Partial linear space

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

Let $S=(𝒫,ℒ,\mathbf{\text{𝐈}})$ an incidence structure, for which the elements of $𝒫$ are called points and the elements of $ℒ$ 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 shows that in any finite linear space $S=(𝒫,ℒ,\mathbf{\text{𝐈}})$ which is not a single point or a single line, we have $|𝒫|\le |ℒ|$.

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: Incidence Geometry. Lecture notes (archived)

External links

• partial linear space at the University of Kiel
• Partial linear space at PlanetMath.