  • Each point is incident with lines.
  • If a point and a line are not incident, there are exactly pairs, such that is incident with and is incident with.
A partial geometry with these parameters is denoted by.

Properties

  • The number of points is given by and the number of lines by.
  • The point graph of a is a strongly regular graph:.
  • Partial geometries are dualizable structures: the dual of a is simply a.

    Special cases

  • The generalized quadrangles are exactly those partial geometries with.
  • The Steiner systems are precisely those partial geometries with.

    Generalisations

A partial linear space of order is called a semipartial geometry if there are integers such that:
  • If a point and a line are not incident, there are either or exactly pairs, such that is incident with and is incident with.
  • Every pair of non-collinear points have exactly common neighbours.
A semipartial geometry is a partial geometry if and only if.
It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters
A nice example of such a geometry is obtained by taking the affine points of and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters.