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An [[Incidence system|incidence system]] to axiomatically describe chain geometries (cf. [[Chain geometry|Chain geometry]]).
 
An [[Incidence system|incidence system]] to axiomatically describe chain geometries (cf. [[Chain geometry|Chain geometry]]).
  
A weak chain space is an incidence system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101701.png" /> satisfying the three conditions below. Here, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101702.png" /> are called chains and two different points (i.e., elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101703.png" />) are called distant if they are incident with a common chain.
+
A weak chain space is an incidence system $  {\mathcal I} = ( P, \mathfrak C,I ) $
 +
satisfying the three conditions below. Here, the elements of $  \mathfrak C $
 +
are called chains and two different points (i.e., elements of $  P $)  
 +
are called distant if they are incident with a common chain.
  
 
i) any three pairwise distant points are contained in exactly one chain;
 
i) any three pairwise distant points are contained in exactly one chain;
Line 9: Line 24:
 
iii) any point lies in at least one chain.
 
iii) any point lies in at least one chain.
  
For a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101704.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101705.png" /> be the set of all points distant to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101706.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101707.png" />. Then the incidence system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101708.png" /> is called the residual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c1101709.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017010.png" />.
+
For a point $  p $,  
 +
let $  D _ {p} $
 +
be the set of all points distant to $  p $
 +
and let $  ( p ) = \{ {C \in \mathfrak C } : {pIC } \} $.  
 +
Then the incidence system $  {\mathcal I} _ {p} = ( D _ {p} , ( p ) ,I ) $
 +
is called the residual space of $  {\mathcal I} $
 +
at $  p $.
  
A partial parallel structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017011.png" /> is an incidence system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017012.png" /> together with an [[Equivalence|equivalence]] relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017014.png" /> satisfying the two conditions below. Here, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017015.png" /> are called lines.
+
A partial parallel structure $  ( P, \mathfrak B,I, \| ) $
 +
is an incidence system $  ( P, \mathfrak B,I ) $
 +
together with an [[Equivalence|equivalence]] relation $  \| $
 +
on $  \mathfrak B $
 +
satisfying the two conditions below. Here, the elements of $  \mathfrak B $
 +
are called lines.
  
 
a) two different points are incident with at most one line;
 
a) two different points are incident with at most one line;
  
b) for a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017016.png" /> and point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017017.png" />, there is exactly one line, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017018.png" />, incident with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017019.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017020.png" />. Condition b) is the Euclid parallel axiom.
+
b) for a line $  L $
 +
and point $  p $,  
 +
there is exactly one line, $  L  ^  \prime  $,  
 +
incident with $  p $
 +
and such that $  L \| L  ^  \prime  $.  
 +
Condition b) is the Euclid parallel axiom.
  
A partial parallel structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017021.png" /> is called a partial affine space if there is an [[Affine space|affine space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017023.png" /> is the set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017025.png" /> is the set of straight lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017027.png" /> is the natural parallelism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017028.png" />.
+
A partial parallel structure $  ( P, \mathfrak B,I, \| ) $
 +
is called a partial affine space if there is an [[Affine space|affine space]] $  {\mathcal A} $
 +
such that $  P $
 +
is the set of points of $  {\mathcal A} $,  
 +
$  \mathfrak B $
 +
is the set of straight lines of $  {\mathcal A} $
 +
and $  \| $
 +
is the natural parallelism on $  {\mathcal A} $.
  
 
A weak chain space is called a chain space if all residual spaces of it are partial affine spaces.
 
A weak chain space is called a chain space if all residual spaces of it are partial affine spaces.
Line 23: Line 61:
 
Every proper [[Chain geometry|chain geometry]] is a chain space. Conversely, the chain spaces that are proper chain geometries can be characterized by suitable automorphism groups [[#References|[a1]]].
 
Every proper [[Chain geometry|chain geometry]] is a chain space. Conversely, the chain spaces that are proper chain geometries can be characterized by suitable automorphism groups [[#References|[a1]]].
  
