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An [[Incidence system|incidence system]] constructed by means of an [[Algebra|algebra]]. Originally (1842), Ch. von Staudt defined, on the projective line/plane over  $  \mathbf C $, 
a chain ( "Kette" ) in a synthetic way as a subline/plane over  $  \mathbf R $. 
Much later, in 1973, W. Benz [[#References|[a1]]] gave a common frame for rather different phenomena (the geometries of Möbius, Laguerre and Lie, and Minkowsky; cf. [[Benz plane|Benz plane]]) using the concept of a geometry over an algebra. A recent survey of the development of this theory is [[#References|[a3]]].

==Fundamental concepts.==
Let  $  A $
be a [[Ring|ring]] (associative with one) and let  $  A  ^ {*} $
be its group of units. To define the projective line  $  \mathbf P ( A ) $
over  $  A $, 
one introduces an [[Equivalence|equivalence]] relation  $  \equiv $
on  $  A \times A $, 
as for the [[Projective straight line|projective straight line]] over a skew-field:

$$ 
( a,b ) \equiv ( a  ^  \prime  ,b  ^  \prime  )  \iff  \exists u \in A  ^ {*} : a  ^  \prime  = ua  \&  b  ^  \prime  = ub.
$$

Let  $  [ a,b ] $
denote the equivalence class of  $  ( a,b ) $. 
Then

$$ 
\mathbf P ( A ) =
$$

$$ 
= 
\left \{ {[ a,b ] } : {a,b \in A, \exists x,y \in A: \left (

\begin{array}{cc}
 a  & b  \\
 x  & y  \\
\end{array}

\right ) \in { \mathop{\rm GL} } _ {2} ( A ) } \right \} .
$$

One says that  $  [ a,b ] $
is distant to  $  [ c,d ] $
if

$$ 
\left (

\begin{array}{cc}
 a  & b  \\
 c  & d  \\
\end{array}

\right ) \in { \mathop{\rm GL} } _ {2} ( A ) .
$$

Let  $  K $
be a commutative ring and  $  A $
a  $  K $-
algebra, where  $  K $
is imbedded in the centre of  $  A $
via the mapping  $  k \mapsto k \cdot 1 $. 
Regard  $  \mathbf P ( K ) $
as a subset of  $  \mathbf P ( A ) $, 
and define  $  \mathfrak C _ {K} ( A ) = \{ {\mathbf P ( K )  ^  \gamma  } : {\gamma \in { \mathop{\rm GL} } _ {2} ( A ) } \} $. 
Then the incidence system  $  \Sigma ( K,A ) = ( \mathbf P ( A ) , \mathfrak C _ {K} ( A ) ) $, 
with  $  \in $
as incidence relation, is called a chain geometry. The elements of  $  \mathfrak C _ {K} ( A ) $
are called chains. Any three pairwise distant points belong to exactly one chain.  $  { \mathop{\rm PGL} } _ {2} ( A ) $
is a group of automorphisms of  $  \Sigma ( K,A ) $; 
it is transitive on the set of triples of pairwise distant points, and hence transitive on the set of chains. Four pairwise distant points belong to a common chain, provided their [[Cross ratio|cross ratio]] belongs to  $  K $.

If  $  K $
is a field,  $  \Sigma ( K,A ) $
is said to be a proper chain geometry. A proper chain geometry is a [[Chain space|chain space]].

Below,  $  K $
denotes a [[Field|field]].

==Affine case.==
Let  $  D = \{ {[ 1,x ] } : {x \in A } \} $. 
Then  $  D $
is the set of all points of  $  \mathbf P ( A ) $
distant to  $  [ 0,1 ] $. 
Consider the traces of the chains in  $  D $:

$$ 
{\widehat{\mathfrak C}  } _ {K} ( A ) = \left \{ {C \cap D } : {C \in \mathfrak C _ {K} ( A ) , \left | {C \cap D } \right | \geq  3 } \right \} .
$$

There is a natural bijection from  $  D $
into the [[Affine space|affine space]]  $  A $
over  $  K $
via the mapping  $  [ 1,a ] \mapsto a $. 
Under this mapping,  $  {\widehat{\mathfrak C}  } _ {K} ( A ) $
becomes the set

$$ 
{\mathcal K} _ {K} ( A ) = \left \{ {\mathbf K ( a,b,c ) } : {a \in A  ^ {*} , b,c \in A, \left | {\mathbf K ( a,b,c ) } \right | \geq  3 } \right \}
$$

of affine chains, defined by

$$ 
\mathbf K ( a,b,c ) = \left \{ {( a t + b ) ^ {- 1 } + c } : {t \in K, a t + b \in A  ^ {*} } \right \} \cup \{ c \} .
$$

This trace geometry is called the affine chain geometry, denoted by  $  {\mathcal A} ( K,A ) = ( A, {\mathcal K} _ {K} ( A ) ) $. 
For  $  a \in A $, 
the set  $  \{ {\mathbf K ( a,0,c ) } : {c \in A } \} $
is a bundle of parallel straight lines in the affine space  $  A $. 
For an algebraic (especially, finite-dimensional)  $  K $-
algebra  $  A $(
cf. also [[Algebraic algebra|Algebraic algebra]]), the affine chain geometry  $  {\mathcal A} ( K,A ) $
is an affine chain space (cf. [[Chain space|Chain space]]).

