Rational Series and Their Languages (EATCS Monographs on Theoretical Computer Science)

2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640

D1 (x) = (1 − Bx)(1 − 2641 2642

d−1 X i=1



Gi xi ) − Gd xd

with d ≥ 1, B, Gi ∈ K+ and B < λ1 . Since an+1 /an ∼ λ1 we see that un+1 = an+1 − Ban ≥ 0 for n large enough. 3. Let X T = tn xn = D1 S . n≥0

2643 2644 2645 2646

Suppose first that λ1 isPsimple, that is m = 1. Then T is a polynomial and Lemma 3.5 shows that n≥0 an+h xn is K+ -rational for h large enough. Hence S is K+ -rational. Suppose next that m ≥ 2 and argue by induction on m. Note that S, D1−1 and T have the dominating eigenvalue λ1 , the latter with

132 2647 2648 2649 2650 2651

2652 2653 2654 2655 2656

Chapter VIII. Positive Series in One Variable

multiplicity m − 1. Lemma 3.3(iii) and (ii) show that D1−1 and S have positive dominating coefficient. Thus by Lemma 3.3(i), since D1−1 T = S, the series T also has positive dominating coefficient. This implies that T P has ultimately positive coefficients and thus that for h large enough, the series n≥0 tn+h+d xn is K+ -rational, by induction on m. Thus tn+h+d = νN n γ for some representation (ν, N, γ) over K+ . Define a representation (ℓ, M, c) over K+ by   ah  uh+1      P Q   ℓ = (1, 0, . . . , 0) , M = , c =  ...  0 N   uh+d−1  γ

where h is chosen large enough and where all rows of Q are 0 except the last which is ν. We prove that   ah+n  uh+n+1      .. n M c=  .   uh+n+d−1  N nγ .

This is true for n = 0 by definition. Admitting it holds for n, the equality for n + 1 follows from Lemma 3.5 (where n is replaced by n + h), since QN n γ is a n column vector whose components are all 0 exceptP the last one which P is νN γ = h−1 n n i h tn+h+d . We deduce that ℓM c = an+h and S = i=0 ai x + x n≥0 an+h x is therefore K+ -rational. 

2657

4

Series of star height 2

2658

We consider now the star height of K+ -rational series. Theorem 4.1 Let K be a subfield of R or K = Z. Any K+ -rational series is in the subsemiring of K+ [[x]] generated by K+ [x] and by the series of the form (Bxp )∗

or

d−1 X i=1

Gi xi + Gd xd (Bxp )∗

∗

2659

with p, d ≥ 1, B, Gi ∈ K+ . In particular, they have star height at most 2.

2660

Proof. Denote by L this semiring. It is clearly closed under the substitution x 7→ αxq for q ≥ 1, α ∈ K+ . Thus it is also closed under the merge of series. So, if we follow the proof of Soittola’s theorem, wePmay pursue after steps 1. and 2. We start with a notation. Given a series V = n≥0 vn xn and an integer P P h ≥ 0, we write V (h) = n>h vn xn and V(h) = n≤h vn xn . Thus it follows from U = (1 − Bx)S that

2661

U (h) = S (h) − BxS (h−1) = S (h) (1 − Bx) − Bah xh+1

U(h) = S(h) − BxS(h−1) = S(h−1) (1 − Bx) + ah xh .

4. Series of star height 2

133

We show below the existence of a polynomial Ph with coefficients in K+ , for h large enough, such that   U (h) = Ph + T (h) + ah Gd xh+d (Bx)∗ H ∗

where

H = G + Gd xd (Bx)∗ . If m = 1, we take h large enough and T (h) = 0. If m ≥ 2, we conclude by induction on m that T (h) is in L. Thus the series U (h) is in L, and since (1 − Bx)S (h) = Bah xh+1 + U (h) the series S=

h X

ai xi + (Bx)∗ (Bah xh+1 + U (h) ) .

i=0

2662

is in L. Now from T = D1 S = (1 − Bx)(1 − H)S = U (1 − H) , we get

Next


(h)
(h)
(h) T (h) = U (1 − H) = U (h) (1 − H) + U(h) (1 − H)
(h) = U (h) (1 − H) + U(h) − U(h) H
(h) = U (h) (1 − H) − U(h) H . U(h) H

Recall that G = U(h) H


(h)


(h)
(h) = U(h) G + U(h) Gd xd (Bx)∗

Pd−1 i=1


(h)

=

Gi xi . The first term of the right-hand side is X

uj Gℓ xj+ℓ .

0≤j≤h 0<ℓh

Setting j + ℓ = h + i with 0 < i < d, this rewrites as X uj Gℓ , wi =

Pd−1 i=1

wi xh+i with

0≤j≤h 0<ℓ
2663 2664 2665

Now note that in this sum, since ℓ < d, we have j > h − d, hence uj ≥ 0 for h
(h) large enough. This shows that U(h) H is a polynomial with coefficients in K+ . To compute the second term, recall that U(h) = S(h−1) (1 − Bx) + ah xh . Consequently U(h) (Bx)∗ = S(h−1) + ah xh (Bx)∗ .

134

Chapter VIII. Positive Series in One Variable

So the term (U(h) Gd xd (Bx)∗ )(h) reduces to the sum of a polynomial with coefficients in K+ and of the series Gd ah xh+d (Bx)∗ . Thus we obtain, for h large enough T (h) = U (h) (1 − H) − Gd ah xh+d (Bx)∗ − Ph 2666

with Ph ∈ K+ [x].

2667

Exercises for Chapter VIII

2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678



P 1.1 a) Let θ be a real number. Show that the series S = n≥0 (cos2 nθ)xn is a C-rational series. (Give an expression for S as a rational function by using the formula cos nθ = 1/2(einθ + e−inθ ).) b) Let 0 < a < c be integers and let θ be a real number with 0 < θ < π/2, such that cos θ = a/c. that the numbers cn cos nθ are integers. Show P Show 2n that the series T = (c cos2 nθ)xn is Z-rational with coefficients in N. c) Show that if c 6= a, then z = eiθ is not a root of unity (use the fact that z is an algebraic number of degree ≤ 2, and that the assumption that z is a root of unity of order p implies that φ(p) ≤ p, where φ is Euler’s function). Show that T is not R+ -rational (use Theorem 1.1) (see Berstel 1971, and also Eilenberg 1974). 1.2 Show that the Z-rational series X x + 5x2 = (2· 5n − (3 + 4i)n − (3 − 4i)n )xn 1 + x − 5x2 − 125x3 n≥0

= x + 4x2 + x3 + 144x4 + · · ·

2679

has positive coefficients but is not N-rational. √ 1.3 Let c > d be integers such that d ± i c2 − d2 are not roots of unity, and define a sequence an by  n  n p p an = b1 cn + b2 d + i c2 − d2 + b3 d − i c2 − d2

P for integers b1 ≥ b2 +b3 . Show that an xn is Z-rational with nonnegative coefficients and is not N-rational. Example: for c = 3, d = 2, b1 = 2, b2 = b3 = 1, one gets X

2680 2681 2682 2683

an xn =

4 − 12x + 24x2 = 4 + 8x + 4x2 + 8x3 + · · · 1 − 5x + 15x2 − 27x3

P 1.4 Let S = an xn = P (x)/Q(x) be a rational series over R, where P (x) and Q(x) have no common root, and Q(x) is a polynomial of degree 2 with Q(0) = 1. Set Q(x) = 1 − ax − bx2 and P (x) = c − dx. Set further Q(x) = (1 − αx)(1 − βx). a) Show that a0 = c, a1 = ac − d and for n ≥ 2 
 1 (αc − d)αn − (βc − d)β n if α 6= β , an = α − β αn−1 (αc − d)n + αc
if α = β .

4. Series of star height 2 2684 2685 2686 2687 2688 2689

3.1

2690 2691 2692 2693 2694

3.2

2695 2696 2697 2698 2699 2700

3.3

2701

3.4

2702 2703

3.5

2704 2705 2706 2707

3.6

2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2721

3.7

2722

4.1

135

b) Assuming that an ≥ 0 for n ≥ 0, show successively that c ≥ 0, ac−d ≥ 0, a ≥ 0, a2 + 4b ≥ 0 and αc − d > 0. c) Show that conversely, if these five conditions are fulfilled, then an ≥ 0 for n ≥ 0. P Let S = an xn = P (x)/Q(x) be a rational series over R, where P (x) and Q(x) have no common root, and Q(x) is a polynomial of degree 2 with Q(0) = 1. Show that a S is R+ -rational if and only if all coefficients an are nonnegative. Hint: Set Q(x) = (1 − αx)(1 − βx) and use the Exercise 1.4 to show that if all an are nonnegative, then α and β are real, and that at least one is positive. Then, use Soittola’s theorem. Let K be a subring of some field and P ∈ K[x] with P (0) = 1. Let M be the companion matrix of P . With the notations of Lemma 3.3, show that Pn = det(1 − M n x). Deduce that the coefficients of Pn are in the subring generated by the coefficients of P . Show that the characteristic polynomial of a Perrin companion matrix is the corresponding Soittola polynomial (see Perrin (1992)). Show that the inverse of a Soittola denominator is an R+ -rational series 1 (multiply by (Bx)∗ ). Show that = (Bx)∗ (G(x) + Gd xd (Bx)∗ )∗ . D(x) Let M be a square matrix over some subsemiring K of a commutative ring. Show that det(1 − M x)−1 is a K-rational series. Hint: let Mi be the submatrix corresponding to the first i rows and columns. Show that det(1 − Mi−1 )/ det(1 − Mi x) is K-rational and then take the product. P a) Let S = n≥0 an xn ∈ C[[x]] be rational with a dominating eigenvalue λ. Let S = P/Q(1 − λx), with P, Q ∈ C[x] and Q(0) = 1, in lowest terms. Show that (x−n S)Q(1 − λx) is a polynomial of degree ultimately equal to deg(Q) and that limn→∞ (x−n S)Q(1 − λx)/an = Q, with coefficientwise limit. b) Modify Lemma 3.4 so that the conclusion includes the property that Qd (Bx)∗ i=2 (1 − λi x) has positive coefficients. c) Let S(x) = N (x)/D(x), with D(x) equal to the Soittola denominator (3.3), with the condition that (Bx)∗ E has positive coefficients, where D(x) = (1 − λx)E(x) and λ is the dominating root. Define x−n S = an Rn (x)/D(x). Show that (Bx)∗ Rn (x) has positive coefficients for n large enough. Deduce that S is K+ -rational. d) Deduce an alternative proof of Soittola’s theorem in the case where the dominant eigenvalue is simple. See Katayama et al. (1978). By drawing the weighted automaton associated to a Perrin companion matrix, give another proof of Theorem 4.1, see (Perrin 1992). Let A = {a, b}. A Dyck word over A is a word w such that |w|a = |w|b and |u|a ≥ |u|b for each prefix u of w. The height of a Dyck word w is max{|u|a − |u|b }, where u ranges over the prefixes of w. The first Dyck words are 1, ab, aabb, abab, aaabbb, aababb, aabbab, abaabb, ababab, . . .

