Partial linear space explained

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=({lP},{lL},bf{I})

an incidence structure, for which the elements of

{lP}

are called points and the elements of

{lL}

are called lines. S is a partial linear space, if the following axioms hold:

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=({lP},{lL},bf{I})

which is not a single point or a single line, we have

|l{P}|\leq|l{L}|

.

Examples

References

Combinatorics of Finite Geometries. Cambridge University Press 1986,, p. 1-22

External links