An
incidence structure consists of points
, lines
, and flags
where a point
is said to be incident with a line
if
. It is a (
finite) partial geometry if there are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s
such that:
* For any pair of distinct points
and
, there is at most one line incident with both of them.
* Each line is incident with
points.
* Each point is incident with
lines.
* If a point
and a line
are not incident, there are exactly
pairs
, such that
is incident with
and
is incident with
.
A partial geometry with these parameters is denoted by
.
Properties
* The number of points is given by
and the number of lines by
.
* The point graph (also known as the
collinearity graph) of a
is a
strongly regular graph:
.
* Partial geometries are dual structures: the dual of a
is simply a
.
Special case
* The
generalized quadrangles are exactly those partial geometries
with
.
* The
Steiner systems
are precisely those partial geometries
with
.
Generalisations
A
partial linear space of order
is called a semipartial geometry if there are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s
such that:
* If a point
and a line
are not incident, there are either
or exactly
pairs
, such that
is incident with
and
is incident with
.
* Every pair of non-collinear points have exactly
common neighbours.
A semipartial geometry is a partial geometry if and only if
.
It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters
.
A nice example of such a geometry is obtained by taking the affine points of
and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters
.
See also
*
Strongly regular graph
*
Maximal arc
References
*
*
*
*
*
{{DEFAULTSORT:Partial Geometry
Incidence geometry