2723 2724

The words aabb, aababb, abaabb have height 2. Denote by D the set of Dyck words over A.

136 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735

2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747

Chapter VIII. Positive Series in One Variable a) Show that D = 1 + aDbD. b) Denote by Dh the set of Dyck words of height at most h. In particular D0 = {1} is just composed of the empty word. Show that for h ≥ 0 aDh bDh+1 . Dh+1 = 1 +P P Set f (x) = n≥0 Card(D∩A2n )xn , and fh (x) = n≥0 Card(Dh ∩A2n )xn . These are the generating functions of the number of Dyck words (Dyck words of height at most h). c) Show that f = (xf )∗ and that fh+1 = (xfh )∗ for h ≥ 0. d) Show that fh = qh−1 /qh for h ≥ 0, where qh+1 = qh − xqh−1 for h ≥ 0, with q0 = q−1 = 1. e) Give an expression of star height at most 2 for f3 , f4 , f5 .

Notes to Chapter VIII A proof of Theorem 1.1 based on the Perron-Frobenius theorem has been given by Fliess (1975). The proof of Theorem 3.1 given here is based on Soittola (1976), Perrin (1992). The proof of Theorem 3.1 by Katayama et al. (1978) seems to have a serious gap, see the final comments in Berstel and Reutenauer (2007); however it works in the case of a simple dominant eigenvalue, and this is summarized in Exercise 3.6. Recently, algorithmic aspects of the construction have been considered in Barcucci et al. (2001) and in Koutschan (2005, 2006). The example of Exercise 1.2 is from Gessel (2003), Exercise 1.3 is from Koutschan (2006). Exercises 1.4 and 3.1 are from an unpublished paper of late C. Birger, 1971, see also (Salomaa and Soittola 1978). A related result is in (Halava et al. 2006).

2748

2749

2750

Chapter IX

Matrix Semigroups and Applications

2755

In the first section, we be bounded (Theorem matrix semigroup. As rational series is finite. are studied.

2756

1

2751 2752 2753 2754

2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776

show that the size of a finite semigroup of matrices can 1.1). This implies that the finiteness is decidable for a a consequence, one can decide whether the image of a To complete the chapter, series with polynomial growth

Finite matrix semigroups and the Burnside problem

We first give a result concerning finite monoids of matrices. Recall that for a given word w, we denote by w∗ the submonoid generated by w. Theorem 1.1 (Jacob 1978, Mandel and Simon 1977) Let µ : A∗ → Qn×n be a monoid morphism such that, for all w ∈ A∗ , the monoid µw∗ is finite. Then there exists an effectively computable integer N depending only on Card A and n such that Card µ(A∗ ) ≤ N . As we shall see, the function (Card A, n) 7→ N grows extremely rapidly. There exists however one case where there is a reasonable bound (which moreover does not depend on Card A), namely the case described in the lemma below. A set E of matrices in Qn×n is called irreducible if there is no subspace of 1×n Q other than 0 and Q1×n invariant for all matrices in E (the matrices act on the right on Q1×n ). Lemma 1.2 (Sch¨ utzenberger 1962c) Let M ⊂ Qn×n be an irreducible monoid of matrices such that all nonvanishing eigenvalues of matrices in M are roots 2 of unity. Then Card M ≤ (2n + 1)n . Proof. Let m ∈ M . The eigenvalues 6= 0 of m are roots of unity, whence algebraic integers over Z. Hence tr(m) is an algebraic integer. Since tr(m) ∈ Q and Z is integrally closed, this implies that tr(m) ∈ Z. The norm of each eigenvalue is 0 137

138 2777 2778

Chapter IX. Matrix Semigroups and Applications

or 1. Thus | tr(m)| ≤ n. This shows that tr(m) takes at most 2n + 1 distinct values for m ∈ M . Let m1 , . . . , mk ∈ M be a basis of the subspace N of Qn×n generated by M . Clearly k ≤ n2 . Define an equivalence relation ∼ on M by m ∼ m′ ⇐⇒ tr(mmi ) = tr(m′ mi ) for i = 1, . . . , k .

2779 2780

The number of equivalence classes of this relation is at most (2n + 1)k . In order to prove the lemma, it suffices to show that m ∼ m′ implies m = m′ . Let m, m′ ∈ M be such that m ∼ m′ . Set p = m − m′ , and assume p 6= 0. There exists a vector v ∈ Q1×n such that vp 6= 0. It follows that the subspace vpN of Q1×n is not the null space. Since it is invariant under M and M is irreducible, one has vpN = Q1×n . Consequently, there exists some q ∈ N such that vpq = v. This shows that pq has the eigenvalue 1. Now, for all integers j ≥ 1, tr((pq)j ) = tr(pq(pq)j−1 ) = 0

2781 2782 2783 2784

2785 2786 2787

because q(pq)j−1 is a linear combination of the matrices m1 , . . . , mk , and by assumption tr(pr) = 0 for r ∈ M . Newton’s formulas imply that all eigenvalues of pq vanish. This yields a contradiction.  For the proof of Theorem 1.1, we need another lemma. Lemma 1.3 (Sch¨ utzenberger 1962c) (i) Let α be a morphism from A∗ into a finite monoid M . Then, for each word w of length ≥ Card(M )2 , there exists a factorization w = x′ zx′′ with z 6= 1, αx′ = α(x′ z) and α(zx′′ ) = αx′′ .   µ′ ν (ii) Let µ : A∗ → Qn×n be a multiplicative morphism of the form , 0 µ′′ and let w = x′ zx′′ ∈ A∗ be such that µ′ x′ = µ′ (x′ z) and µ′′ (zx′′ ) = µ′′ x′′ . Then for any n in N, µ′ x′ νz n µ′′ x′′ = n µ′ x′ νzµ′′ x′′ ν(x′ z n x′′ ) = ν(x′ x′′ ) + n µ′ x′ νzµ′′ x′′ .

(1.1)

Proof. (i) Indeed, the set {(x, y) ∈ (A∗ )2 | w = xy} has at least 1 + Card(M )2 elements, and therefore there exist two distinct factorizations w = x′ y ′ = y ′′ x′′ such that αx′ = αy ′′ 2788 2789

and αy ′ = αx′′ .

We may assume that |x′ | < |y ′′ |. Then there is a word z 6= 1 such that y ′′ = x′ z and y ′ = zx′′ . Thus w = x′ zx′′ with the required properties. (ii) One has the identity   n  n P ak bcℓ a a b k+ℓ=n−1 . = 0 c 0 cn

1. Finite matrix semigroups and the Burnside problem

139

Thus ν(z n ) =

X

µ′ (z k )νzµ′′ (z ℓ ) .

k+ℓ=n−1

Multiplying on the left by µ′ x′ and on the right by µ′′ x′′ , we obtain X µ′ x′ νz n µ′′ x′′ = µ′ x′ µ′ (z k )νzµ′′ (z ℓ )µ′′ x′′ X = µ′ (x′ z k )νzµ′′ (z ℓ x′′ ) = n µ′ x′ νzµ′′ x′′ . Finally by considering the product µ′ x′ µz n µ′′ x′′ , we obtain

ν(x′ z n x′′ ) = νx′ µ′′ (z n x′′ ) + µ′ x′ ν(z n )µ′′ x′′ + µ′ (x′ z n )νx′′ = µx′ µ′′ x′′ + n µ′ x′ νzµ′′ x′′ + µ′ x′ νx′′ = ν(x′ x′′ ) + n µ′ νzµ′′ x′′ .



Corollary 1.4 (Sch¨ utzenberger 1962c) Let µ : A∗ → Qn×n be a morphism into a monoid of matrices which are triangular by blocks  ′  µ ν µ= . 0 µ′′ Assume that µ′ A∗ and µ′′ A∗ are finite, and that µw∗ is finite for any word w. Then X Card(νA∗ ) ≤ Card Ai , 0≤i<(H ′ H ′′ )2

2790

where H ′ = Card µ′ A∗ and H ′′ = Card µ′′ A∗ .

2791

Proof. In Lemma 1.3(i), take α = (µ′ , µ′′ ). Then each word w of length ≥ (H ′ H ′′ )2 has a factorization w = x′ zx′′ with z 6= 1 and the relations (1.1) hold. Thus, since µz ∗ is finite, ν(x′ z ∗ x′′ ) is also finite and we must have µ′ x′ νzµ′′ x′′ = 0 and νw = ν(x′ x′′ ). Since |x′ x′′ | < |w|, the corollary follows. 

2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803

Proof of Theorem 1.1. Assume first that the monoid µA∗ is irreducible, and consider any matrix µw ∈ µA∗ . Since µz ∗ is finite, there are integers 0 ≤ i < j with µwi = µwj . But this implies that the eigenvalues of w are 0 or roots of unity. The theorem thus follows from Lemma 1.2. If µA∗ is not irreducible, there is some subspace V of Q1×n which is invariant under µA∗ . Consider a supplementary space W of V . In a basis which is adapted to the decomposition Q1×n = W ⊕V , the morphism µ admits the form described in Lemma 1.3. Arguing by induction on the dimension of the representation, the result follows from Lemma 1.3. 

2805 2806

We say that an element s of a semigroup S is torsion if s generates a finite subsemigroup of S; equivalently, sl = sℓ for some 1 ≤ k < ℓ. We say that S is a torsion semigroup if each element in S is torsion.

2807 2808

Corollary 1.5 (McNaughton and Zalcstein 1975) Every finitely generated torsion semigroup of square matrices over Q is finite. 

2804

140

Chapter IX. Matrix Semigroups and Applications

2809

Recall that a ray is a subset of A∗ of the form uv ∗ w, with u, v, w ∈ A∗ .

2810 2811

Corollary 1.6 (Reutenauer 1977b) Let S ∈ QhhAii be a rational series such that for any ray R, the set {(S, w) | w ∈ R} is finite. Then the set of coefficients of S is finite.

2812

Proof. Let (λ, µ, γ) be a reduced linear representation of S. By Corollary II.2.3, there exist polynomials P1 , . . . , Pn , Q1 , . . . , Qn such that for all words w, µw = ((S, Pi wQj ))1≤i,j≤n . By assumption, the set {(S, uwm v) | m ∈ N} is finite for all words u, v, w. The same holds for the set {(S, P wm Q) | m ∈ N} where P, Q are polynomials. This shows that µw∗ is finite for any word w. By Corollary 1.5, the monoid µA∗ is finite, and in particular {(S, w) | w ∈ A∗ } 2813

is finite, since (S, w) = λµwγ.

2814 2815

Corollary 1.7 (Jacob 1978) It is decidable whether a finite set of matrices over Q generates a finite monoid.

2816

2819 2820

Proof. By Theorem 1.1, there is an upper bound on the size of such a monoid if it is finite. Let E be a finite set of matrices, M the monoid generated by E, and let N be the upper bound given in Theorem 1.1. Then M is finite if and only if every product of N matrices in E equals a product of at most N − 1 matrices in E. This last condition is clearly decidable. 

2821

Recall that the image of a series is the set of its coefficients.

