By Joseph V. Collins

Excerpt from An hassle-free Exposition of Grassmann's Ausdehnungslehre, or conception of Extension

The sum qf any variety of vectors is located via becoming a member of the start element of the second one vector to the top aspect of the 1st, the start aspect of the 3rd to the top aspect of the second one. and so forth; the vector from the start aspect of the 1st vector to the top 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

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Suppose a, b ∈ α ; β. Then there is a c ∈ G such that a α c β b. The claim is that a = ac−1 c ≡β ac−1 b ≡α cc−1 b = b. week 8 41 ac−1 ≡β ac−1 and c ≡β b imply (ac−1 )c ≡β (ac−1)b, and a ≡α c and c−1 b ≡α c−1 b imply a(c−1b) ≡α c(c−1b). So a, b ∈ β ; α. Hence α ; β ⊆ β ; α. The permutability of congruence relations on groups is a reflection of the fact that normal subuniverses permute under complex product, and hence that the relative product of two normal subuniverses is a normal subuniverse (recall the correspondence between congruences and normal subuniverses).

Mn a1 a2 a3 an−1 an Figure 16 To see this first let α = ΘM n (ai , aj ) with i = j. 1 = ai ∨ aj ≡α ai ∨ ai = ai = ai ∧ ai ≡α ai ∧ aj = 0. So ΘM n (ai, aj ) = ∇Mn . Now let α = ΘM n (ai, 0) and choose any j = i. 1 = ai ∨ aj ≡α 0 ∨ aj = aj . So ΘM n (aj , 1) ⊆ ΘM n (ai, 0) for all j = i. If i, j, k are all distinct, then aj ≡α 1 ≡α ak , and hence ∇Mn = ΘM n (aj , ak ) ⊆ ΘM n (ai , 0). Similarly ΘM n (ai , 1) = ∇Mn . (5) The only simple mono-unary algebras are cycles of prime order (exercise). The proof of the following theorem is also left as an exercise.

Ii) A ∼ = B × C implies either B or C is trivial. (iii) A has exactly two factor congruence relations, more precisely, the only two factor congruences of A are ∆A and ∇A . Proof. (i) =⇒ (ii): trivial (ii) =⇒ (iii). Let α and α ˆ be complementary congruences of A. Then A ∼ ˆ = A/α ×A/α. By assumption A/α or A/α ˆ is trivial. In the first case we have α = ∇A and hence ˆ =α∩α ˆ = ∆A . If A/α ˆ is trivial, then α ˆ = ∇A and α = ∆A . So ∆A and ∇A α ˆ = ∇A ∩ α are the only factor congruences of A. (iii) =⇒ (i).

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An elementary exposition of Grassmann's Ausdehnungslehre, or Theory of extension by Joseph V. Collins

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