Friday, September 26, 2014

Construction Algorithm

Given a finite linear space, the Constructibility Theorem proves that there exists a set of isomorphic linear surfaces. However, the Constructibility Theorem does not provide an algorithm for producing sample isomorphic linear surfaces. Here I will present such an algorithm. The input to the algorithm is a set of linear surfaces B in a space of dimension n, and a set of regions R isomorphic to a linear section. The output is a linear surface dividing the regions in R. Choose a linear surface b0 and n noncoincidental points p0, p1, ... pn-1 on b0. Let f map each linear surface b to a point p with coordinates equal to the lengths of the segments perpendicular to b0 from b to p0, p1, ... pn-1. I claim that f maps linear surfaces through the same point to points on a linear surface. Thus f determines mapping g from points to linear surfaces. Since b0 provides an orientation, two linear surfaces can be opposite or on the same side of a point. Let Q be the intersection points of each n-tuple from B, and D be the linear surfaces mapped to from Q by g. D forms a finite linear space with regions corresponding to sections in B. To find a linear surface dividing the regions in R, find which side of R each point in Q is on. Use those sidednesses to determine a region x in D. Choose a point y in x. Apply the inverse of f to y to obtain a linear surface dividing the regions in R.

I must prove that the map g from point to linear surface is well defined.

Consider a linear surface b through point p that f maps to coordinates c0, c1, ... cn-1. Since b goes through p, one coordinate ci depends on the other coordinates. If every coordinate except ci and one other cj is held fixed, then by similar triangles, ci = dcj + e, for some scalars d and e. Thus, the dependence of ci upon the other coordinates is the affine composition of affine functions. Therefore, the coordinates lie on a linear surface.

Implications of this algorithm for abstract linear spaces include that every linear space of the same number of boundaries has the same number of sections, and that the notion of sidedness of n-tuples of boundaries wrt boundaries is well defined.

Section Space Theorem

If space S0n is linear, then there exists linear space S1n, unique up to isomorphism, with boundaries corresponding to n-tuples of boundaries from S0n and regions corresponding to sections of S0n.

The proof follows from the algorithm above.

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