Partial linear space
Type of incidence structure
title: "Partial linear space" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["incidence-geometry"] description: "Type of incidence structure" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Partial_linear_space" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Type of incidence structure ::
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 L}, \textbf{I}) an incidence structure, for which the elements of {\mathcal P} are called points and the elements of {\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=({\mathcal P},{\mathcal L}, \textbf{I}) which is not a single point or a single line, we have |\mathcal{P}| \leq |\mathcal{L}|.
Examples
References
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{{citation | last = Shult | first1 = Ernest E. | doi = 10.1007/978-3-642-15627-4 | publisher = Springer | title = Points and Lines | isbn = 978-3-642-15626-7 | year = 2011 | series =Universitext}}.
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Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, , p. 1-22
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Lynn Batten and Albrecht Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
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Eric Moorhouse: Incidence Geometry. Lecture notes (archived)
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