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The collection of all subspaces of an [[Incidence system|incidence system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753501.png" />, where the elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753502.png" /> are called points, the elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753503.png" /> are called lines and I is the incidence relation. A subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753504.png" /> is defined to be a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753506.png" /> for which the following condition holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753508.png" />, then the set of points of the line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p0753509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535010.png" /> also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535011.png" />. The incidence system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535012.png" /> satisfies the following requirements:
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1) for any two different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535014.png" /> there exists a unique line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535017.png" />;
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{{TEX|auto}}
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{{TEX|done}}
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 +
The collection of all subspaces of an [[Incidence system|incidence system]]  $  \pi = \{ {\mathcal P} , {\mathcal L} , \textrm{ I } \} $,
 +
where the elements of the set  $  {\mathcal P} $
 +
are called points, the elements of the set  $  {\mathcal L} $
 +
are called lines and I is the incidence relation. A subspace of  $  \pi $
 +
is defined to be a subset  $  S $
 +
of  $  {\mathcal P} $
 +
for which the following condition holds: If  $  p , q \in S $
 +
and  $  p \neq q $,
 +
then the set of points of the line passing through  $  p $
 +
and  $  q $
 +
also belongs to  $  S $.
 +
The incidence system  $  \pi $
 +
satisfies the following requirements:
 +
 
 +
1) for any two different points p $
 +
and $  q $
 +
there exists a unique line $  L $
 +
such that p \textrm{ I } L $
 +
and $  q \textrm{ I } L $;
  
 
2) every line is incident to at least three points;
 
2) every line is incident to at least three points;
  
3) if two different lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535019.png" /> intersect at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535020.png" /> and if the following four relations hold: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535024.png" />, then the straight lines passing through the pairs of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535026.png" /> intersect.
+
3) if two different lines $  L $
 +
and $  M $
 +
intersect at a point p $
 +
and if the following four relations hold: $  q \textrm{ I } L $,  
 +
$  r \textrm{ I } L $,  
 +
$  s \textrm{ I } M $,  
 +
$  l \textrm{ I } M $,  
 +
then the straight lines passing through the pairs of points $  r , l $
 +
and $  s , q $
 +
intersect.
  
A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535027.png" /> is generated by a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535028.png" /> of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535029.png" /> (written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535030.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535031.png" /> is the intersection of all subspaces containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535032.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535033.png" /> of points is said to be independent if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535034.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535035.png" />. An ordered maximal and independent set of points of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535036.png" /> is called a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535037.png" />, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535038.png" /> of its elements is called the dimension of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535039.png" />. A subspace of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535040.png" /> is a point, a subspace of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535041.png" /> is a [[Projective straight line|projective straight line]], a subspace of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535042.png" /> is called a [[Projective plane|projective plane]].
+
A subspace $  S $
 +
is generated by a set $  s $
 +
of points in $  {\mathcal P} $(
 +
written $  S = \langle  s \rangle $)  
 +
if $  S $
 +
is the intersection of all subspaces containing $  s $.  
 +
A set $  s $
 +
of points is said to be independent if for any $  x \in s $
 +
one has $  x \notin \langle  s \setminus  \{ x \} \rangle $.  
 +
An ordered maximal and independent set of points of a subspace $  S $
 +
is called a basis of $  S $,  
 +
and the number $  d ( S) $
 +
of its elements is called the dimension of the subspace $  S $.  
 +
A subspace of dimension 0 $
 +
is a point, a subspace of dimension $  1 $
 +
is a [[Projective straight line|projective straight line]], a subspace of dimension $  2 $
 +
is called a [[Projective plane|projective plane]].
  
