Editing Incidence structure (section)

==Examples==
{{main|Incidence geometry}}

{{Gallery|title=Some examples of incidence structures|width=180|height=170|align=center|
File:Fano plane.svg|alt1=Fano plane|1. [[Fano plane]]|
File:Incidencestructure.svg|alt2=A non-uniform incidence structure|2. Non-uniform structure|
File:GQ(2,2), the Doily.svg|alt3=Generalized Quadrangle called the Doily|3. [[Generalized quadrangle]]|
File:Möbius–Kantor configuration.svg|alt4=8 point and 8 line configuration|4. [[Möbius–Kantor configuration]]|
File:Pappusconfig.svg|alt5=Configuration of Pappus theorem|5. [[Pappus configuration]] }}

An incidence structure is ''uniform'' if each line is incident with the same number of points. Each of these examples, except the second, is uniform with three points per line.

===Graphs===
Any [[Graph (discrete mathematics)|graph]] (which need not be [[simple graph|simple]]; [[Loop (graph theory)|loops]] and [[multiple edges]] are allowed) is a uniform incidence structure with two points per line. For these examples, the vertices of the graph form the point set, the edges of the graph form the line set, and incidence means that a vertex is an endpoint of an edge.

===Linear spaces===
Incidence structures are seldom studied in their full generality; it is typical to study incidence structures that satisfy some additional axioms. For instance, a ''[[partial linear space]]'' is an incidence structure that satisfies:
# Any two distinct points are incident with at most one common line, and
# Every line is incident with at least two points.
If the first axiom above is replaced by the stronger:
# <li value="3"> Any two distinct points are incident with exactly one common line,</li>
the incidence structure is called a ''[[linear space (geometry)|linear space]]''.<ref>The term ''linear space'' is also used to refer to vector spaces, but this will rarely cause confusion.</ref><ref>{{harvnb|Moorhouse|2014|page=5}}</ref>

===Nets===
A more specialized example is a {{mvar|k}}'''-net'''.  This is an incidence structure in which the lines fall into {{mvar|k}} '''parallel classes''', so that two lines in the same parallel class have no common points, but two lines in different classes have exactly one common point, and each point belongs to exactly one line from each parallel class.  An example of a {{mvar|k}}-net is the set of points of an [[affine plane]] together with {{mvar|k}} parallel classes of affine lines.