By Joseph V. Collins

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

The sum qf any variety of vectors is located by means of becoming a member of the start element of the second one vector to the top aspect of the 1st, the start element of the 3rd to the tip aspect of the second one. etc; the vector from the start aspect of the 1st vector to the tip aspect 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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**Sample text**

Y <_ x Then and This is not yet a P, since we have not required that singletons be covers, or that the local axiom is satisfied. It is a system of generators for a topology. Now consider the locale the subset of and P(P)xp(P) R e Cov(x). P(P) of downward closed subsets of consisting of pairs (R, +(x)) where P, and x e P Making this inf-stable, we obtain the set of pairs GALOIS THEORY 25 (R A S , +(x) A S) where S e P(P) and R e Cov(x) . By Proposition 4 the locale quotient Q of P(P) by the generated congruence relation is Q = {T e P(P) |Vx e P VR e Cov(x) VS e P(P) R A S C T iff + (x) A S C T} .

Then We have to show: enough to show is open, and x u e 0 (X) u = satisfies \/ u1-- ieJ Vi e I, u. £ V / u . But u. <_ p~31u - , so it is 1 1 jeJ 3 u i A p"3 U i <_ X / u . J j eJ 3 u . < p*(u. ) 1 x jeJ 3 By the usual argument, it is enough to show that if the left hand side of the bottom line equals 1, then the right hand side equals 1, but this is obvious. Proof of Theorem 1: An atom of X is an open subspace a c — > X such that a*a CZ A and 3 a = 1- Let A be the set of atoms. Each atom a defines a point of X, since a = 1 by Lemma 1.

Let M e s£(S ). The sup-lattice M(l) e s£(S) i s A. JOYAL § M. TIERNEY 48 equipped with a canonical Z-module structure, and putting defines an equivalence of categories T: sl(SL Moreover, for any pair M,N e si(S zop zop TM = M(l) ) -> Mod(Z) ) , we have a natural isomorphism T(Hom(M,N)) « Homz(TM,TN). e. M: Z o p -• si(S) is a functor satisfying conditions 1) and 2) of Proposition 1. If a <_ b, denote the morphism M(b) •* M(a) by p , and write E^ for its left adjoint. Then, the multiplication by a on TM = M(l) is given by the composite Ia p a : M(l) ->• M(l).