In a projective space the operations of addition and intersection of spaces are defined. The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535043.png" /> of two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535045.png" /> is defined to be the smallest of the subspaces containing both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535047.png" />. The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535048.png" /> of two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535050.png" /> is defined to be the largest of the subspaces contained in both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535052.png" />. The dimensions of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535054.png" />, of their sum, and of their intersection are connected by the relation
+
In a projective space the operations of addition and intersection of spaces are defined. The sum $  P _ {m} + P _ {k} $
 +
of two subspaces $  P _ {m} $
 +
and $  P _ {k} $
 +
is defined to be the smallest of the subspaces containing both $  P _ {m} $
 +
and $  P _ {k} $.  
 +
The intersection $  P _ {m} \cap P _ {k} $
 +
of two subspaces $  P _ {m} $
 +
and $  P _ {k} $
 +
is defined to be the largest of the subspaces contained in both $  P _ {m} $
 +
and $  P _ {k} $.  
 +
The dimensions of the subspaces $  P _ {m} $,  
 +
$  P _ {k} $,  
 +
of their sum, and of their intersection are connected by the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535055.png" /></td> </tr></table>
+
$$
 +
m + k  = d ( P _ {m} \cap P _ {k} ) + d ( P _ {m} + P _ {k} ) .
 +
$$
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535056.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535059.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535060.png" /> is a complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535062.png" />), and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535063.png" />, then
+
For any $  P _ {m} $
 +
there is a $  P _ {n-m- 1} $
 +
such that $  P _ {m} \cap P _ {n-m- 1}= P _ {-1} = \emptyset $
 +
and $  P _ {m} + P _ {n-m- 1} = P _ {n} $(
 +
$  P _ {n-m- 1} $
 +
is a complement of $  P _ {m} $
 +
in $  P _ {n} $),  
 +
and if $  P _ {m} \subset  P _ {r} $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535064.png" /></td> </tr></table>
+
$$
 +
( P _ {m} + P _ {k} ) \cap P _ {r} = P _ {m} + P _ {k} \cap P _ {r}  $$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535065.png" /> (Dedekind's rule), that is, with respect to the operation just introduced the projective space is a complemented [[Modular lattice|modular lattice]].
+
for any $  P _ {k} $(
 +
Dedekind's rule), that is, with respect to the operation just introduced the projective space is a complemented [[Modular lattice|modular lattice]].
  
A projective space of dimension exceeding two is Desarguesian (see [[Desargues assumption|Desargues assumption]]) and hence is isomorphic to a projective space (left or right) over a suitable [[Skew-field|skew-field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535066.png" />. The (for example) left projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535067.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535068.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535069.png" /> is the collection of linear subspaces of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535070.png" />-dimensional left linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535071.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535072.png" />; the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535073.png" /> are the lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535074.png" />, i.e. the left equivalence classes of rows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535075.png" /> consisting of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535076.png" /> which are not simultaneously equal to zero (two rows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535078.png" /> are left equivalent if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535079.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535081.png" />); the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535083.png" />, are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535084.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535085.png" />. It is possible to establish a correspondence between a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535086.png" /> and a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535087.png" /> projective space under which to a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535088.png" /> corresponds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535089.png" /> (the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535091.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535092.png" />, then the corresponding assertion is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535093.png" />. This correspondence between the properties of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535095.png" /> is called the duality principle for projective spaces (see [[#References|[2]]]).
+
A projective space of dimension exceeding two is Desarguesian (see [[Desargues assumption|Desargues assumption]]) and hence is isomorphic to a projective space (left or right) over a suitable [[Skew-field|skew-field]] $  k $.  
 +
The (for example) left projective space $  P _ {n}  ^ {l} ( k) $
 +
of dimension $  n $
 +
over a skew-field $  k $
 +
is the collection of linear subspaces of an $  ( n+ 1) $-
 +
dimensional left linear space $  A _ {n+1}  ^ {l} ( k) $
 +
over $  k $;  
 +
the points of $  P _ {n}  ^ {l} ( k) $
 +
are the lines of $  A _ {n+1}  ^ {l} ( k) $,  
 +
i.e. the left equivalence classes of rows $  ( x _ {0} \dots x _ {n} ) $
 +
consisting of elements of $  k $
 +
which are not simultaneously equal to zero (two rows $  ( x _ {0} \dots x _ {n} ) $
 +
and $  ( y _ {0} \dots y _ {n} ) $
 +
are left equivalent if there is a $  \lambda \in k $
 +
such that $  x _ {i} = \lambda y _ {i} $,  
 +
$  i = 0 \dots n $);  
 +
the subspaces $  P _ {m}  ^ {l} ( k) $,  
 +
$  m = 1 \dots n $,  
 +
are the $  ( m+ 1) $-
 +
dimensional subspaces $  A _ {m+1}^ {l} ( k) $.  
 +
It is possible to establish a correspondence between a left $  P _ {n}  ^ {l} ( k) $
 +
and a right $  P _ {n}  ^ {r} ( k) $
 +
projective space under which to a subspace $  P _ {s}  ^ {l} ( k) $
 +
corresponds $  P _ {n-s- 1}  ^ {r} ( k) $(
 +
the subspaces $  P _ {s}  ^ {l} ( k) $
 +
and $  P _ {n-s- 1}  ^ {r} ( k) $
 +
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 $  P _ {n}  ^ {l} ( k) $,  
 +
then the corresponding assertion is true for $  P _ {n}  ^ {r} ( k) $.  
 +
This correspondence between the properties of the spaces $  P _ {n}  ^ {r} ( k) $
 +
and $  P _ {n}  ^ {l} ( k) $
 +
is called the duality principle for projective spaces (see [[#References|[2]]]).
  
