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 $$S=({\mathcal P},{\mathcal L}, \textbf{I}) $$ 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

 * Projective space
 * Affine space
 * Polar space
 * Generalized quadrangle
 * Generalized polygon
 * Near polygon