|
If we begin with a set of axioms for a geometry, such as
PROJECTIVE PLANES
1. Two distinct points are contained in a unique line.
2. Two distinct lines intersect a unique point.
3. There exist four points of which no three are incident with the same line.
|
along with a model for that geometry:
then one could take subsets of both points and lines from the specific geometry, such as
and ask if this also gives a projective plane (this particular example is not a projective plane, and can can you see why?). Such subexamples would be called sub-geometries if they also satisfied the axioms of our original geometry.
For the particular case where we are working with near-linear spaces
NEAR-LINEAR SPACES
1. Any line has at least two points.
2. Two points are on at most one line.
|
and in keeping with the lingo of linear algebra, what could we mean by subspace? We will mean something more than just a sub-geometry:
Definition. A subspace of a near-linear space (P, L) is a subset P' of P and a subset L' of L such that whenever p and q are points of P' which are on a line pq of L, then the entire line pq is in L'. Also, any line belong to L' must contain at least two points belong to P'.
|
Of course any near-linear space is a subspace of itself, and we always have the trivial subspace whose point and line sets each consist only of the empty set. A subspace that is different from the near linear geometry will be called a proper subspace. For example, given the near-linear space (also, this is an affine plane of order 3)
then the following is not a (proper) subspace, as any pair of points in a subspace that lie on a line in the original space must line on that same line in the subspace.
 .
Since we have begun using linear algebra lingo, it would be nice if a line, along with the points incident with that line, formed a subsapce, and nicely it does. Similarly any single point (with no lines) forms a subspace. What other nice features of subspaces of a linear vector space carry over to subspaces of a near-linear geometry?
|
PROBLEM 7: (a) Show that the intersection of any number of subspaces (so there could be infinitely many, and perhaps uncountably infinitely many) is also a subspace. Solution
7(b). Define a hyperplane H of a near linear space L to be a subspace such that there is no subspace S of L
with S a proper subspace of L and H a proper subspace of S. Find the hyperplanes of the near-linear space below. Solution
|
References: Batten, Batten & Beutelspacher, Beutelspacer and Rosenbaum, Poster.
|
