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 , I ) {\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})} an incidence structure, for which the elements of P {\displaystyle {\mathcal {P}}} are called points and the elements of L {\displaystyle {\mathcal {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 , I ) {\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})} which is not a single point or a single line, we have | P | ≤ | L | {\displaystyle |{\mathcal {P}}|\leq |{\mathcal {L}}|} .
Examples
- 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 the University of Kiel
- partial linear space at PlanetMath