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 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 which is not a single point or a single line, we have .
Examples Projective space Affine space Polar space Generalized quadrangle Generalized polygon Near polygon
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