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 $S=(,, \textbf)$ an incidence structure, for which the elements of $$ are called ''points'' and the elements of $$ 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 $S=(,, \textbf)$ which is not a single point or a single line, we have $, \mathcal, \leq , \mathcal,$.

Examples

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

References

* .

* 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

at the University of Kiel

at PlanetMath

Geometry