2822 2823

Corollary 1.8 (Jacob 1978) It is decidable whether a rational series has a finite image. 

2824

2

2817 2818



Polynomial growth

We now turn our attention to questions concerning growth of rational series over Z. Recall that a series S ∈ ZhhAii has polynomial growth or is polynomially bounded if there exist a real number q ≥ 0 and a real number C such that |(S, w)| ≤ C|w|q

2825 2826 2827

for all nonempty words w. The smallest of these q, if it exists, is called the degree of growth of S. Observe that series with degree of growth 0 are precisely the series with finite image. In the sequel, we shall consider morphisms µ : A∗ → Qn×n which have the block-triangular form   µ0 ν1 ∗ · · · ∗  .  0 µ1 . . . . . . ..   . . .  (2.1) µ=  .. 0 . . . . ∗    . .  . . . . νq  0 · · · 0 µq

141

2. Polynomial growth 2828

Observe that each µi is itself a morphism.

2829

Theorem 2.1 Let S ∈ ZhhAii be a rational series and let (λ, µ, γ) be a reduced linear representation of S. Then S has polynomial growth if and only if the set {tr(µw) | w ∈ A∗ } is finite.

2830 2831

Proof. Suppose first that S has polynomial growth. Then there exist, by Corollary II.2.3, real numbers C, q such that for all i, j, |(µw)i,j | ≤ C|w|q for all words w. Thus, for any r ∈ N, we have |(µwr )i,j | ≤ Crq |w|q . Consequently, for every eigenvalue ρ of µw one has |ρ|r ≤ C ′ rq 2832 2833 2834 2835 2836

for some constant C ′ . Thus |ρ| ≤ 1. This implies that −n ≤ tr(µw) ≤ n, where n is the dimension of µ. Since S is Z-rational, there exists a reduced linear representation with coefficients in Z (Theorem VII.1.1). This representation is similar to (λ, µ, γ) by Theorem II.2.4 and consequently, the trace of any matrix µw is an integer. Thus each tr(µw) is in {−n, . . . , n}. Conversely, suppose that the set {tr(µw) | w ∈ A∗ } is finite. Let w be a word and let λ1 , . . . , λn be the eigenvalues of µw with their multiplicities. The sequence X p ap = λi = tr(µwp ) 1≤i≤n

takes only a finite number of distinct values. Since it satisfies a linear recurrence relation, it is ultimately periodic, and there is a relation ap+h = ap+k 2837 2838

p≥0

for some h, k ∈ N,Ph > k. The minimal polynomial (see Section VI.1) of ap xp divides the polynomial xh − xk . Consequently, the the rational series p∈N

2839 2840 2841

2842 2843 2844 2845

eigenvalues of this series (in the sense defined in Section VI.1) are roots of unity or 0. In view of the uniqueness of the exponential polynomial (Section VI.2), the λi are therefore roots of unity or 0. Next, if the monoid µA∗ is not irreducible, then µ can be put, by changing the basis, into the form  ′  µ ν µ= 0 µ′′ Arguing by induction, µ is equivalent to a morphism of the form (2.1) with each µi A∗ irreducible. By Lemma 1.2 and by our computations, all monoids µi A∗ are finite. To complete the proof, it suffices to apply the following two lemmas.  Lemma 2.2 Let K be a commutative semiring. (i) Let  ′  µ ν µ= 0 µ′′

142 2846 2847 2848 2849 2850 2851

Chapter IX. Matrix Semigroups and Applications

be a morphism A∗ → K n×n . Every series recognized by µ is a linear combination of series recognized by µ′ or by µ′′ and of series of the form S ′ aS ′′ , where S ′ is recognized by µ′ , a ∈ A and S ′′ is recognized by µ′′ . (ii) If µ : A∗ → K n×n has the form (2.1) with each µi of finite image, then each series recognized by µ is a linear combination of products of at most k + 1 characteristic series of rational languages. Proof. (i)A series recognized by µ is a linear combinations of series of the form X

(µw)i,j w

(2.2)

w

with 0 ≤ i, j ≤ n. It suffices to show that when i, j are coordinates corresponding to ν, the series (2.2) is a linear combination of series of the form S ′ aS ′′ . This is a consequence of the formula X νw = µ′ xνaµ′′ y . w=xay

2852 2853 2854 2855 2856 2857

2858 2859 2860 2861 2862

(ii) Using (i) iteratively, we see that a series recognized by µ is a K-linear combination of series of the form S0 a1 S1 a2 · · · aℓ Sℓ , with ℓ ≤ k, where ai ∈ A and each Si is recognized by some µj . Since µj (A∗ ) is a finite monoid, each language µ−1 j (m) is rational by Theorem III.1.1 (Kleene’s theorem). Hence a series recognized by µj is a linear combination of characteristic series of rational languages and this concludes the proof.  Lemma 2.3 (i) Let S, T be two series over R and p, q ∈ N.. If S has degree of growth q and T and has degree of growth p, then ST has degree of growth at most p + q + 1. (ii) The product of q + 1 characteristic series of rational languages has degree of growth at most q.

Proof. (i) We have |(S, w)| ≤ C |w|+q and (T, w)| ≤ D |w|+p for suitable q p P constants C, D. Since (ST, w) = w=uv (S, u)(T, v), it follows that X |u| + q |v| + p |(ST, w)| ≤ CD . q p w=uv

The summation is equal to the coefficient of x|w| in the product X i + q  X j + p xi xj . q p i j

2865


i
P q+1 Since i i+q , we obtain that this coefficient is |w|+p+q+1 . q x = 1/(1 − x) p+q+1 Since this is a polynomial in |w| of degree p + q + 1, the assertion follows. (ii) follows from (i) by induction. 

2866 2867

Corollary 2.4 It is decidable whether a rational series S ∈ ZhhAii has polynomial growth.

2863 2864

2. Polynomial growth

143

Proof. A reduced linear representation (λ, µ, γ) of S can effectively be computed. Then according to Theorem 2.1, the series S has polynomial growth if and only if the series X tr(µw)w w

2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887

has a finite image. This series is rational (Lemma II.1.3) and it is decidable, by Corollary 1.8 whether a rational series has a finite image.  The main result of this section is the following theorem. Theorem 2.5 (Sch¨ utzenberger 1962c) Let S be a Z-rational series which has polynomial growth. Then S has a minimal linear representation (λ, µ, γ) whose coefficients are in Z, and such that µ has the block-triangular form (2.1) where each µi A∗ is a finite monoid. Moreover, let q be the smallest integer for which this holds. Then the degree of growth of S exists and is equal to q and there exist words x0 , . . . , xq , y1 , . . . , yq such that (S, x0 y1n x1 · · · yqn xq ) is a polynomial in n of degree q. Corollary 2.6 (Sch¨ utzenberger 1962c) The degree of growth of a polynomially bounded Z-rational series S is equal to the smallest integer q such that S belongs to the submodule of ZhhAii spanned by the products of at most q+1 characteristic series of rational languages. Proof. Suppose that the degree of growth of S is q. Then, by the theorem, there exists a linear representation (λ, µ, γ) of S with µ of the form (2.1). By Lemma 2.2(ii), we get that the series S is a Z-linear combination of no more than q + 1 characteristic series of rational languages. Conversely, suppose that S is of this form. Then by Lemma 2.3 S has degree of growth ≤ q, and this proves the second assertion.  Recall that, given a ring K, two representations µ, µ′ : A∗ → K n×n are called similar if, for some invertible matrix P over K, one has µ′ w = P −1 µwP

2888 2889 2890

for any word w. In other words, µ′ is obtained from µ after a change of basis over K. When several rings occur, we will emphasize this by saying similar over K. Lemma 2.7 Let µ : A∗ → Zn×n be a representation. Suppose that µ is similar over Q to a representation µ′ : A∗ → Qn×n which has the block-triangular form   µ0 ∗ · · · ∗  . . ..   0 µ µ′ =  . . 1 . . .   .. . . . . ∗  0 · · · 0 µq

2891 2892 2893

Then µ is similar over Z to a representation ν : A∗ → Zn×n having the same form and such that the corresponding diagonal blocks of µ′ and ν are similar over Q.

144 2894 2895 2896 2897 2898 2899 2900 2901 2902 2903 2904 2905 2906 2907 2908 2909 2910 2911 2912 2913 2914 2915 2916 2917 2918

Chapter IX. Matrix Semigroups and Applications

Proof. The hypothesis means that there is a basis of the Q-vector space Qn×1 of column vectors of the form B0 ∪ · · · ∪ Bq such that for any word w, the matrix µw sends the subspace Ei spanned by B0 ∪ · · · ∪ Bi into itself, and that µi w represents the action of µw on Bi modulo Ei−1 . We put E−1 = 0. It suffices therefore to show the existence of a Z-basis of Zn×1 of the form C0 ∪ · · · ∪ Cq such that Ei is also spanned over Q by C0 ∪ · · · ∪ Ci . Then Ci , as is Bi , will be a Q-basis of Ei modulo Ei−1 and therefore the diagonal blocks will be similar over Q, as in the statement. Recall that if V is a submodule of Zn , then it has a basis d1 e1 , . . . , dk ek for some basis e1 , . . . , en of Zn and some nonzero integers d1 , . . . , dk (see Lang (1984), Theorem III.7.8, knowing that Z is a principal ring). If V is divisible (that is, dv ∈ V and d ∈ Z, d 6= 0 imply v ∈ V ), then one may choose d1 = · · · = dk = 1. In other words, given a divisible submodule V of a finitely generated free Z-module F , there exists a free submodule W such that F = V ⊕ W . Let Vi = Ei ∩ Zn×1 . These submodules of Zn×1 are all divisible and 0 = V−1 ⊆ V0 ⊆ · · · ⊆ Vq = Zn×1 . Thus we may find free submodules Wi of Zn×1 such that Vi = Vi−1 ⊕ Wi for i = 0, . . . , q. Let Ci be a Z-basis of Wi . Then C0 ∪· · ·∪Ci is a Z-basis of Vi and therefore Ei is spanned over Q by C0 ∪· · ·∪Ci .  Proof of Theorem 2.5, first part. Let S ∈ ZhhAii be a rational series having polynomial growth, and let (λ, µ, γ) be a reduced linear representation of S. We may assume, by Theorem VII.1.1, that (λ, µ, γ) has integral coefficients. The second part of the proof of Theorem 2.1 shows that, after a change of the basis of Q1×n , µ has a decomposition of the form (2.1) where each µi A∗ is finite. In fact, by Lemma 2.7, the change of basis can be done in Z1×n .  Lemma 2.8 (Sch¨ utzenberger 1962c) Let µ : A∗ → Zn×n be a representation of the form  ′  µ ν µ= , 0 µ′′ where µ′ , µ′′ have finite image. If (νA∗ )v is finite for some nonnull vector v, then µ is similar over Z to a representation   µ1 ν , µ= 0 µ2

2919

where µ1 and µ2 have finite image and with dim(µ1 ) > dim(µ′ ). Proof. By Lemma 2.7, we may work over Q. Let F = {u ∈ Qn×1 | (µA∗ )ufinite}. Then F is invariant under each µw. Let also E ′ , E ′′ be the subspaces of Qn×1 corresponding to µ′ and µ′′ . Then E ′ ⊆ F . Moreover, E ′′ is a direct sum E ′′ = (E ′′ ∩ F ) ⊕ E1′′ . Taking a basis of E ′′ corresponding this direct sum,  to µ′′1 ν ′ ′′ we see that µ is similar to a representation of the form . Thus µ is 0 µ′′2 similar to a representation of the form   ′ µ ν1 ν2  0 µ′′1 ν ′  . 0 0 µ′′2

2. Polynomial growth

145

We have F = E ′ ⊕ (E ′′ ∩ F ) ,

2920 2921 2922 2923

(2.3)

′ n×1 ′′ since = E′ ⊕ Thus, for any vector u in F , the set  ′ ∗E ⊆ ∗F and Q  E′ .  µ A ν1 A µ ν1 u is finite. Thus has finite image. Moreover, µ′′2 has 0 µ′′1 A∗ 0 µ′′1 also finite image, since it is a part of µ′′ . Taking  ′    µ ν1 ν2 µ1 = , ν= , µ2 = µ′′2 , 0 µ′′1 ν′   µ1 ν we see that µ is similar to . 0 µ2 Now, if (νA∗ )v is finite for some nonnull vector v, we see that F is strictly larger than E ′ and consequently dim(µ1 ) = dim(µ′ ) + dim(µ′′1 ) > dim(µ′ ) since dim(µ′′1 ) = dim(E ′′ ∩ F ) > 0 by (2.3). 

