# .

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 $$S=({\mathcal P},{\mathcal B}, \textbf{I})$$ an incidence structure, for which the elements of $${\mathcal P}$$ are called points and the elements of $${\mathcal B}$$ are called lines. S is a partial linear space, if the following axioms hold:

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

Examples

Projective space
Affine space
Polar space
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 Margaret Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, ISBN 0-521-31857-2, p. 1-22
L.M. Batten, A. Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
Eric Moorhouse: "Incidence Geometry". http://www.uwyo.edu/moorhouse/handouts/incidence_geometry.pdf

partial linear space at a website of the university of Kiel
partial linear space at PlanetMath