A contact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017029.png" /> is a weak chain space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017030.png" /> together with a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017032.png" /> is an equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017033.png" /> with the following properties:
+
A contact space $  {\mathcal C} = ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P }  ) $
 +
is a weak chain space $  ( P, \mathfrak B,I ) $
 +
together with a family $  ( \rho _ {p} ) _ {p \in P }  $,  
 +
where $  \rho _ {p} $
 +
is an equivalence relation on $  ( p ) $
 +
with the following properties:
  
1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017035.png" /> is the only point common to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017037.png" />;
+
1) if $  C \rho _ {p} C  ^  \prime  $,  
 +
then $  p $
 +
is the only point common to $  C $
 +
and $  C  ^  \prime  $;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017039.png" /> is a point distant to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017040.png" />, then there is a unique chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017041.png" /> incident with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017044.png" />.
+
2) if $  pIC $
 +
and $  q $
 +
is a point distant to $  p $,  
 +
then there is a unique chain $  C  ^  \prime  $
 +
incident with $  p $
 +
and $  q $
 +
for which $  C \rho _ {p} C  ^  \prime  $.
  
Clearly, for a contact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017045.png" /> any residual space of the incidence system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017046.png" /> gives rise to a partial parallel structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017047.png" />. Conversely, any chain space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017048.png" /> is a contact space (taking for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017049.png" /> the natural parallelism of the affine space underlying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017050.png" />). One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [[#References|[a3]]].
+
Clearly, for a contact space $  ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P }  ) $
 +
any residual space of the incidence system $  ( P, \mathfrak B,I ) $
 +
gives rise to a partial parallel structure $  ( D _ {p} , ( p ) ,I, \rho _ {p} ) $.  
 +
Conversely, any chain space $  {\mathcal I} $
 +
is a contact space (taking for $  \rho _ {p} $
 +
the natural parallelism of the affine space underlying $  {\mathcal I} $).  
 +
One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [[#References|[a3]]].
  
An affine chain space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017051.png" /> is a contact space, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017052.png" /> is the point set of an affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017053.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017054.png" /> are called affine chains and are normal rational curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017055.png" />, i.e., affine parts of curves which are a Veronese variety (cf. [[Veronese mapping|Veronese mapping]]). For the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017056.png" /> of all affine chains that are straight lines, the structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017057.png" /> is a partial affine space. Affine chain spaces can be constructed by means of Jordan algebras [[#References|[a2]]] (cf. also [[Jordan algebra|Jordan algebra]]).
+
An affine chain space $  \mathfrak A = ( P, \mathfrak C,I ) $
 +
is a contact space, where $  P $
 +
is the point set of an affine space $  {\mathcal A} $.  
 +
The elements of $  \mathfrak C $
 +
are called affine chains and are normal rational curves in $  {\mathcal A} $,  
 +
i.e., affine parts of curves which are a Veronese variety (cf. [[Veronese mapping|Veronese mapping]]). For the set $  \mathfrak L $
 +
of all affine chains that are straight lines, the structure $  ( P, \mathfrak L,I, \| ) $
 +
is a partial affine space. Affine chain spaces can be constructed by means of Jordan algebras [[#References|[a2]]] (cf. also [[Jordan algebra|Jordan algebra]]).
  
A classical example is the quadric model of a chain space, constructed on a [[Quadric|quadric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017058.png" /> by means of plane sections. Moreover, the [[Stereographic projection|stereographic projection]] from a simple point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017059.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017060.png" /> (to a hyperplane different from the tangent plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017061.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110170/c11017062.png" />) then gives rise to an affine chain space (cf. also [[Benz plane|Benz plane]]).
+
A classical example is the quadric model of a chain space, constructed on a [[Quadric|quadric]] $  Q $
 +
by means of plane sections. Moreover, the [[Stereographic projection|stereographic projection]] from a simple point $  p $
 +
of $  Q $(
 +
to a hyperplane different from the tangent plane of $  Q $
 +
at $  p $)  
 +
then gives rise to an affine chain space (cf. also [[Benz plane|Benz plane]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Herzer,  "Chain geometries"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Herzer,  "Affine Kettengeometrien über Jordan-Algebren"  ''Geom. Dedicata'' , '''59'''  (1996)  pp. 181–195</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Meuren,  A. Herzer,  "Ein Axiomsystem für partielle affine Räume"  ''J. Geom.'' , '''50'''  (1994)  pp. 124–142</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Herzer,  "Chain geometries"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Herzer,  "Affine Kettengeometrien über Jordan-Algebren"  ''Geom. Dedicata'' , '''59'''  (1996)  pp. 181–195</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Meuren,  A. Herzer,  "Ein Axiomsystem für partielle affine Räume"  ''J. Geom.'' , '''50'''  (1994)  pp. 124–142</TD></TR></table>

Latest revision as of 16:43, 4 June 2020


An incidence system to axiomatically describe chain geometries (cf. Chain geometry).

