# 101 Short Cuts in Math Anyone Can Do by Gordon Rockmaker

By Gordon Rockmaker

A hundred and one brief Cuts in Math somebody Can Do will liberate the secrets and techniques of the artwork of calculation. it's going to bring up your energy of computation and thereby provide help to get extra out of the maths you presently recognize. you'll soon be surprised at your skill to resolve as soon as advanced difficulties quick.

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Vn-i in yi(7(A),D), we have Pigivo^. /n-i) o T) = 0^(507^(^(307, . . , y ^ _ i 07)) a s 7 f ^ = idD = 0^(^7(2/007, . P{yn-i)) = g{P{yo)^'"^P{yn-i)) as a preserves g^ as7f^ -=id/^. Therefore /3 preserves G. It is also easy to show that /? , and that P is continuous. So /? : yi(7(A), D) -^ D preserves the structure induced by D. Since D dualises D, the map /? must be given by evaluation at some b G 7(^) Q A. We will show that a is given by evaluation at b on a~^{D), Let X G a~^{D). For all a G A, we have x{j{a)) — 7(x(a)) G D, as 7 is a term function.

So we may regard Mo as the zeroth approximation to a dualising structure for M. 1 Lattice operations 27 to try to find afinitefamily R of algebraic relations on M such that the structure M =^ (M; G, R, T) yields a duality on ISP(M). In this section, we give a beautiful illustration of this approach. We show that a finite algebra must be dualisable if it has a pair of binary homomorphisms that are lattice operations. 1 Theorem Let M be a finite algebra that has binary homomorphisms V and A such that (M; V, A) is a lattice.

Assume that each s E S is a strong idempotent of a map in G'. Assume further that, for all k G M\D and t G M \ 5 , the sets G' and S fl End(M)(^) distinguish t within M. Then M yields a duality on I§P(M). Proof Let A be a finite algebra in yi : - ISP(M) and let a : D(A) -^ M be a morphism. 3 Dualisable term retracts 43 evaluation. So we can assume that there exists k E a(yi(A, M))\I?. 2, the map a is given by evaluation at some a e ^ on the set a~^{S), Since k G a ( y i ( A , M ) ) and a preserves End(M), we have End(M)(fc) C a ( y i ( A , M ) ) .