A finite skew-field is necessarily commutative; consequently, a finite projective space of dimension exceeding two and of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535096.png" /> is isomorphic to the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535097.png" /> over the [[Galois field|Galois field]]. The finite projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535098.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p07535099.png" /> points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350100.png" /> subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350101.png" /> (see [[#References|[4]]]).
+
A finite skew-field is necessarily commutative; consequently, a finite projective space of dimension exceeding two and of order $  q $
 +
is isomorphic to the projective space $  \mathop{\rm PG} ( n , q ) $
 +
over the [[Galois field|Galois field]]. The finite projective space $  \mathop{\rm PG} ( n , q ) $
 +
contains $  ( q  ^ {n+1} - 1 ) / ( q - 1 ) $
 +
points and $  \prod_{i=0}^ {r} ( q  ^ {n+1- i} - 1 ) / ( q  ^ {r+1- i} - 1 ) $
 +
subspaces of dimension $  r $(
 +
see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350102.png" /> has order
+
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 $  \mathop{\rm PG} ( n , p ^ {h} ) $
 +
has order
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350103.png" /></td> </tr></table>
+
$$
 +
hp  ^ {hn( n+ 1)/2} \prod_{i=1} ^ { n+1} ( p  ^ {hi} - 1 ) .
 +
$$
  
Every projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350104.png" /> admits a cyclic transitive group of collineations (see [[#References|[3]]]).
+
Every projective space $  \mathop{\rm PG} ( n , q ) $
 +
admits a cyclic transitive group of collineations (see [[#References|[3]]]).
  