A weak chain space is an incidence system $ {\mathcal I} = ( P, \mathfrak C,I ) $ satisfying the three conditions below. Here, the elements of $ \mathfrak C $ are called chains and two different points (i.e., elements of $ P $) are called distant if they are incident with a common chain.

i) any three pairwise distant points are contained in exactly one chain;

ii) any chain contains at least three points;

iii) any point lies in at least one chain.

For a point $ p $, let $ D _ {p} $ be the set of all points distant to $ p $ and let $ ( p ) = \{ {C \in \mathfrak C } : {pIC } \} $. Then the incidence system $ {\mathcal I} _ {p} = ( D _ {p} , ( p ) ,I ) $ is called the residual space of $ {\mathcal I} $ at $ p $.

A partial parallel structure $ ( P, \mathfrak B,I, \| ) $ is an incidence system $ ( P, \mathfrak B,I ) $ together with an equivalence relation $ \| $ on $ \mathfrak B $ satisfying the two conditions below. Here, the elements of $ \mathfrak B $ are called lines.

a) two different points are incident with at most one line;

b) for a line $ L $ and point $ p $, there is exactly one line, $ L ^ \prime $, incident with $ p $ and such that $ L \| L ^ \prime $. Condition b) is the Euclid parallel axiom.

A partial parallel structure $ ( P, \mathfrak B,I, \| ) $ is called a partial affine space if there is an affine space $ {\mathcal A} $ such that $ P $ is the set of points of $ {\mathcal A} $, $ \mathfrak B $ is the set of straight lines of $ {\mathcal A} $ and $ \| $ is the natural parallelism on $ {\mathcal A} $.

A weak chain space is called a chain space if all residual spaces of it are partial affine spaces.

Every proper chain geometry is a chain space. Conversely, the chain spaces that are proper chain geometries can be characterized by suitable automorphism groups [a1].

A contact space $ {\mathcal C} = ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P } ) $ is a weak chain space $ ( P, \mathfrak B,I ) $ together with a family $ ( \rho _ {p} ) _ {p \in P } $, where $ \rho _ {p} $ is an equivalence relation on $ ( p ) $ with the following properties:

1) if $ C \rho _ {p} C ^ \prime $, then $ p $ is the only point common to $ C $ and $ C ^ \prime $;

2) if $ pIC $ and $ q $ is a point distant to $ p $, then there is a unique chain $ C ^ \prime $ incident with $ p $ and $ q $ for which $ C \rho _ {p} C ^ \prime $.

Clearly, for a contact space $ ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P } ) $ any residual space of the incidence system $ ( P, \mathfrak B,I ) $ gives rise to a partial parallel structure $ ( D _ {p} , ( p ) ,I, \rho _ {p} ) $. Conversely, any chain space $ {\mathcal I} $ is a contact space (taking for $ \rho _ {p} $ the natural parallelism of the affine space underlying $ {\mathcal I} $). One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [a3].

An affine chain space $ \mathfrak A = ( P, \mathfrak C,I ) $ is a contact space, where $ P $ is the point set of an affine space $ {\mathcal A} $. The elements of $ \mathfrak C $ are called affine chains and are normal rational curves in $ {\mathcal A} $, i.e., affine parts of curves which are a Veronese variety (cf. Veronese mapping). For the set $ \mathfrak L $ of all affine chains that are straight lines, the structure $ ( P, \mathfrak L,I, \| ) $ is a partial affine space. Affine chain spaces can be constructed by means of Jordan algebras [a2] (cf. also Jordan algebra).

A classical example is the quadric model of a chain space, constructed on a quadric $ Q $ by means of plane sections. Moreover, the stereographic projection from a simple point $ p $ of $ Q $( to a hyperplane different from the tangent plane of $ Q $ at $ p $) then gives rise to an affine chain space (cf. also Benz plane).

References

[a1] A. Herzer, "Chain geometries" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995)
[a2] A. Herzer, "Affine Kettengeometrien über Jordan-Algebren" Geom. Dedicata , 59 (1996) pp. 181–195
[a3] S. Meuren, A. Herzer, "Ein Axiomsystem für partielle affine Räume" J. Geom. , 50 (1994) pp. 124–142
How to Cite This Entry:
Chain space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain_space&oldid=12810
This article was adapted from an original article by A. Herzer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article