Lemma 2.9 (Sch¨ utzenberger 1962c) Let µ : A∗ → Qn×n be a representation of the form  ′  µ ν µ= , 0 µ′′ 2924 2925 2926 2927 2928 2929 2930 2931 2932 2933 2934 2935 2936 2937

where µ′ , µ′′ have finite image, and let α : A∗ → M be a morphism of A∗ into a finite monoid M . Suppose that (νA∗ )v is infinite for any nonnull vector of the
0 form v in Qn×1 . Then, for any such vector, there exist words x′ , z, x′′ in A∗ such that µ′ x′ νzµ′′ x′′ v 6= 0, α(x′ z) = αx′ , α(zx′′ ) = αx′′ and α(z 2 ) = αz. Proof. We claim that for each vector v with (νA∗ )v infinite, there exist words x′ , z, x′′ in A∗ such that α(x′ z) = αx′ , α(zx′′ ) = αx′′ and µ′ x′ νzµ′′ x′′ v 6= 0. Indeed, arguing by contradiction, let w be a word of length greater than or equal to Card(M ) Card(µ′ A∗ ) Card(µ′′ A∗ ). Then by Lemma 1.3(i), there exists a factorization w = x′ zx′′ with z nonempty and ϕ(x′ z) = ϕ(x′ ), ϕ(zx′′ ) = ϕ(x′′ ), where ϕ = (α, µ′ , µ′′ ). Then, by assumption, we have µ′ x′ νzµ′′ x′′ v = 0. By Lemma 1.3(ii), ν(w)v = ν(x′ zx′′ )v = ν(x′ x′′ )v, and since x′ x′′ is shorter than w, we contradict the hypothesis that (νA∗ )v is infinite, and the claim is proved. Now α(z n ) is idempotent for some n ≥ 1. Since µ′ x′ νz n µ′′ x′′ = n µ′ x′ νzµ′′ x′′ by Lemma 1.3(ii), the lemma is proved by replacing z by z n .  In the sequel, we will consider matrices having an upper triangular form   m0,0 m0,1 · · · m0,q ..    0 m1,1 .  (2.4) m= .  . . . . . . . ..   .. 0 · · · 0 mq,q

where each mi,j is a matrix of fixed size depending on i and j, with mi,i square. We denote by M this set of matrices. In what follows, we call matrix polynomial in n over Q a matrix of the form m0 + nm1 + · · · + nd md ,

146 2938 2939 2940 2941 2942 2943 2944 2945 2946 2947

Chapter IX. Matrix Semigroups and Applications

where the mi are matrices of the same size. If md 6= 0, then d is the degree of this matrix polynomial. If d = 0 we say that the polynomial is constant. More generally, we consider also matrix polynomials in several commuting variables n, n1 , n2 , . . .. We denote by P the set of matrices m ∈ M such that each mi,j is a matrix polynomial in n over Q of degree at most j − i. Lemma 2.10 (i) P is a ring. (k) (ii) Let M1 , . . . , Mq ∈ P. Write Mk = (mi,j ) in accordance with (2.4). Then the block of coordinate 0, q of the product M (nn1 ) · · · Mq (nnq ) is a matrix polynomial in n, n1 , . . . , nq and the coefficient of nq n1 · · · nq in this polynomial (1) (2) (q−1) is m0,1 m1,2 · · · mq−1,q .

2948

The proof is left to the reader.

2949

Lemma 2.11 (Sch¨ utzenberger 1962c) Let a, b, c in M be such that ai,i bi,i = ai,i , b2i,i = bi,i , bi,i ci,i = ci,i . Set m(n) = abn c. Then m(n) ∈ P and its i, i + 1

2950 2951

(n)

block is mi,i+1 = nai,i bi,i+1 ci+1,i+1 + C, where C is some constant.

Proof. (i) We compute the n-th power of the matrix b. We first compute its block of coordinates 0, q. The latter is the sum of all labels of paths of length n from 0 to q in the directed graph with vertices 0, 1, . . . , q and edges i → j, for i ≤ j, labelled bi,j . Such a path has a unique decomposition (abusing slightly the notation) 0 bn0,0 b0,i1 bni11,i1 bi1 ,i2 · · · bik−1 ,q bnq,qk ,

2952 2953 2954 2955 2956 2957 2958 2959 2960

(2.5)

for some vertices 0 < i1 < i2 < · · · < ik−1 < q, 0 ≤ k ≤ q, and some exponents n0 , n1 , . . . , nk with n0 + n1 + · · · + nk + k = n. Note that bhi,i = bi,i for h ≥ 1. Hence, for a fixed k, the sum of the labels of the paths (2.5) is matrix polynomial of degree ≤ k (see Exercise 2.1). Hence the sum of all labels is a polynomial of degree at most q. 0 1 Assume now that q = 1. Then the paths of (2.5) are of the form bn0,0 b0,1 bn1,1 n with n0 +1+n1 = n. Hence this block of b is equal to nb0,0 b0,1 b1,1 + a constant. Finally, it is easy to generalize this: the i, j-block of bn is a matrix polynomial of degree ≤ j−i, and if j = i+1, it is equal to nbi,i bi,i+1 bi+1,i+1 + some constant. (ii) We now compute the product m(n) = abn c. Set bn = (di,j ). Then the u, v-block of the product is X m(n) au,i di,j cj,v , u,v = u≤i≤j≤v

which is a sum of matrix polynomials of degree ≤ j − i ≤ v − u, and we are done. In the special case v = u + 1, the sum is au,u du,u cu,u+1 + au,u du,u+1 cu+1,u+1 + au,u+1 du+1,u+1 cu+1,u+1 . The two extreme terms are constants and the middle term is au,u (nbu,u bu,u+1 bu+1,u+1 + C)cu+1,u+1 = nau,u bu,u+1 cu+1,u+1 + C ′ 2961

for some constants C and C ′ , since ai,i bi,i = ai,i and bi,i ci,i = ci,i .



3. Limited languages and the tropical semiring 2962 2963 2964 2965 2966 2967 2968 2969 2970 2971

Proof of Theorem 2.5, second part. We may choose, among the linear minimal representations of S having the form (2.1) and coefficients in Z, a representation having, in lexicographic order from left to right, the largest possible vector (dim µ0 , dim µ1 , . . . , dim  µq ). This µi νi+1 shows, in view of Lemma 2.8, that for i = 1, . . . , q, all the morphisms 0 µi+1
0 , the set (νi A∗ )vi+1 is have the property that, for any nonnull vector vi+1 infinite. Hence, for any such vi+1 , there exist by Lemma 2.9, some words x′i , zi+1 , x′′i+1 such that µi x′i ·νi+1 zi+1 µi+1 x′′i+1 vi+1 6= 0, and µ(x′i zi+1 ) = µx′i , µ(zi+1 x′′i+1 ) = 2 µx′′i+1 , µ(zi+1 ) = µzi+1 , where µ = (µ0 , . . . , µq ). Let vq be some nonzero vector corresponding to the last block. Then we know from the preceding argument the existence of words x′q−1 , zq , x′′q such that vq−1 = µq−1 x′q−1 νq zq µq x′′q vq 6= 0. Suppose we have defined vi+1 , x′i , zi+1 , x′′i+1 such that vi = µi x′i νi+1 zi+1 µi+1 x′′i+1 vi+1 6= 0. We thus find x′i−1 , zi , x′′i with the above properties such that vi−1 = µi−1 x′i−1 νi zi µi x′′i vi 6= 0. Finally, we obtain the existence of words x′0 , . . . , x′q−1 , z1 , . . . , zq , x′′1 , . . . , x′′q such that µ0 x′0 ν1 z1 µ1 x′′1 µ1 x′1 ν2 z2 · · · µq−1 x′q−1 νq zq µq x′′q 6= 0 .

2972 2973 2974

147

(2.6)

n By Lemma 2.11, the matrix µi x′i νi+1 zi+1 µi+1 x′′i+1 is in P, and its i, i+1-block is ′ ′′ equal to nµi xi νi+1 zi+1 µi+1 xi+1 + some constant. This is still true if we replace n by nni , with ni ≥ 1. Choose some q-tuple (n1 , . . . , nq ) of positive integers and form the product

µx′0 µz1nn1 µx′′1 µx′1 µz2nn2 µx′′2 µx′2 · · · µx′q−1 µzqnnq µx′′q .

2989

Since P is closed under product, this matrix is in P. Consider its 0, q-block, which is the only one that can have degree q exactly. Viewing it as a matrix polynomial in n, n1 , . . . , nq , we see by Lemma 2.10(ii) that the coefficient of nq n1 n2 · · · nq is the left-hand side of (2.6). Thus, we may choose n1 , . . . , nq in such a way that this block has degree q exactly in n. Now, let yi = zini for i = 1, . . . , q and xi = x′′i x′i for i = 1, . . . , q − 1. Then µ(x′0 y1n x1 · · · yqn x′′q ) is a matrix polynomial of degree q exactly, and it follows that (S, x′0 y1n x1 · · · yqn x′′q ) is a polynomial in n of degree ≤ q. Moreover, for any words u, v, µ(ux′0 y1n x1 · · · yqn x′′q v) is a matrix polynomial of degree ≤ q and therefore (S, ux′0 y1n x1 · · · yqn x′′q v) is a polynomial of degree ≤ q. Now, µ(x′0 y1n x1 · · · yqn x′′q ) is, in view of Corollary II.2.3, a linear combination of (S, ux′0 y1n x1 · · · yqn x′′q v) for some words u, v. Hence one of these polynomials in n must have degree exactly q, and we put x0 = ux′0 , xq = x′′q v. This shows that S has degree of growth at least q, and to conclude the proof, we use Lemma 2.2(ii) and Lemma 2.3(ii). 

