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

Examples

• Projective space
• Affine space
• Polar space
• Generalized quadrangle
• Generalized polygon
• Near polygon

References

• Shult, Ernest E. (2011), Points and Lines, Universitext, Springer, doi:10.1007/978-3-642-15627-4, ISBN 978-3-642-15626-7.

• Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, ISBN 0-521-31857-2, 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 a website of the university of Kiel
• partial linear space at PlanetMath