A linear space is an incidence structure
which satisfies the following two axioms.
Obviously, a linear space is also a partial
linear space. Conversely, given any partial
linear space, we may construct a linear space by just joining each pair
of non-collinear points with a line of length 2. A proper linear space is a
linear space with at least three points on each line. These are the real linear
spaces, not arising from a smaller partial linear space.
- Each two distinct points are on a unique common line.
- Each line has at least two points.
is a finite linear space, we denote the number of points and lines of
by v and b, respectively. Using this notation, the following important
and b=v if and only if
is an, eventually degenerated, projective plane.
In fact, all linear spaces with b not too far distant from v are
known, in some sense. These are the so called restricted linear spaces.
Contributed by Hauke Klein
Version $Id: linspace.html,v 1.1 2000/12/26 18:30:36 hauke Exp $