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3 Cancellation theory for RF (G) 23 Proof The equation f = f1 ∗ f2 with L( f1 ) = β and L( f2 ) = α − β is equivalent to the conjunction of f1 (ξ ) = f (ξ ), 0 ≤ ξ < β, f2 (ξ ) = f (ξ + β ), 0 < ξ ≤ α −β, and f1 (β ) f2 (0) = f (β ). Hence f1 and f2 living on the domains speciﬁed above and solving the equation f = f1 ∗ f2 do exist; they are uniquely determined once one of the values f1 (β ), f2 (0) has been speciﬁed, and one of these values can be chosen arbitrarily. 8. 15 (Visibility of cancellation) Let f , g ∈ RF (G) be reduced functions.

Analysing f (gh) in a similar way, we ﬁnd that f (gh) = f (g2 ◦ h1 ) = ( f1 ◦ u)((u−1 ◦ g3 ) ◦ h1 ) = ( f1 ◦ u)(u−1 ◦ (g3 ◦ h1 )) = f1 (g3 ◦ h1 ) = f1 ◦ (g3 ◦ h1 ). 17(ii) plus ε0 ( f1 , g3 ) = ε0 (g3 , h1 ) = 0 in the last step. 18). Case 2: L(u) ≥ L(g2 ). 14 together with the equations g = u−1 ◦ g1 and g = g2 ◦ v to decompose u−1 as u−1 = g2 ◦ u1 , for some u1 ∈ RF (G). Case 2(a): L(u1 ) > 0. 18 (associativity of the circle product), we write −1 −1 −1 f = f 1 ◦ u = f1 ◦ (u−1 1 ◦ g2 ) = ( f 1 ◦ u1 ) ◦ g2 ; in particular, ε0 ( f1 , u−1 1 ) = 0.

We conclude that ( f g)g−1 = gg−1 = 1G = f = f (gg−1 ), which shows that reduced multiplication is indeed not associative on F (G). 3 Cancellation theory for RF (G) Our principal aim for the remainder of this chapter is to show that the restriction of reduced multiplication to the subset RF (G) is associative, so that we have the following. 13 For every group G, the set RF (G) forms a group under reduced multiplication. 13; they will also be needed many times in the rest of this book. 4. 14 (Dissection of reduced functions) Let f : [0, α] → G be a reduced function, and let β be a real number such that 0 ≤ β ≤ α.