A correlation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350105.png" /> of a projective space is a permutation of subspaces that reverses inclusions, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350106.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350107.png" />. 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|Polarity]]).
+
A correlation $  \delta $
 +
of a projective space is a permutation of subspaces that reverses inclusions, that is, if $  S \subset  T $,  
 +
then $  S  ^  \delta  \supset T  ^  \delta  $.  
 +
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|Polarity]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Artin,   "Geometric algebra" , Interscience (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge,   D. Pedoe,   "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Dembowski,   "Finite geometries" , Springer (1968) pp. 254</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B. Segre,   "Lectures on modern geometry" , Cremonese (1961)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.02101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947) {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Dembowski, "Finite geometries" , Springer (1968) pp. 254 {{MR|0233275}} {{ZBL|0159.50001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B. Segre, "Lectures on modern geometry" , Cremonese (1961) {{MR|0131192}} {{ZBL|0095.14802}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
The real and complex projective spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350108.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350109.png" />, of all real, respectively complex, lines through the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350110.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350111.png" />, are the Grassmann manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350113.png" /> (cf. [[Grassmann manifold|Grassmann manifold]]).
+
The real and complex projective spaces $  \mathbf P _ {n} ( \mathbf R ) $,  
 +
respectively $  \mathbf P _ {n} ( \mathbf C ) $,  
 +
of all real, respectively complex, lines through the origin in $  \mathbf R  ^ {n+1} $,  
 +
respectively $  \mathbf C  ^ {n+1} $,  
 +
are the Grassmann manifolds $  G _ {n+1,1} ( \mathbf R ) = \mathop{\rm Gr} _ {1} ( \mathbf R  ^ {n+1} ) $,
 +
$  G _ {n+1,1} ( \mathbf C ) = \mathop{\rm Gr} _ {1} ( \mathbf C  ^ {n+1} ) $(
 +
cf. [[Grassmann manifold|Grassmann manifold]]).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350114.png" /> has a CW-decomposition of exactly one cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350115.png" /> in each even dimension. Consequently, its homology is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350116.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350118.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350119.png" />.
+
$  \mathbf P _ {n} ( \mathbf C ) $
 +
has a CW-decomposition of exactly one cell $  e _ {2m} $
 +
in each even dimension. Consequently, its homology is $  H _ {2i} ( \mathbf P _ {n} ( \mathbf C ) ;  \mathbf Z ) = \mathbf Z $
 +
for $  i = 0 \dots n $
 +
and $  H _ {2i+1} ( \mathbf P _ {n} ( \mathbf C ) ;  \mathbf Z ) = 0 $
 +
for $  i = 0 \dots n- 1 $.
  
Real projective space has a CW-decomposition with exactly one cell in each dimension. For odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350120.png" /> the homology groups are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350123.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350124.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350125.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350126.png" />. For even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350127.png" /> the homology groups are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350128.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350130.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350132.png" />.
+
Real projective space has a CW-decomposition with exactly one cell in each dimension. For odd $  n = 2m + 1 $
 +
the homology groups are: $  H _ {2i} ( \mathbf P _ {2m+1} ( \mathbf R ) ) = 0 $,  
 +
$  i = 1 \dots m $,  
 +
$  H _ {0} ( \mathbf P _ {2m+1} ( \mathbf R )) = \mathbf Z $;  
 +
$  H _ {2m+1} ( \mathbf P _ {2m+1} ( \mathbf R )) = \mathbf Z $;  
 +
$  H _ {2i+1} ( \mathbf P _ {2m+1} ( \mathbf R )) = \mathbf Z / ( 2) $
 +
for $  i = 0 \dots m- 1 $.  
 +
For even $  n = 2m $
 +
the homology groups are: $  H _ {0} ( \mathbf P _ {2m} ( \mathbf R )) = \mathbf Z $;  
 +
$  H _ {2i} ( \mathbf P _ {2m} ( \mathbf R )) = 0 $,  
 +
$  i = 1 \dots m $;  
 +
$  H _ {2i+1} ( \mathbf P _ {2m} ( \mathbf R )) = \mathbf Z / ( 2) $,  
 +
$  i= 0 \dots m- 1 $.
  
