By Joseph V. Collins
Excerpt from An uncomplicated Exposition of Grassmann's Ausdehnungslehre, or idea of Extension
The sum qf any variety of vectors is located by means of becoming a member of the start aspect of the second one vector to the tip aspect of the 1st, the start element of the 3rd to the tip element of the second one. etc; the vector from the start element of the 1st vector to the tip element of the final is the sum required.
The sum and distinction of 2 vectors are the diagonals of the parallelogram whose adjoining facets are the given vectors.
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Extra info for An elementary exposition of Grassmann's Ausdehnungslehre, or Theory of extension
It is easy to check that the neighbors can be reached by one step in one of the four directions. The left neighbor of an element in the first column is the last element in the same row, and so on. These diagrams are clearly arranged for n = 4 . For n 4 , we even obtain smaller diagrams. For n = 5 , we use two of the diagrams above, one for e = 0 and one for e = 1 . Then the fifth neighbor may be found at the same position of the other diagram. For n = 6 , we already have to work with 4 of these diagrams.
For n = 1 nothing has to be done. Let n 1. x⌈n 2⌉ and A0(⌊n 2⌋) A0(n) : In parallel we apply A1 (⌈n 2⌉) to x1 to x⌈n 2⌉+1 xn . Afterwards pi is computed for i ≤ ⌈n 2⌉ . All ⌈n 2⌉ may be pi = (x1 ◦ · · · ◦ x⌈n 2⌉) ◦ (x⌈n 2⌉+1 ◦ · · · ◦ xi ) for i computed in one step each in parallel. Ak (n) (k ≥ 1) : In parallel we compute the ⌊n 2⌋ pairs x1 ◦ x2 , x3 ◦ x4 . Afterwards we apply Ak−1(⌈n 2⌉) to these pairs and, if n is odd, xn . We compute all p2i p1 and pn . The missing ⌊n 2⌋ − 1 prefixes p2i+1 = p2i ◦ x2i+1 can be computed in parallel.
We have already seen that minimal polynomials consist of implicants only. Obviously the sum of all implicants computes f . If m and m′ are implicants, but m is a proper submonom of m′ , m ∨ m′ = m by the law of simplification, and we may cancel m′ . 4 : An implicant m ∈ I(f) is called prime implicant if no proper submonom of m is an implicant of f . PI(f) is the set of all prime implicants of f . 1 : implicants. Minimal polynomials for f consist only of prime All algorithms for the minimization of Boolean functions start with the computation of all prime implicants.
An elementary exposition of Grassmann's Ausdehnungslehre, or Theory of extension by Joseph V. Collins