The collection of all subspaces of an
incidence system
,
where the elements of the set
are called
points,
the elements of the set
are called
lines
and I is the
incidence relation.
A
subspace
of
is defined to be a subset
of
for which the following condition holds: If
and
,
then the set of points of the line passing through
and
also belongs to
.
The incidence system
satisfies the following requirements:
1)
for any two different points
and
there exists a unique line
such that
and
;
2)
every line is incident to at least three points;
3)
if two different lines
and
intersect at a point
and if the following four relations hold:
,
,
,
,
then the straight lines passing through the pairs of points
and
intersect.
A subspace
is
generated
by a set
of points in
(written
)
if
is the intersection of all subspaces containing
.
A set
of points is said to be
independent
if for any
one has
.
An ordered maximal and independent set of points of a subspace
is called a
basis
of
,
and the number
of its elements is called the
dimension
of the subspace
.
A subspace of dimension
is a point, a subspace of dimension
is a
projective straight line,
a subspace of dimension
is called a
projective plane.
In a projective space the operations of addition
and intersection of spaces are defined. The sum
of two subspaces
and
is defined to be the smallest of the subspaces containing both
and
.
The intersection
of two subspaces
and
is defined to be the largest of the subspaces contained in both
and
.
The dimensions of the subspaces
,
,
of their sum, and of their intersection are connected by the relation
For any
there is a
such that
and
(
is a complement of
in
),
and if
,
then
for any
(Dedekind's rule),
that is, with respect to the operation just
introduced the projective space is a complemented
modular lattice.
A projective space of dimension exceeding two is Desarguesian (see
Desargues assumption)
and hence is isomorphic to a projective space (left or right) over a suitable
skew-field
.
The (for example) left projective space
of dimension
over a skew-field
is the collection of linear subspaces of an
-dimensional
left linear space
over
;
the points of
are the lines of
,
i.e. the
left equivalence classes of rows
consisting of elements of
which are not simultaneously equal to zero (two rows
and
are left equivalent if there is a
such that
,
);
the subspaces
,
,
are the
-dimensional
subspaces
.
It is possible to establish a correspondence between a left
and a right
projective space under which to a subspace
corresponds
(the subspaces
and
are called
dual
to one another), to an intersection of subspaces corresponds a sum, and to a sum
corresponds an intersection. If an assertion based only on properties
of linear subspaces, their intersections and sums is true for
,
then the corresponding assertion is true for
.
This correspondence between the properties of the spaces
and
is called the
duality principle
for projective spaces (see
[2]).
A finite skew-field is necessarily commutative; consequently, a finite
projective space of dimension exceeding two and of order
is isomorphic to the projective space
over the
Galois field.
The finite projective space
contains
points and
subspaces of dimension
(see
[4]).
A
collineation
of a projective space is a permutation of its
points that maps lines to lines so that subspaces are
mapped to subspaces. A non-trivial collineation of the projective space has at
most one centre and at most one axis. The
group of collineations of a finite projective space
has order
Every projective space
admits a cyclic transitive group of collineations (see
[3]).
A
correlation
of a projective space is a permutation of subspaces that reverses inclusions, that is, if
,
then
.
A projective space admits a correlation only if it is finite-dimensional. An important
role in projective geometry is played by the correlations
of order two, also called polarities
(Polarity).