The real projective plane can be obtained by glueing a disc along its boundary to the boundary of a crosscap (i.e. a [[Möbius strip|Möbius strip]]). An easy way to see this is to view <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350133.png" /> as obtained from a disc by identifying diametrically-opposite boundary points. Now remove a central disc and cut and glue as indicated below.
+
The real projective plane can be obtained by glueing a disc along its boundary to the boundary of a crosscap (i.e. a [[Möbius strip]]). An easy way to see this is to view $  \mathbf P _ {2} ( \mathbf R ) $
 +
as obtained from a disc by identifying diametrically-opposite boundary points. Now remove a central disc and cut and glue as indicated below.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075350a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075350a.gif" />
Line 49: Line 191:
 
Figure: p075350a
 
Figure: p075350a
  
The real projective plane cannot be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350134.png" />, but can be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075350/p075350135.png" />. Its [[Euler characteristic|Euler characteristic]] is 1.
+
The real projective plane cannot be imbedded in $  \mathbf R  ^ {3} $,  
 +
but can be imbedded in $  \mathbf R  ^ {4} $.  
 +
Its [[Euler characteristic|Euler characteristic]] is 1.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Veblen,   J.W. Young,   "Projective geometry" , '''1–2''' , Blaisdell (1938–1946)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Baer,   "Linear algebra and projective geometry" , Acad. Press (1952)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Veblen, J.W. Young, "Projective geometry" , '''1–2''' , Blaisdell (1938–1946) {{MR|0179667}} {{MR|0179666}} {{MR|1519256}} {{MR|1506049}} {{MR|1500790}} {{MR|1500747}} {{ZBL|0127.37604}} {{ZBL|0018.32604}} {{ZBL|63.0693.02}} {{ZBL|55.0413.02}} {{ZBL|52.0732.01}} {{ZBL|51.0591.05}} {{ZBL|51.0569.04}} {{ZBL|49.0547.01}} {{ZBL|48.0843.04}} {{ZBL|47.0582.08}} {{ZBL|41.0606.06}} {{ZBL|39.0606.01}} {{ZBL|38.0562.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) {{MR|0052795}} {{ZBL|0049.38103}} </TD></TR></table>
 +
 
 +
{{OldImage}}

Latest revision as of 18:38, 13 January 2024


The collection of all subspaces of an incidence system $ \pi = \{ {\mathcal P} , {\mathcal L} , \textrm{ I } \} $, where the elements of the set $ {\mathcal P} $ are called points, the elements of the set $ {\mathcal L} $ are called lines and I is the incidence relation. A subspace of $ \pi $ is defined to be a subset $ S $ of $ {\mathcal P} $ for which the following condition holds: If $ p , q \in S $ and $ p \neq q $, then the set of points of the line passing through $ p $ and $ q $ also belongs to $ S $. The incidence system $ \pi $ satisfies the following requirements:

1) for any two different points $ p $ and $ q $ there exists a unique line $ L $ such that $ p \textrm{ I } L $ and $ q \textrm{ I } L $;

2) every line is incident to at least three points;

3) if two different lines $ L $ and $ M $ intersect at a point $ p $ and if the following four relations hold: $ q \textrm{ I } L $, $ r \textrm{ I } L $, $ s \textrm{ I } M $, $ l \textrm{ I } M $, then the straight lines passing through the pairs of points $ r , l $ and $ s , q $ intersect.

A subspace $ S $ is generated by a set $ s $ of points in $ {\mathcal P} $( written $ S = \langle s \rangle $) if $ S $ is the intersection of all subspaces containing $ s $. A set $ s $ of points is said to be independent if for any $ x \in s $ one has $ x \notin \langle s \setminus \{ x \} \rangle $. An ordered maximal and independent set of points of a subspace $ S $ is called a basis of $ S $, and the number $ d ( S) $ of its elements is called the dimension of the subspace $ S $. A subspace of dimension $ 0 $ is a point, a subspace of dimension $ 1 $ is a projective straight line, a subspace of dimension $ 2 $ is called a projective plane.

