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.

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}| . ==References== . Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, , 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 Kiel University

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