CHAPTER 1 The Calculus of Sweedler Powers 1.1. Let us begin by introducing some notation that will be used throughout this chapter. For a natural number n, which is by definition at least 1, we use the nota- tion In := { 1 , . . . , n}. Suppose now that mi,m2,... , mk are several natural num- bers. On the product I-mi x/m,2 x • • • xl77lk, we introduce the so-called lexicographical ordering. This means that, for two /c-tuples (ii,22, • • •, ik) and (ji,J2, • • • ,jk) m this set, we define (zi, z-2,... ,ik) (JI1J2, • • • ,ifc) if and only if there is an index I k such that i\ — .71,22 = J2 • • • ,2/_i = Jz—1, but i\ ji. As this is a total ordering, there is a unique strictly monotone map V^mi,...,?7ifc • J-mi ^ -^1712 X . . . X J-rnk * -*n where n := mi • rri2 • ... • mfe. Explicitly, this map is given by the formula ^m1,...,mfc(H,-.-,u) = (n - l)n2 + fe ~ l)n 3 + .. . + (ifc-i - l)nfc + ik where rii := m^ • m^+i • ... • m^, so that nk = vfik and rt\ = n. This holds since if ir = jr for r = 1,..., / — 1, but i/ ^ , then we have (ii - l)n/ + i + ... + (ik-i - l)nk + ik {ji ~ 1)^+1 + • • • + (jk-i - 1 W + j * as the maximal value that (ij+i — l)n/ + 2 + .. . -j- (ifc-i — l)n/~ + ik can attain is (rai+i - l)nj+2 + .. . + K _ i - \)nk + mk = raz+in/+2 = ^z+i and jjt 1. Considering this calculation for I = 0, we also see that the map is well-defined. Obviously, we have pn = id/n if k = 1. Now suppose that we have another set of natural numbers m^, ra2,..., m[, for a different Z, and let n' := m!x- m2 • ... • m[. On the one hand, we can then look at the composition (T T \ (T T \ P m i " • , m f c X V ? m l'-' m i , T T Vn,*\ j- i-'mi X . . . X lrrik) X V^m^ X . . . X J-rn'j^) -*n x -*n' * ^nn' On the other hand, we have the map ( ^mi,...,mfc,m/1,...,mj • -*mi X . . . X J-rnk * ^mj X . . . X i m ' • ^nn' It is clear from the definition of the lexicographical ordering that we have (ii,... ,ifc,ii,... ,i[) (ii,.. . JkJi, • • -Ji) if and only if we have (i x ,..., ik) (ji,... ,jk) or (ii,... , ik) = (ji,... ,jfc) and (i[,..., i[) (j(,..., j^). This shows that, if all appearing product sets are ordered lexicographically, both maps considered above are strictly monotone, and therefore must be equal. This proves the following statement: 5

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