# Chain space

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).

How to Cite This Entry:
Chain space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain_space&oldid=46302
This article was adapted from an original article by A. Herzer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article