In a projective space the operations of addition and intersection of spaces are defined. The sum $ P _ {m} + P _ {k} $ of two subspaces $ P _ {m} $ and $ P _ {k} $ is defined to be the smallest of the subspaces containing both $ P _ {m} $ and $ P _ {k} $. The intersection $ P _ {m} \cap P _ {k} $ of two subspaces $ P _ {m} $ and $ P _ {k} $ is defined to be the largest of the subspaces contained in both $ P _ {m} $ and $ P _ {k} $. The dimensions of the subspaces $ P _ {m} $, $ P _ {k} $, of their sum, and of their intersection are connected by the relation

$$ m + k = d ( P _ {m} \cap P _ {k} ) + d ( P _ {m} + P _ {k} ) . $$

For any $ P _ {m} $ there is a $ P _ {n-m- 1} $ such that $ P _ {m} \cap P _ {n-m- 1}= P _ {-1} = \emptyset $ and $ P _ {m} + P _ {n-m- 1} = P _ {n} $( $ P _ {n-m- 1} $ is a complement of $ P _ {m} $ in $ P _ {n} $), and if $ P _ {m} \subset P _ {r} $, then

$$ ( P _ {m} + P _ {k} ) \cap P _ {r} = P _ {m} + P _ {k} \cap P _ {r} $$

for any $ P _ {k} $( 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 $ k $. The (for example) left projective space $ P _ {n} ^ {l} ( k) $ of dimension $ n $ over a skew-field $ k $ is the collection of linear subspaces of an $ ( n+ 1) $- dimensional left linear space $ A _ {n+1} ^ {l} ( k) $ over $ k $; the points of $ P _ {n} ^ {l} ( k) $ are the lines of $ A _ {n+1} ^ {l} ( k) $, i.e. the left equivalence classes of rows $ ( x _ {0} \dots x _ {n} ) $ consisting of elements of $ k $ which are not simultaneously equal to zero (two rows $ ( x _ {0} \dots x _ {n} ) $ and $ ( y _ {0} \dots y _ {n} ) $ are left equivalent if there is a $ \lambda \in k $ such that $ x _ {i} = \lambda y _ {i} $, $ i = 0 \dots n $); the subspaces $ P _ {m} ^ {l} ( k) $, $ m = 1 \dots n $, are the $ ( m+ 1) $- dimensional subspaces $ A _ {m+1}^ {l} ( k) $. It is possible to establish a correspondence between a left $ P _ {n} ^ {l} ( k) $ and a right $ P _ {n} ^ {r} ( k) $ projective space under which to a subspace $ P _ {s} ^ {l} ( k) $ corresponds $ P _ {n-s- 1} ^ {r} ( k) $( the subspaces $ P _ {s} ^ {l} ( k) $ and $ P _ {n-s- 1} ^ {r} ( k) $ 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 $ P _ {n} ^ {l} ( k) $, then the corresponding assertion is true for $ P _ {n} ^ {r} ( k) $. This correspondence between the properties of the spaces $ P _ {n} ^ {r} ( k) $ and $ P _ {n} ^ {l} ( k) $ 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 $ q $ is isomorphic to the projective space $ \mathop{\rm PG} ( n , q ) $ over the Galois field. The finite projective space $ \mathop{\rm PG} ( n , q ) $ contains $ ( q ^ {n+1} - 1 ) / ( q - 1 ) $ points and $ \prod_{i=0}^ {r} ( q ^ {n+1- i} - 1 ) / ( q ^ {r+1- i} - 1 ) $ subspaces of dimension $ r $( 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 $ \mathop{\rm PG} ( n , p ^ {h} ) $ has order

$$ hp ^ {hn( n+ 1)/2} \prod_{i=1} ^ { n+1} ( p ^ {hi} - 1 ) . $$

Every projective space $ \mathop{\rm PG} ( n , q ) $ admits a cyclic transitive group of collineations (see [3]).

A correlation $ \delta $ of a projective space is a permutation of subspaces that reverses inclusions, that is, if $ S \subset T $, then $ S ^ \delta \supset T ^ \delta $. 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).