2990

3

2975 2976 2977 2978 2979 2980 2981 2982 2983 2984 2985 2986 2987 2988

Limited languages and the tropical semiring

Let L ⊂ A∗ be a language. Recall that L∗ denotes the submonoid generated by S L. Equivalently, L = n≥0 Ln . The language L is called limited if there exists m ≥ 0 such that L∗ = 1 ∪ L ∪ · · · ∪ Lm .

148 2991 2992 2993 2994 2995 2996 2997 2998 2999 3000 3001 3002 3003 3004 3005 3006

3007 3008 3009

Chapter IX. Matrix Semigroups and Applications

Suppose that L is a recognizable language, recognized by the automaton A = (Q, I, E, T ), where I, T (the initial and terminal states) are subsets of Q and E is a subset of Q × A × Q. Let q0 be a new state, set Q0 = q0 ∪ Q and let A∗ = (Q0 , q0 , E0 , q0 ) be the automaton defined by (i) (ii) (iii) (iv)

E0 contains E; a a for each edge p −→ q in A with q ∈ T , p −→ q0 is an edge in A∗ ; a a for each edge p −→ q in A with p ∈ I, q0 −→ q is an edge in A∗ ; a a for each edge p −→ q in A∗ in A with p ∈ I, q ∈ T , q0 −→ q0 is an edge in A∗ .

It is easily verified that A∗ recognizes the language L∗ . We show now how to encode the limitedness problem for L into a finiteness problem for a certain semigroup of matrices over the tropical semiring. First, we define the latter. It is the semiring, denoted T, whose underlying set is N ∪ ∞, with addition (a, b) 7→ min(a, b) and product (a, b) 7→ a + b with the evident meaning for a + ∞. Addition and multiplication in T are commutative and have respective neutral elements ∞ and 0. Coming back to the previous automaton, we associate to it a monoid morphism α from A∗ into the multiplicative monoid TQ0 ×Q0 of square matrices over T indexed by Q0 , defined as follows. For a letter a,  a ∗  ∞ if p −→ q is not an edge of A ; a ∗ (αa)p,q = 0 if p −→ q is an edge of A and q 6= q0 ;  a  1 if p −→ q is an edge of A∗ and q = q0 . With these notations and definitions, one has the following result.

Proposition 3.1 A rational language is limited if and only if the associated representation α has finite image. Proof 1. We define the weight ω of a path c in A∗ as the number of edges in c that end at q0 . In particular, the weight of any empty path is 0. We claim that for any word w in A∗ , and any p, q ∈ Q0 , w

(αw)p,q = min{ω(c) | c : p −→ q} , 3010 3011

(3.1)

that is, the minimum of the weights of the paths labeled w from p to q (we use here the convention that min(∅) = ∞). Indeed, if w is the empty word, then the right-hand side of (3.1) is ∞ if p 6= q, and is 0 if p = q, and this proves (3.1) in this case. If w = a ∈ A, then a a the right-hand side of (3.1) is ∞ if p −→ q is not an edge in A∗ , it is 0 if p −→ q is an edge and q 6= q0 , and is 1 if it is an edge and q = q0 ; this is exactly the definition of (αa)p,q . Now, let w = uv, where u, v are shorter that w, so by induction Equation (3.1) holds for u and v. Then, translating into N ∪ ∞ the operations in T, we have
(αw)p,q = min (αu)p,r + (αv)r,q . r∈Q0

By induction, this is equal to


u v min min{ω(d) | d : p −→ r} + min{ω(e) | e : r −→ q} .

r∈Q0

149

3. Limited languages and the tropical semiring

Since the minimum is distributive with respect to addition, and since the weight of a path de is the sum of the weights of the paths d and e, we obtain that u

v

(αw)p,q = min {ω(de) | d : p −→ r , e : r −→ q} , r∈Q0

3012

and this is equal to the right-hand side of (3.1), as was to be shown. 2. From Equation (3.1), it follows that (αw)q0 ,q0 is equal to the least m such that w ∈ Lm , and is ∞ if w ∈ / L∗ . Thus L is limited if and only if the set {(αw)q0 ,q0 | w ∈ A∗ }

3013 3014 3015 3016 3017 3018 3019 3020 3021 3022 3023

(3.2)

is finite. Now, let p, q ∈ Q0 and suppose that (αw)p,q = m 6= ∞. By (3.1), this means w that there is a path p −→ q in A∗ having m edges ending in q0 , and that no u w other path p −→ q has fewer such edges. Hence, we find a subpath q0 −→ q0 , for some factor u of w, having m − 1 such edges, and such that no other path u q0 −→ q0 has fewer such edges. This implies by (3.1) that (αu)q0 ,q0 = m − 1. We conclude that if the set (3.2) is finite, then so is the set {(αw)p,q | w ∈ A∗ }. Thus L is limited if and only if α(A∗ ) is finite.  We need to consider another semiring, denoted T0 , whose underlying set is {0, 1, ∞}, with the same operations ans T, that is: addition in T0 is the min(a, b) operation, and multiplication is the usual addition. Let ψ : T → T0 be the mapping which sends 0 to 0, ∞ to ∞ and any a ∈ T\{0, ∞} to 1. It is easily verified that ψ is a semiring morphism. Moreover, let ι be the injective mapping that sends 0, 1 an ∞ in T0 to themselves in T. Note that ι is not a semiring morphism. However ψι = idT0 .

3024

The mappings ψ and ι are naturally extended to matrices over T and T0 .

3025 3026

Theorem 3.2 (Simon 1978) The following conditions are equivalent for a finitely generated subsemigroup S of Tn×n :

3027 3028 3029 3030 3031 3032 3033 3034 3035 3036 3037 3038 3039

(i) S is finite; (ii) S is a torsion semigroup; (iii) for any idempotent e in ψS, one has (ιe)2 = (ιe)3 . Corollary 3.3 It is decidable whether a finite subset of Tn×n generates a finite subsemigroup, and whether a rational language is limited. Proof. Since ψ is a monoid morphism and since T0n×n is finite, condition (iii) of the theorem is decidable. For a rational language L, the limitedness problem is reduced by Proposition 3.1 to the finiteness of a certain finitely generated submonoid of Tn×n , hence to the preceding question.  We use the natural ordering ≤ on T that extends the natural ordering of N, together with the natural condition that t ≤ ∞ for all t ∈ T. This ordering is compatible with the semiring structure since if a ≤ b, then min(a, x) ≤ min(b, x)

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Chapter IX. Matrix Semigroups and Applications

and a + x ≤ b + x. We extend this ordering to matrices over T, by setting (aij ) ≤ (bij ) if and only if aij ≤ bij for all i, j. Then again, this ordering is compatible with sum and product of matrices over T. For any subset X of a semigroup S, we denote by X + the subsemigroup of S generated by X. Lemma 3.4 Let X be a finite subset of the multiplicative semigroup Tn×n and let Y = ιψX. Then X + is finite if and only Y + is finite. Note that y = ιψx is obtained from x by replacing each nonzero finite entry in x by 1, 0 and ∞ being unchanged. Hence, the entries equal to 0 or ∞ in x and y are the same. Proof. We may assume that some entry of some matrix in X is finite. Let M be the maximum of these finite entries. Let x1 , . . . , xp ∈ X, set yk = ιψxk . We show below that for i, j ∈ {1, . . . , n}, the following hold. (i) (x1 · · · xp )i,j = ∞ ⇐⇒ (y1 · · · yp )i,j = ∞; (ii) if the entries (x1 · · · xp )i,j and (y1 · · · yp )i,j are finite, then (y1 · · · yp )i,j ≤ (x1 · · · xp )i,j ≤ M (y1 · · · yp )i,j ,

3054 3055

3056 3057 3058 3059 3060 3061 3062 3063 3064 3065 3066 3067 3068 3069 3070 3071 3072

where the right-hand side product is taken in N. These two properties imply the lemma. For the proof of (i), observe that, by definition of T
(x1 · · · xp )i,j = min (x1 )i,k1 + (x2 )k1 ,k2 + · · · + (xp )kp−1 ,j ,

(3.3)

where the minimum is taken over all k1 , . . . , kp−1 in {1, . . . , n} and the sum is taken in N ∪ ∞. A similar formula holds for the yk ’s. Now, if (x1 · · · xp )i,j = ∞, then for each k1 , . . . , kp−1 , the sum in the righthand side of (3.3) must be ∞ and therefore at least one term (xj )kj−1 ,kj is equal to ∞; by the definition of ψ and ι, we obtain that (y1 · · · yp )i,j = ∞. The converse is similar, implying (i). For (ii), the first inequality follows from the properties of the order ≤ on Tn×n and the fact that ιψx ≤ x. For the second, knowing that (x1 · · · xp )i,j is finite, we may restrict the minimum in (3.3) to those k1 , . . . , kp−1 such that the sum in the right-hand side is finite. Then each term (xℓ )kj−1 ,kj is finite and therefore is less or equal to M (yℓ )kj−1 ,kj by the definition of ψ and ι. This implies the second equality in (ii).  Lemma 3.5 Let e be idempotent in the multiplicative monoid T0n×n and set f = ιe. For any i, j in {1, . . . , n}, one of the following statements holds. (i) (f m )i,j = fi,j for any m ≥ 1; (ii) fi,j = 1 and (f m )i,j = 2 for any m ≥ 2; (iii) (f m )i,j = m for any m ≥ 1.

Proof 1. Note that fi,j ∈ {0, 1, ∞}. We have e = ψιe = ψf , hence for m ≥ 1, ψ(f m ) = ψ(f )m = em = e, and therefore ei,j = 0 ⇐⇒ (f m )i,j = 0 ;

ei,j = 1 ⇐⇒ (f m )i,j = 1, 2, 3 . . . ; ei,j = ∞ ⇐⇒ (f m )i,j = ∞ .

