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2 Let K be a commutative semiring. Denote by K A ⊗K A the complete tensor product, which is the set of infinite linear combinations over K of the elements u ⊗ v with u, v ∈ A∗ . If S, T ∈ K A , then S ⊗ T denotes the element S⊗T = u,v∈A∗ (S, u)(T, v)u ⊗ v . ¯ Define a mapping ∆ : K A → K A ⊗K A by ∆(S) = u,v∈A∗ 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 (S, uv)u ⊗ v . a) Show that the series S is recognizable if and only if ∆(S) is a finite sum 1≤i≤r Si ⊗ Ti , with Si , Ti ∈ K A .

Then w = cx for some c ∈ C. We choose w in such a way that the corresponding word x has minimal length. 4), (S, cx) = αc,p (S, px) , p∈P 777 778 779 780 and by the choice of x, one has (S, px) = 0 for all p ∈ P : indeed, either px ∈ P , or px = c′ y for some c′ ∈ C and y shorter than x. Thus (S, cx) = 0, a contradiction. A subset T of A∗ is suffix-closed if xy ∈ T implies y ∈ T for all words x and y. 7 Let S be a rational series of rank n. There exists a prefix-closed set P and a suffix-closed set T , both with n elements, such that det((S, pt))p∈P,t∈T = 0 .

Show that the inverse of a rational series is in general not rational, by considering the series n≥0 1/(n + 1)an in Q a (use Eisenstein’s criterion). 3 Let w = a1 · · · an be a word (ai ∈ A). For any subset I = {i1 < · · · < ik } of {1, . . , n}, define w|I to be the word ai1 · · · aik . Given two words x and y of length n and p respectively, define their shuffle product x ⊔⊔ y to be the polynomial x ⊔⊔ y = w(I, J) , where the sum is over all couples (I, J) with {1, 2, . . , n + p} = I ∪ J, |I| = n, |J| = p, and where w(I, J) is defined by w(I, J)|I = x, w(I, J)|J = y.

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