==Structure of morphisms.==
Let  $  A $, 
$  A  ^  \prime  $
be  $  K $-
algebras. A  $  K $-
Jordan homomorphism  $  \alpha : A \rightarrow {A  ^  \prime  } $
is a  $  K $-
semi-linear mapping satisfying: i)  $  1  ^  \alpha  = 1 $; 
and ii) for all  $  a,b \in A $
one has  $  ( aba )  ^  \alpha  = a  ^  \alpha  b  ^  \alpha  a  ^  \alpha  $. 
For a  "strong"  algebra  $  A $(
strongness guarantees a great richness in units for  $  A $), 
any point of  $  \mathbf P ( A ) $
can be written as  $  [ 1 + ab,a ] $
for suitable  $  a,b \in A $. 
Then a  $  K $-
Jordan homomorphism  $  \alpha : A \rightarrow {A  ^  \prime  } $
induces a well-defined mapping  $  \sigma : {\mathbf P ( A ) } \rightarrow {\mathbf P ( A  ^  \prime  ) } $, 
$  [ 1 + ab,a ] \mapsto [ 1 + a  ^  \alpha  b  ^  \alpha  ,a  ^  \alpha  ] $, 
which preserves pairs of distant points and maps chains to chains. Moreover, under  $  \sigma $: 
$  [ 1,0 ] \mapsto [ 1,0 ] $, 
$  [ 0,1 ] \mapsto [ 0,1 ] $, 
$  [ 1,1 ] \mapsto [ 1,1 ] $. 
Such a mapping is called a fundamental morphism from  $  \Sigma ( K,A ) $
to  $  \Sigma ( K,A  ^  \prime  ) $.

Conversely, any fundamental morphism  $  \Sigma ( K,A ) \rightarrow \Sigma ( K,A  ^  \prime  ) $
having more than one chain in its image can be obtained in this manner (see [[#References|[a2]]] for a more general context).

Let  $  { \mathop{\rm Aut} } _ {K} ( A ) $
be the group of  $  K $-
Jordan automorphisms of  $  A $, 
and let  $  F ( K,A ) $
be the group of fundamental automorphisms of  $  \Sigma ( K,A ) $. 
Then  $  { \mathop{\rm Aut} } _ {K} ( A ) \simeq F ( K,A ) $
and  $  { \mathop{\rm Aut} } _ {K} \Sigma ( K,A ) \simeq F ( K,A ) \cdot { \mathop{\rm PGL} } _ {2} ( A ) $.

==Rational representations.==
As in the affine case one tries to find kinds of representations for chain geometries  $  \Sigma ( K,A ) $
on a part of a projective space where the chains become curves, at least when  $  A $
is finite dimensional. In this way one has discovered incidence systems  $  ( V \setminus  W, \mathfrak C ) $
isomorphic to  $  \Sigma ( K,A ) $, 
where  $  V $
is a projective variety (cf. also [[Projective scheme|Projective scheme]]),  $  W $
is a (Zariski-) closed subset of  $  V $
and  $  \mathfrak C $
consists of rational curves on  $  V \setminus  W $(
cf. also [[Incidence system|Incidence system]]; [[Rational curve|Rational curve]]).

For a  $  K $-
algebra of  $  K $-
dimension  $  n $, 
a representation of  $  \Sigma ( K,A ) $
on a part of the [[Grassmann manifold|Grassmann manifold]]  $  G _ {2n,n }  ( K ) $
is obtained as follows. For  $  [ a,b ] \in \mathbf P ( A ) $, 
the set  $  A ( a,b ) = \{ {( xa,xb ) } : {x \in A } \} $
is an  $  n $-
dimensional subspace of the  $  K $-
vector space  $  A \times A $
of dimension  $  2n $. 
Then  $  A ( a,b ) $
is mapped to a point of  $  G _ {2n,n }  ( K ) $(
see [[Exterior algebra|Exterior algebra]]). By this procedure, chains are mapped to normal rational curves of order  $  n $, 
i.e., to images of  $  \mathbf P ( K ) $
under the [[Veronese mapping|Veronese mapping]]  $  v _ {n} $, 
and  $  V $
is the intersection of  $  G _ {2n,n }  ( K ) $
with a linear subspace. Other examples can be obtained from this by suitable projection.

A quadratic algebra  $  A $(
i.e., any element of  $  A \setminus  K $
has a quadratic minimal polynomial; cf. also [[Extension of a field|Extension of a field]]) has a representation as a quadric model (cf. [[Chain space|Chain space]]). Here,  $  V = Q $
is the [[Quadric|quadric]] and  $  W $
is its set of singular points; the chains are conics.

== $  n $-chain geometries.==
These are generalizations of chain geometries ( $  n = 1 $) 
to higher dimensions. E.g., let  $  A $
be a quadratic field extension of  $  K $(
cf. [[Extension of a field|Extension of a field]]). Then, in the projective plane over  $  A $
the  $  2 $-
chains are the subplanes over  $  K $; 
these are better known as Baer subplanes, especially in finite geometries (cf. also [[Geometry|Geometry]]).

A Burau geometry is a [[Projective space|projective space]] over  $  A $(
again a quadratic extension of  $  K $) 
together with all projective sublines over  $  K $. 
It can be characterized by the property that the incidence system consisting of a projective line  $  L $
over  $  A $(
as a point set) and all sublines over  $  K $
contained in  $  L $(
considered as blocks) for a [[Möbius plane|Möbius plane]]. A more general concept can be found in [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Benz,   "Vorlesungen über Geometrie der Algebren" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Blunk,   "Chain spaces over Jordan systems"  ''Abh. Math. Sem. Hamburg'' , '''64'''  (1994)  pp. 33–49</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Herzer,   "Chain geometries"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Pieconkowski,   "Projektive Räume über Schiefkörperpaaren" , W und T, Wiss.- und Technik-Verl.  (1994)</TD></TR></table>