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 $S=({{\mathcal P}},{{\mathcal B}},{\textbf {I}})$ 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 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 $S=({\mathcal {P}},{\mathcal {B}},{\textbf {I}})$ which is not a single point or a single line, we have $|{\mathcal {P}}|\leq |{\mathcal {L}}|$ .

Examples

References

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

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

External links

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

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