3. Limited languages and the tropical semiring 3073 3074 3075 3076 3077 3078 3079 3080 3081 3082 3083 3084 3085 3086 3087

by definition of ψ. 2. Suppose that (f p )i,j = 0 for some p ≥ 1. Then by step 1 one has ei,j = 0 and therefore (f m )i,j = 0 for all m ≥ 1. 3. Suppose next that (f p )i,j = 1 for some p ≥ 2. Then eij = 1 by step 1, hence fi,j = 1 since f = ιe. Moreover, we have (f m )i,j 6= 0 for any m ≥ 1 by step 2. Since f p = f p−1 f , there exists an index k such that either (f p−1 )i,k = 0 and fk,j = 1 or (f p−1 )i,k = 1 and fk,j = 0. In the first case, (f m )i,k = 0 for any m ≥ 1 by step 2. Thus (f m )i,j ≤ m−1 (f )i,k + fk,j ≤ 1 for all m ≥ 2. In the second case, we have (f m )k,j = 0 for any m ≥ 1 by step 2, and by step 1 we get fi,k = 1. Hence (f m )i,j ≤ fi,k + (f m−1 )k,j ≤ 1 for all m ≥ 2. Thus in all cases (f m )i,j = 1 for any m ≥ 1. 4. We now show that if 2 ≤ (f p )i,j < p for some p ≥ 3, then (f m )i,j = 2 for any m ≥ 2 and moreover fi,j = 1. This latter equality follows from step 1 and the equality f = ιe, since we must have ei,j = 1, hence fi,j = 1. Let q = (f p )i,j . By the definition of the operations in T and Tn×n we have (with addition in N ∪ ∞) q = fk0 ,k1 + fk1 ,k2 + · · · + fkp−1 ,kp

3088 3089 3090 3091 3092 3093 3094 3095 3096 3097 3098 3099 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3112 3113 3114 3115 3116

151

(3.4)

for some i = k0 , k1 , . . . , kp−1 , kp = j. Since q < ∞, each term in (3.4) is 0 or 1. Let 0 < h < p. Then we deduce that (f h )k0 kh < ∞, hence fk0 ,kh < ∞ by step 1, and it follows that fk0 ,kh ≤ 1; similarly fkh ,kp ≤ 1. Moreover, q < p hence (3.4) implies that fkℓ ,kℓ+1 = 0 for some 0 ≤ ℓ < p. Then (f m )kℓ ,kℓ+1 = 0 for any m ≥ 1 by step 2. Suppose that ℓ = 0. Then (f p−1 )k0 ,k1 = 0 and fk1 ,kp ≤ 1 imply that (f p )i,j = (f p )k0 ,kp ≤ 1, a contradiction; likewise ℓ = p − 1 implies this contradiction. Hence 0 < ℓ < p − 1. We deduce that for any m ≥ 3, (f m )i,j = (f m )k0 ,kp ≤ fk0 ,kℓ +(f m−2 )kℓ ,kℓ+1 + fkℓ+1 ,kp ≤ 2. Also (f 2 )i,j = (f 2 )k0 ,kp ≤ fk0 ,k1 + fk1 ,kp ≤ 2. Now, we cannot have (f m )i,j ≤ 1 for some m ≥ 2 since this would imply, by steps 2 and 3, that (f p )i,j ≤ 1. Thus (f m )i,j = 2 for any m ≥ 2 and fi,j = 1. 5. Suppose now that neither (i) nor (ii) holds. This implies, by steps 2–4 that (f p )i,j ≥ p for all p ≥ 1. Indeed, if (f p )i,j < p for some p ≥ 1, then either (f p )i,j = 0 and (i) holds by step 1, or (f p )i,j ≥ 1, hence p ≥ 2; then either (f p )i,j = 1 and (i) holds by step 2, or (f p )i,j ≥ 2, hence p ≥ 3; then (ii) holds by step 4. Since the finite entries of f are equal to 0 or 1, the finite entries of f p are ≤ p. Hence they are equal to p. Now assume that (f p )i,j = ∞ for some p ≥ 1. Then, by step 1, ei,j = ∞. If (f m )i,j 6= ∞ for some m ≥ 1, then again by step 1, ei,j 6= ∞. Thus (f m )i,j = ∞ for all m ≥ 1, contradicting that (i) does not hold, and (iii) follows.  Proof of Theorem 3.2. The implication (i) =⇒ (ii) is clear. (ii) =⇒ (iii). We have e = ψs for some s ∈ S. Then ιe = ιψs. Since s is torsion, so is ιe by Lemma 3.4. Let i, j ∈ {1, . . . , n}. Then by Lemma 3.5, condition (iii) of this lemma cannot hold. Hence (i) or (ii) holds and consequently (ιe)2 = (ιe)3 . (iii) =⇒ (i). In view of Brown’s theorem (see the Appendix), it is enough to show that for any idempotent e in T0n×n , the semigroup ψ −1 (e) ∩ S is locally finite. So, consider a finite subset X of ψ −1 (e) ∩ S. We may suppose that e is

152 3117 3118 3119

3120 3121 3122 3123 3124

Chapter IX. Matrix Semigroups and Applications

in ψ(S). Then by hypothesis (ιe)2 = (ιe)3 . Let Y = ιψX. Since ψX = {e}, we have Y = {ιe} and consequently Y + is finite. Hence X + is finite by Lemma 3.4, and we can conclude that ψ −1 (e) ∩ S is locally finite. 

Appendix : Brown’s theorem A semigroup S is called locally finite if each finite subset of S generates a finite subsemigroup. Let ϕ : S → T be a morphism of semigroups such that (i) T is locally finite; (ii) for each idempotent e in T , the semigroup ϕ(e) is locally finite.

3125

Then S is locally finite. See Brown (1971).

3126

Exercises for Chapter IX

3127 3128 3129 3130 3131 3132 3133 3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146 3147 3148

3149 3150 3151 3152 3153 3154 3155 3156

1.1 Let S ∈ QhhAii be a rational series such that, for every ray R, almost all coefficients (S, w), w ∈ R, vanish. Show that S is a polynomial. 1.2 Let S ∈ NhhAii be an N-rational series having a polynomial growth. Show that S is in the N-subalgebra of NhhAii generated by the characteristic series of rational languages (use a rational expression for S and the fact that if T ∈ NhhAii is not the characteristic series of a code, then the growth of T ∗ is not polynomial). 1.3 Show that Corollary 2.6 holds when Z is replaced by N. 2.1 A composition of m of length k is a k-tuple of positive integers (m1 , . . . , mk ) such that m1
+ · · · + mk = m. Show that the number of such compositions is m−1 k−1 . Hint: associate to the composition the subset {m1 , m1 + m2 , . . . , m1 + · · · + mk−1 } of {1, . . . , m − 1}. 3.1 Show that T is indeed a semiring by verifying all the axioms given in Section I.1. 3.2 Show that L = a ∪ (a2 )∗ ∪ (a∗ b)∗ is limited and find the smallest m such that L∗ = 1 ∪ L ∪ · · · ∪ Lm . 3.3 Show that T0 is indeed a semiring and that ψ : T → T0 is a semiring morphism. 3.4 Show that ι is not a semiring morphism and that ψι = idT0 . 3.5 Show that the ordering of matrices over T is compatible with sum and product. P 3.6 Show that n≥0 nan ∈ Thhaii is equal to (1a)∗ .

Notes to Chapter IX Most of the results of Section 1 hold in arbitrary fields. Theorem 1.1 can be extended, but the bound N then also depends on the field considered. Corollaries 1.5, 1.6 hold in arbitrary fields, and Lemma 1.2 holds in fields of charac2 2 teristic 0, provided the bound (2n + 1)n is replaced by rn , where r is the size of the set {tr(m) | m ∈ M }. This set is always finite (under the assumptions of the lemma) for a finite monoid M . Corollaries 1.7, 1.8 extend to “computable” fields.

3. Limited languages and the tropical semiring 3157 3158 3159 3160 3161 3162

153

The results and proofs of Section 3 are all due to Simon (1978); he shows also that a rational language L is not limited if and only if there exists a word w in L∗ such that for any m ≥ 1, wm ∈ / 1 ∪ L ∪ · · · ∪ Lm . Krob has shown that it is undecidable whether two rational series over T are equal, see Krob (1994). It is also decidable whether a rational series over the tropical semiring has finite image, see Hashiguchi (1982), Leung (1988), Simon (1988, 1994).

154

Chapter IX. Matrix Semigroups and Applications

3163

3164

3165

Chapter X

Noncommutative Polynomials

3181 3182

This chapter deals with algebraic properties of noncommutative polynomials. They are of independent interest, but most of them will be of use in the next chapter. In contrast to commutative polynomials, the algebra of noncommutative polynomials is not Euclidean, and not even factorial. However, there are many interesting results concerning factorization of noncommutative polynomials: this is one of the major topics of the present chapter. The basic tool is Cohn’s weak algorithm (Theorem 1.1) which is the subject of Section 1. This operation constitutes a natural generalization of the classical Euclidean algorithm. Section 2 deals with continuant polynomials which describe the multiplicative relations between noncommutative polynomials (Theorem 2.2). We introduce in Section 3 cancellative modules over the ring of polynomials. We characterize these modules (Theorem 3.1) and obtain, as consequences, results on full matrices, factorization of polynomials, and inertia. The main result of Section 4 is the (easy) extension of Gauss’s lemma to noncommutative polynomials.

3183

1

3166 3167 3168 3169 3170 3171 3172 3173 3174 3175 3176 3177 3178 3179 3180

The weak algorithm

Let K be a commutative field and let A be an alphabet. Recall that the degree of a polynomial P in KhAi was defined in Section I.2: we will denote it by deg(P ). We recall the usual facts about the degree, that is deg(0) = −∞

deg(P + Q) ≤ max(deg(P ), deg(Q)) deg(P + Q) = deg(P ), if deg(Q) < deg(P )

(1.1)

deg(P Q) = deg(P ) + deg(Q) .

(1.2)

Note that the last equality shows that KhAi is an integral domain, that is P Q = 0 implies

P = 0 or Q = 0 . 155

156

Chapter X. Noncommutative Polynomials

Definition A finite family P1 , . . . , Pn of polynomials in KhAi is (right) dependent if either some Pi = 0 or if there exist polynomials Q1 , . . . , Qn such that X
Pi Qi < max(deg(Pi Qi )) . deg i

i

Definition A polynomial P is (right) dependent family!dependent – on the family P1 , . . . , Pn if either P = 0 or if there exist polynomials Q1 , . . . , Qn such that X
Pi Qi < deg(P ) deg P − i

and if furthermore for any i = 1, . . . , n deg(Pi Qi ) ≤ deg(P ) . 3184 3185

Note that if P is dependent on P1 , . . . , Pn then the family P, P1 , . . . , Pn is dependent. The converse is given by the following theorem. Theorem 1.1 (Cohn 1961) Let P1 , . . . , Pn be a dependent family of polynomials with deg(P1 ) ≤ · · · ≤ deg(Pn ) .

3186

Then some Pi is dependent on P1 , . . . , Pi−1 . Let P be a polynomial and let u be a word in A∗ . We define the polynomial P u as X P u−1 = (P, wu)w . −1

w∈A∗

The operator P 7→ P u−1 is symmetric to the operator P 7→ u−1 P which was introduced in Section I.5. It is easy to verify that this operator is linear, and that the following relations hold: deg(P u−1 ) ≤ deg(P ) − |u| −1

P (uv)

= (P v

−1

)u

−1

(1.3) (1.4)

Moreover, for any letter a, (P Q)a−1 = P (Qa−1 ) + (Q, 1)P a−1 3187 3188

(1.5)

where (Q, 1) denotes as usual the constant term of Q. The last equality is simply the symmetric equivalent of Lemma I.7.2. Lemma 1.2 If P, Q are polynomials and w is a word, then there exists a polynomial P ′ such that (P Q)w−1 = P (Qw−1 ) + P ′

3189

with either P = P ′ = 0 or deg(P ′ ) < deg(P ).

