Chain space
An incidence system to axiomatically describe chain geometries (cf. Chain geometry).
A weak chain space is an incidence system satisfying the three conditions below. Here, the elements of
are called chains and two different points (i.e., elements of
) 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 , let
be the set of all points distant to
and let
. Then the incidence system
is called the residual space of
at
.
A partial parallel structure is an incidence system
together with an equivalence relation
on
satisfying the two conditions below. Here, the elements of
are called lines.
a) two different points are incident with at most one line;
b) for a line and point
, there is exactly one line,
, incident with
and such that
. Condition b) is the Euclid parallel axiom.
A partial parallel structure is called a partial affine space if there is an affine space
such that
is the set of points of
,
is the set of straight lines of
and
is the natural parallelism on
.
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 is a weak chain space
together with a family
, where
is an equivalence relation on
with the following properties:
1) if , then
is the only point common to
and
;
2) if and
is a point distant to
, then there is a unique chain
incident with
and
for which
.
Clearly, for a contact space any residual space of the incidence system
gives rise to a partial parallel structure
. Conversely, any chain space
is a contact space (taking for
the natural parallelism of the affine space underlying
). One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [a3].
An affine chain space is a contact space, where
is the point set of an affine space
. The elements of
are called affine chains and are normal rational curves in
, i.e., affine parts of curves which are a Veronese variety (cf. Veronese mapping). For the set
of all affine chains that are straight lines, the structure
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 by means of plane sections. Moreover, the stereographic projection from a simple point
of
(to a hyperplane different from the tangent plane of
at
) 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 |
Chain space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain_space&oldid=46302