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 ${\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})}$ an incidence structure, for which the elements of ${\displaystyle {\mathcal {P}}}$ are called points and the elements of ${\displaystyle {\mathcal {L}}}$ 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 ${\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})}$ which is not a single point or a single line, we have ${\displaystyle |{\mathcal {P}}|\leq |{\mathcal {L}}|}$.

Examples

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