157

1. The weak algorithm 3190 3191

Proof. We may assume P 6= 0. If w is the empty word, then (P Q)w−1 = P Q and Qw−1 = Q, so that (P Q)w−1 = P (Qw−1 ) and the proof is complete. Let w = au with a a letter. Then by induction one has (P Q)u−1 = P (Qu−1 ) + P ′ deg(P ′ ) < deg(P ) Now, by Eq. (1.4), one has (P Q)w−1 = ((P Q)u−1 )a−1 = (P (Qu−1 ))a−1 + P ′ a−1 . Thus, by Eqs.(1.5) and (1.4), we have (P Q)w−1 = P ((Qu−1 )a−1 ) + (Qu−1 , 1)P a−1 + P ′ a−1 = P (Qw−1 ) + P ′′

3192 3193 3194 3195 3196 3197

with P ′′ = (Qu−1 , 1)P a−1 + P ′ a−1 . Next, by Eq. (1.3), deg(P a−1 ) < deg(P ) and deg(P ′ a−1 ) ≤ deg(P ′ )−|a| < deg(P ). Hence deg(P ′′ ) < deg(P ), as desired.  ProofP of Theorem 1.1. We may suppose that no Pi is equal to 0. Hence deg( Pi Qi ) < maxi (deg(Pi Qi )). Let r P = maxi (deg(Pi Qi )) and let I = {i | Pi Qi has degree deg(R) < r. Let deg(Pi Qi ) = r}. The polynomial R = i∈I

3198 3199 3200

k = sup(I); then i ∈ I =⇒ deg(Pi ) ≤ deg(Pk ). Let w be a word such that |w| = deg(Qk ) and 0 6= (Qk , w) = α−1 ∈ K: such a word exists because Qk 6= 0 (otherwise deg(R) < r = deg(Pk Qk ) = −∞). By Lemma 1.2, we have X X Rw−1 = Pi (Qi w−1 ) + Pi′ i∈I

i∈I

Pi′

for some polynomials with deg(Pi′ ) < deg(Pi ). Since Qk w−1 = α−1 , X X Pi (Qi w−1 ) = αRw−1 − α Pi′ . Pk + α i∈I\k

(1.6)

i∈I

Now, by Eq. (1.3) deg(Rw−1 ) ≤ deg(R) − |w| < r − |w|

= deg(Pk Qk ) − deg(Qk ) = deg(Pk ) .

Furthermore, deg(Pi′ ) < deg(Pi ) ≤ deg(Pk ). Consequently, by Eq. (1.1), the degree of the right-hand side of Eq. (1.6) is < deg(Pk ). Moreover, deg(Pi (Qi w−1 )) = deg(Pi ) + deg(Qi w−1 ) ≤ deg(Pi ) + deg(Qi ) − deg(Qk )

3201 3202 3203 3204 3205 3206 3207

by Eq. (1.3). So we have deg(Pi (Qi w−1 )) ≤ r − deg(Qk ) = deg(Pk ). This shows that Pk is dependent on Pi , i ∈ I \ k; hence Pk also is dependent on P1 , . . . , Pk−1 .  For two polynomials X, Y in KhAi, the (left) Euclidean division of X and Y (that is the problem of finding polynomials Q and R such that X = Y Q + R and deg(R) < deg(Y )) is not always possible. However, the next result gives a necessary and sufficient condition for this.

158

Chapter X. Noncommutative Polynomials

Corollary 1.3 Let X, Y, P, Q1 , Q2 , R1 be polynomials such that XP + Q1 = Y Q2 + R1 with P 6= 0, deg(Q1 ) ≤ deg(P ), deg(R1 ) < deg(Y ) . Then there exists polynomials Q and R such that X = Y Q + R with deg(R) < deg(Y ) 3208

(that is, Euclidean division of X by Y is possible). Proof. Note that Y 6= 0 (otherwise deg(R1 ) < −∞). If Y ∈ K, the corollary is immediate (take Q = Y −1 X and R = 0). Otherwise, we prove it by induction on deg(X). If deg(X) < deg(Y ), the proof is immediate (take Q = 0 and R = X). Suppose that deg(X) ≥ deg(Y ). Then deg(Q1 ) ≤ deg(P ) < deg(XP ) because 1 ≤ deg(Y ) ≤ deg(X) and deg(R1 ) < deg(Y ) ≤ deg(X) ≤ deg(XP )

3209 3210 3211 3212

because 0 ≤ deg(P ). Thus, deg(Q1 ) and deg(R1 ) are both < max(deg(XP ), deg(Y Q2 )) and by Eq. (1.1), deg(R1 − Q1 ) < max(deg(XP ), deg(Y Q2 )). In view of Theorem 1.1, X is dependent on Y , that is there exist two polynomials Q3 and X1 such that X = Y Q3 + X1 with deg(X1 ) < deg(X). Put this expression for X into the initial equality. This gives X1 P + Q1 = Y (Q2 − Q3 P ) + R1 .

3213 3214

Since deg(X1 ) < deg(X), we have by induction X1 = Y Q4 + R with deg(R) < deg(Y ). Thus X = Y Q3 + Y Q4 + R, which proves the corollary.  The next result is a particular case of the previous one.

3215

3217 3218

Corollary 1.4 If X, Y, X ′ , Y ′ are nonzero polynomials such that XY ′ = Y X ′ , then there exist polynomials Q, R such that X = Y Q + R and deg(R) < deg(Y ). 

3219

2

3216

Continuant polynomials

Definition Let a1 , . . . , an be a finite sequence of polynomials. We define the sequences p0 , . . . , pn of continuant polynomials (with respect to a1 , . . . , an ) in the following way: p0 = 1, p1 = a1 , and for 2 ≤ i ≤ n, pi = pi−1 ai + pi−2 .

2. Continuant polynomials

159

Example 2.1 The first continuant polynomials are p 2 = a1 a2 + 1 p 3 = a1 a2 a3 + a1 + a3 p 4 = a1 a2 a3 a4 + a1 a2 + a1 a4 + a3 a4 + 1 3220

Notation We shall write p(a1 , . . . , ai ) for pi .

3221

It is easy to see that the continuant polynomials may be obtained by the “leap-frog construction”: consider the “word” a1 · · · an and all words obtained by repetitively suppressing some factors of the form ai ai+1 in it. Then p(a1 , . . . , an ) is the sum of all these “words”. Now, we have by definition

3222 3223 3224

p(a1 , . . . , an ) = p(a1 , . . . , an−1 )an + p(a1 , . . . , an−2 ) .

(2.1)

The combinatorial construction sketched above shows that symmetrically p(a1 , . . . , an ) = a1 p(a2 , . . . , an ) + p(a3 , . . . , an ) .

(2.2)

An equivalent but useful relation is p(an , . . . , a1 ) = an p(an−1 , . . . , a1 ) + p(an−2 , . . . , a1 ) .

(2.3)

Proposition 2.1 (Wedderburn 1932) The continuant polynomials satisfy the relation p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 )p(an , . . . , a1 ) .

(2.4)

Proof. This is surely true for n = 1. Suppose n ≥ 2. Then by Eq. (2.1), p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 ) an p(an−1 , . . . , a1 ) + p(a1 , . . . , an−2 )p(an−1 , . . . , a1 ) which is equal by induction to p(a1 , . . . , an−1 ) an p(an−1 , . . . , a1 ) + p(a1 , . . . , an−1 )p(an−2 , . . . , a1 ) . This is equal, by Eq. (2.3), to p(a1 , . . . , an−1 )p(an , . . . , a1 ) 3225

as desired.



Theorem 2.2 (Cohn 1969) Let X, Y, X ′ , Y ′ be nonzero polynomials such that XY ′ = Y X ′ . Then there exists polynomials U, V, a1 , . . . , an with n ≥ 1 such that X = U p(a1 , . . . , an ), Y ′ = p(an−1 , . . . , a1 )V Y = U p(a1 , . . . , an−1 ), X ′ = p(an , . . . , a1 )V . 3226 3227

Moreover, one has deg(a1 ), . . . , deg(an−1 ) ≥ 1, and if deg(X) > deg(Y ), then deg(an ) ≥ 1.

160

Chapter X. Noncommutative Polynomials

Proof. (i) Suppose first that X is a right multiple of Y , that is X = Y Q. Then the theorem is obvious for U = Y , V = Y ′ , n = 1, a1 = Q; then indeed X = Y Q = U p(a1 ), Y ′ = 1 · V, Y = U · 1 3228 3229

and Y X ′ = XY ′ = Y QY ′ , whence X ′ = QY ′ = p(a1 )V . Furthermore, if deg(X) > deg(Y ), then deg(Q) ≥ 1. (ii) Next, we prove the theorem in the case where deg(X) > deg(Y ), by induction on deg(Y ). If deg(Y ) = 0, then X is a right multiple of Y and we may apply (i). Suppose deg(Y ) ≥ 1. By Corollary 1.4, X = Y Q + R for some polynomials Q and R such that deg(R) < deg(Y ). If R = 0, apply (i). Otherwise, we have Y X ′ = XY ′ = Y QY ′ + RY ′ , hence Y (X ′ − QY ′ ) = RY ′ ; note that Y, R, Y ′ 6= 0, hence X ′ − QY ′ 6= 0. Furthermore, deg(R) < deg(Y ), and we may apply the induction hypothesis: there exist polynomials U, V, a1 , . . . , an such that Y = U p(a1 , . . . , an ), X ′ − QY ′ = p(an−1 , . . . , a1 )V

R = U p(a1 , . . . , an−1 ), Y ′ = p(an , . . . , a1 )V deg(a1 ), . . . , deg(an ) ≥ 1 .

(2.5)

Hence
X = Y Q + R = U p(a1 , . . . , an )Q + p(a1 , . . . , an−1 ) = U p(a1 , . . . , an , Q) 3230 3231 3232 3233

by Eq. (2.1). Similarly, X ′ = p(Q, an , . . . , a1 )V . Thus X, Y, X ′ , Y ′ admit the announced expression. Furthermore, deg(Q) ≥ 1; indeed, by Eq. (1.2), deg(X) = deg(Y Q) = deg(Y ) + deg(Q), and hence deg(Q) = deg(X) − deg(Y ) ≥ 1. This prove the theorem in the case where deg(X) > deg(Y ). (iii) In the general case, one has again X = Y Q + R with deg(R) < deg(Y ) (Corollary 1.4). If R = 0, the proof is completed by (i). Otherwise, as above, Y (X ′ − QY ′ ) = RY ′ with deg(Y ) > deg(R). Hence we may apply (ii): there exist U, V, a1 , . . . , an such that Eq. (2.5) holds. Then we obtain, as in (ii): X = U p(a1 , . . . , an , Q), Y ′ = p(an , . . . , a1 )V Y = U p(a1 , . . . , an ), X ′ = p(Q, an , . . . , a1 )V .

3234

This proves the theorem.



Proposition 2.3 Let a1 , . . . , an be polynomials such that a1 , . . . , an−1 have positive degree, and let Y be a polynomial of degree 1 such that p(an−1 , . . . , a1 ) and p(an . . . , a1 ) are both congruent to a scalar modulo the right ideal Y KhAi. Then for i = 1, . . . , n p(ai , . . . , a1 ) ≡ p(a1 , . . . , ai )

mod Y KhAi .