References

[1] E. Artin, "Geometric algebra" , Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[2] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1 , Cambridge Univ. Press (1947) MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502
[3] R. Dembowski, "Finite geometries" , Springer (1968) pp. 254 MR0233275 Zbl 0159.50001
[4] B. Segre, "Lectures on modern geometry" , Cremonese (1961) MR0131192 Zbl 0095.14802

Comments

The real and complex projective spaces $ \mathbf P _ {n} ( \mathbf R ) $, respectively $ \mathbf P _ {n} ( \mathbf C ) $, of all real, respectively complex, lines through the origin in $ \mathbf R ^ {n+1} $, respectively $ \mathbf C ^ {n+1} $, are the Grassmann manifolds $ G _ {n+1,1} ( \mathbf R ) = \mathop{\rm Gr} _ {1} ( \mathbf R ^ {n+1} ) $, $ G _ {n+1,1} ( \mathbf C ) = \mathop{\rm Gr} _ {1} ( \mathbf C ^ {n+1} ) $( cf. Grassmann manifold).

$ \mathbf P _ {n} ( \mathbf C ) $ has a CW-decomposition of exactly one cell $ e _ {2m} $ in each even dimension. Consequently, its homology is $ H _ {2i} ( \mathbf P _ {n} ( \mathbf C ) ; \mathbf Z ) = \mathbf Z $ for $ i = 0 \dots n $ and $ H _ {2i+1} ( \mathbf P _ {n} ( \mathbf C ) ; \mathbf Z ) = 0 $ for $ i = 0 \dots n- 1 $.

Real projective space has a CW-decomposition with exactly one cell in each dimension. For odd $ n = 2m + 1 $ the homology groups are: $ H _ {2i} ( \mathbf P _ {2m+1} ( \mathbf R ) ) = 0 $, $ i = 1 \dots m $, $ H _ {0} ( \mathbf P _ {2m+1} ( \mathbf R )) = \mathbf Z $; $ H _ {2m+1} ( \mathbf P _ {2m+1} ( \mathbf R )) = \mathbf Z $; $ H _ {2i+1} ( \mathbf P _ {2m+1} ( \mathbf R )) = \mathbf Z / ( 2) $ for $ i = 0 \dots m- 1 $. For even $ n = 2m $ the homology groups are: $ H _ {0} ( \mathbf P _ {2m} ( \mathbf R )) = \mathbf Z $; $ H _ {2i} ( \mathbf P _ {2m} ( \mathbf R )) = 0 $, $ i = 1 \dots m $; $ H _ {2i+1} ( \mathbf P _ {2m} ( \mathbf R )) = \mathbf Z / ( 2) $, $ i= 0 \dots m- 1 $.

The real projective plane can be obtained by glueing a disc along its boundary to the boundary of a crosscap (i.e. a Möbius strip). An easy way to see this is to view $ \mathbf P _ {2} ( \mathbf R ) $ as obtained from a disc by identifying diametrically-opposite boundary points. Now remove a central disc and cut and glue as indicated below.

Figure: p075350a

The real projective plane cannot be imbedded in $ \mathbf R ^ {3} $, but can be imbedded in $ \mathbf R ^ {4} $. Its Euler characteristic is 1.

References

[a1] O. Veblen, J.W. Young, "Projective geometry" , 1–2 , Blaisdell (1938–1946) MR0179667 MR0179666 MR1519256 MR1506049 MR1500790 MR1500747 Zbl 0127.37604 Zbl 0018.32604 Zbl 63.0693.02 Zbl 55.0413.02 Zbl 52.0732.01 Zbl 51.0591.05 Zbl 51.0569.04 Zbl 49.0547.01 Zbl 48.0843.04 Zbl 47.0582.08 Zbl 41.0606.06 Zbl 39.0606.01 Zbl 38.0562.01
[a2] R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) MR0052795 Zbl 0049.38103


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How to Cite This Entry:
Projective space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_space&oldid=11959
This article was adapted from an original article by V.V. Afanas'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article