A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph.

Definition

are called points are called lines. S is a partial linear space, if the following axioms hold:

• any line is incident with at least two points

• any pair of distinct points is incident with at most one line

If there is a unique line incident with every pair of distinct points, then we get a linear space.

Properties

The De Bruijn–Erdős theorem.

Examples

• Projective space

• Affine space

• Polar space

• Generalized quadrangle

• Generalized polygon

• Near polygon

References

• .

  Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, , p.1-22

• Lynn Batten and Albrecht Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.

• Eric Moorhouse: . Lecture notes (archived)

External links

• at the University of Kiel