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Index
We will describe an embedding of a near linear space S into another near-linear space S' as a function f that is one-to-one from both the point and line sets of S to those of S'. In addition, if a given point p lies on a line l in S, then the function f takes that point to a point in S' that is incident with f(l).
Note: f(l) is only required to be contained by a line of S', it does not need to equal a line of S'.
This is a much simpler idea than that of a linear map. Recall that a linear map between two near-linear spaces S and S' is a function that maps the point set of S to the point set of S' so that f(l) is in the line set of S' for every line l in S. In other words, for linear maps f, f(l) is a line of S', not just part of a line of S'. In an embedding, our lines do not have to cover the lines to which they are mapped. In other words, in an embedding we can map a line that is incident with 3 points to a line that is incident with 4 or more points. This would not be permissable for a linear function (in a linear function, we can only map a longer line to a shorter one and not vice versa - or of course we can have a linear mapping between lines that are the same size.)
Let's look at an example of an embedding.
To show that our first figure can be embedded into the Fano plane, we will create a mapping between the two. Let's call this mapping f. Let f be defined on the point set of the first figure as follows:
You can check to see that this is in fact an embedding.
Another way that we can show that the first figure can be embedded into the Fano plane is to extend our points and lines such that the figure satisfies the axioms of a projective plane. In other words, we must make sure that any two points determine a unique line, that two lines intersect at a unique point, and of course that any line lies on at least two points. In addition we must be certain that there exists a set of four points, no three of which are collinear. Essentially we extend this linear space into the Fano plane.
We have created the Fano plane! Notice, however, that we can continue to add points and lines (rather than condense the lines and points of the space as done above) such that the projective plane that we create will be of even higher order.
The above method can be used to prove an important theorem relating to embedding into projective spaces. This theorem was proved by Marshall Hall in 1943.
Theorem: A near-linear space S in which there is a set of four points, no three of which are collinear, can be embedded into a projective plane.
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PROBLEM 18: (a) Find an embedding of the following near-linear space into a projective plane of your choice.
18: (b) Recall from Problem 16 the axioms of an affine plane. Show that any affine plane A is embeddable into some projective plane,  .
(Hint: Let the points of A be the points of \ {l } for some line, l, in . Also, keep in mind that parallelism is an equivelance relation on the set of lines in an affine plane, i.e. let [ l ] be the parallel class that contains l. Then if l' is also in [ l ], [ l ] = [ l' ]).
18: (c) Let A be an affine plane embedded in a projective plane such that  . Recall that we can obtain an affine plane A' from a projective plane by removing one line, say  , , along with the points that are incident with that line. If  , then is A isomorphic to A'?
Solutions
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References: Batten, Bennet, Polster.
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