An incidence structure
C
=
(
P
,
L
,
I
)
consists of points
P
, lines
L
, and flags
I
⊆
P
×
L
where a point
p
is said to be incident with a line
l
if
(
p
,
l
)
∈
I
. It is a (finite) partial geometry if there are integers
s
,
t
,
α
≥
1
such that:
For any pair of distinct points
p
and
q
, there is at most one line incident with both of them.
Each line is incident with
s
+
1
points.
Each point is incident with
t
+
1
lines.
If a point
p
and a line
l
are not incident, there are exactly
α
pairs
(
q
,
m
)
∈
I
, such that
p
is incident with
m
and
q
is incident with
l
.
A partial geometry with these parameters is denoted by
p
g
(
s
,
t
,
α
)
.
The number of points is given by
(
s
+
1
)
(
s
t
+
α
)
α
and the number of lines by
(
t
+
1
)
(
s
t
+
α
)
α
.
The point graph of a
p
g
(
s
,
t
,
α
)
is a strongly regular graph :
s
r
g
(
(
s
+
1
)
(
s
t
+
α
)
α
,
s
(
t
+
1
)
,
s
−
1
+
t
(
α
−
1
)
,
α
(
t
+
1
)
)
.
Partial geometries are dual structures : the dual of a
p
g
(
s
,
t
,
α
)
is simply a
p
g
(
t
,
s
,
α
)
.
The generalized quadrangles are exactly those partial geometries
p
g
(
s
,
t
,
α
)
with
α
=
1
.
The Steiner systems are precisely those partial geometries
p
g
(
s
,
t
,
α
)
with
α
=
s
+
1
.
A partial linear space
S
=
(
P
,
L
,
I
)
of order
s
,
t
is called a semipartial geometry if there are integers
α
≥
1
,
μ
such that:
If a point
p
and a line
ℓ
are not incident, there are either
0
or exactly
α
pairs
(
q
,
m
)
∈
I
, such that
p
is incident with
m
and
q
is incident with
ℓ
.
Every pair of non-collinear points have exactly
μ
common neighbours.
A semipartial geometry is a partial geometry if and only if
μ
=
α
(
t
+
1
)
.
It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters
(
1
+
s
(
t
+
1
)
+
s
(
t
+
1
)
t
(
s
−
α
+
1
)
/
μ
,
s
(
t
+
1
)
,
s
−
1
+
t
(
α
−
1
)
,
μ
)
.
A nice example of such a geometry is obtained by taking the affine points of
P
G
(
3
,
q
2
)
and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters
(
s
,
t
,
α
,
μ
)
=
(
q
2
−
1
,
q
2
+
q
,
q
,
q
(
q
+
1
)
)
.