3235

We prove first a lemma.

3236

Lemma 2.4 Let a1 , . . . , an be polynomials such that a1 , . . . , an−1 have positive degree. Then the degrees of 1, p(a1 ), . . . , p(an−1 , . . . , a1 ) are strictly increasing.

3237

161

2. Continuant polynomials Proof. Obviously deg(1) < deg(a1 ). Suppose deg(p(ai−2 , . . . , a1 )) < deg(p(ai−1 , . . . , a1 )) for 2 ≤ i ≤ n − 1. From the relation p(ai , . . . , a1 ) = ai p(ai−1 , . . . , a1 ) + p(ai−2 , . . . , a1 ) ,

it follows that the degree of p(ai , . . . , a1 ) is equal to deg(ai p(ai−1 , . . . , a1 )), and deg(ai p(ai−1 . . . , a1 )) = deg(ai ) + deg(p(ai−1 , . . . , a1 )) > deg(p(ai−1 , . . . , a1 )) 3238

because deg(ai ) ≥ 1. This proves the lemma.

3239

Proof of Proposition 2.3 (Induction on n). When n = 1, the result is evident. Suppose n ≥ 2. Note that if the condition on the degrees is fulfilled for a1 , . . . , an , then a fortiori also a1 , . . . , an−2 have positive degree. By assumption, p(an , . . . , a1 ) is congruent to some scalar α and p(an−1 , . . . , a1 ) is congruent to some scalar β mod. Y KhAi. Suppose p(an−1 , . . . , a1 ) = 0. Then by Eq. (2.3), we have p(an−2 , . . . , a1 ) ≡ α = α − βγ for any γ, because β = 0 in this case. Suppose p(an−1 , . . . , a1 ) 6= 0. Then by Eq. (2.3),

3240 3241 3242 3243 3244 3245



an p(an−1 , . . . , a1 ) + p(an−2 , . . . , a1 ) = Y Q + α 3246 3247 3248 3249 3250 3251 3252 3253

for some polynomial Q. As deg(p(an−2 , . . . , a1 )) < deg(p(an−1 , . . . , a1 )) by Lemma 2.4, we obtain by Corollary 1.3 that an ≡ γ mod Y KhAi for some scalar γ. Using Eq. (2.3) again, and the fact that P ≡ γ, Q ≡ β =⇒ P Q ≡ γβ, we obtain p(an−2 , . . . , a1 ) ≡ α − γβ. In both cases, the induction hypothesis gives p(a1 , . . . , an−2 ) ≡ α − γβ and p(a1 , . . . , an−1 ) ≡ β. Hence, by Eq. (2.1), p(a1 , . . . , an ) ∈ (β + Y KhAi)(γ + Y KhAi) + α − βγ + Y KhAi, and consequently p(a1 , . . . , an ) ≡ βγ + α − γβ ≡ p(an . . . , a1 ), as desired.  Lemma 2.5 Let a1 , . . . , an be polynomials. Then p(a1 , . . . , an ) = 0 ⇐⇒ p(an , . . . , a1 ) = 0 . Proof (Induction on n). The lemma is evidently true for n = 0, 1. Suppose n ≥ 2. It is enough to show that p(a1 , . . . , an ) = 0 implies p(an , . . . , a1 ) = 0. Now, by Eq. (2.4), p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 )p(an , . . . , a1 ) .

3254 3255 3256 3257 3258

Suppose p(a1 , . . . , an ) = 0. If p(a1 , . . . , an−1 ) 6= 0, then p(an , . . . , a1 ) = 0 because KhAi is an integral domain. If p(a1 , . . . , an−1 ) = 0, then p(an−1 , . . . , a1 ) = 0 by induction. Hence, by Eqs. (2.1) and (2.3) p(a1 , . . . , an ) = p(a1 , . . . , an−2 ) and p(an , . . . , a1 ) = p(an−2 , . . . , a1 ). By induction, p(a1 , . . . , an−2 ) and p(an−2 , . . . , a1 ) simultaneously vanish, which proves the lemma. 

162 3259

3

Chapter X. Noncommutative Polynomials

Inertia

Recall that KhAip×q denotes the set of p by q matrices over KhAi. In particular, KhAin×1 is the set of column vectors of order n over KhAi. This set has a natural structure of right KhAi-module. If V is in KhAin×1 , we denote by (V, 1) its constant term, that is, setting   P1   V =  ...  Pn

one has

 (P1 , 1)   (V, 1) =  ...  ∈ KhAin×1 . 

(Pn , 1)

Furthermore, if w is a word in A∗ , we denote by V w−1 the vector   P1 w−1   V w−1 =  ...  . Pn w−1

We have the following relation X V = (V, 1) + (V a−1 )a .

(3.1)

a∈A

3260 3261

Definition A (right) submodule E of KhAin×1 is cancellative if, whenever V ∈ E and (V, 1) = 0, then V a−1 ∈ E for any letter a ∈ A.

3262

This property of vectors of polynomials is closely related to (but weaker than) the property of stability introduced in Section I.5. The next result characterizes cancellative submodules and will be the key to all the results of this section.

3263 3264 3265 3266 3267 3268 3269

Theorem 3.1 A submodule E of KhAin×1 is cancellative if and only if it may be generated, as a right KhAi-module, by p vectors V1 , . . . , Vp such that the matrix ((V1 , 1), . . . , (Vp , 1)) ∈ K n×p is of rank p. In this case, p ≤ n and V1 , . . . , Vp are linearly KhAi-independent. Proof. 1. We begin with the easy part: suppose that E is generated by V1 , . . . , Vp as indicated. Let V ∈ E with (V, 1) = 0. Then X V = Vi Pi (Pi ∈ KhAi) . 1≤i≤p

Taking constant terms, we obtain X 0 = (V, 1) = (Vi , 1)(Pi , 1) .

163

3. Inertia

Because of the rank condition, we have (Pi , 1) = 0 for any i. Hence Pi = P (Pi a−1 )a, which shows that a∈A

V =

X

Vi (Pi a−1 )a .

i, a

By Eq. (3.1) we obtain X Vi (Pi a−1 ) . V a−1 = i

3270 3271 3272 3273 3274 3275 3276 3277 3278 3279 3280 3281

hence V a−1 ∈ E, as desired. 2. Let E be If V ∈ KhAin×1 , V may be P a cancellative submodule of KhAi.n×1 are almost all zero. Let written V = w∈A∗ (V, w)w where (V, w) ∈ KhAi deg(V ) be the maximal length of a word w such that (V, w) 6= 0. Claim. There are vectors V1 , . . . , Vp in E such that

(i) deg(V1 ) ≤ deg(V2 ) ≤ · · · ≤ deg(Vp ) . (ii) The vectors (Vi , 1) form a K-basis of the K-space (E, 1) = {(V, 1) | V ∈ E}. (iii) If V ∈ E and deg(V ) < deg(Vi ) then (V, 1) is a K-linear combination of (V1 , 1), . . . , (Vi−1 , 1). Suppose the claim is true. Then the matrix ((V1 , 1), . . . , (VP p , 1)) has rank p. We show by induction on deg(V ) that each V ∈ E is in E ′ = 1≤i≤p Vi KhAi. If deg(V ) = −∞, that is V = 0, it is obvious. Let deg(V ) ≥ 0 and let i be the smallest integer such that deg(V ) < deg(Vi ) (with i = p + 1 if such an integer does not exist). Then deg(V ) ≥ deg(V1 ), . . . , deg(Vi−1 ). Moreover, if i ≤ p then by (iii) (V, 1) is a linear combination of (V1 , 1), . . . , (Vi−1 , 1), and if i = p + 1 then byP(ii), (V, 1) is also a linear combination of (V1 , 1), . . . , (Vi−1 , 1). Let V ′ = V − 1≤j≤i−1 αj Vj (αj ∈ K) be such that (V ′ , 1) = 0. By the cancellative property of E, V ′ a−1 is in E for any letter a. Now, deg(V ′ ) ≤ max(deg(V ), deg(α1 V1 ), . . . , deg(αi−1 Vi−1 )) = deg(V )

3282 3283

hence deg(V ′ a−1 )P< deg(V ). Hence by induction, V ′ a−1 ∈ E ′ . PNow, by ′ −1 )a, and V ′ is in E ′ . Thus V = V ′ + αj Vj is Eq. (3.1), V ′ = a (V a j

3284

in E ′ as well. 3. Proof of the claim. For d = −1, 0, 1, 2, . . ., let F (d) be the subspace of K n×1 defined by F (d) = {(V, 1) | V ∈ E, deg(V ) ≤ d} . Then 0 = F (−1) ⊂ F (0) ⊂ F (1) ⊂ · · · ⊂ F (d) ⊂ · · · Let 0 ≤ d1 < · · · < dq be such that for any i, F (di − 1) ( F (di ) and such that each F (d) is equal to some F (di ); in other words, one has 0 = F (−1) = · · · = F (d1 − 1) ( F (d1 ) = · · · = F (d2 − 1) ( F (d2 ) ( · · · ( F (dq ) = F (dq + 1) = · · ·

164 3285 3286 3287 3288 3289 3290

Chapter X. Noncommutative Polynomials

In particular, F (dq ) = (E, 1). Now, let B1 be a basis of F (d1 ), B2 be a basis of F (d2 ) mod F (d1 ), . . . , and let Bq be a basis of F (dq ) mod F (dq−1 ). By the definition of the F ’s we may find for each i in {1, . . . , q} vectors Wi,1 , . . . , Wi,ki in E of degree ≤ di such that {(Wi,1 , 1), . . . , (Wi,ki , 1)} = Bi ; in fact, the degree of each Wi,j is exactly di , otherwise (Wi,j , 1) ∈ F (di − 1) = F (di−1 ), which contradicts the fact that Bi is a basis mod F (di−1 ). Define V1 , . . . , Vp by (V1 , . . . , Vp ) = (W1,1 , . . . , W1,k1 , W2,1 , . . . , W2,k2 , . . . , Wq,kq ) .

3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301

Then the condition (i) of the claim is clearly satisfied. Moreover, as F (dq ) = (E, 1), condition (ii) is also satisfied. Let V ∈ E with deg(V ) < deg(Vk ). Then Vk = Wi,j for some i, j, hence deg(V ) < di = deg(Wi,j ), which implies that (V, 1) ∈ F (di − 1) = F (di−1 ) and (V, 1) is a linear combination of W1,1 , . . . , Wi−1,ki−1 , hence of V1 , . . . , Vk−1 . This proves the claim. P 4. We show the last assertion of the theorem. Clearly, p ≤ n. Suppose Vi Pi = 0 where Pi ∈ KhAiPare not all zero; choose such a relation with sup(deg(Pi )) minimum. Then (Vi , 1)(Pi , 1) = 0 which shows as in (1) that (Pi , 1) = 0 for each i. Now Pj is 6= 0, hence Pj a−1 6= 0 for some letter a. P some −1 By Eq. (3.1) we obtain Vi (Pi a ) = 0, which is a new relation contradicting the above minimality. Thus the V ’s are KhAi-independent. 

3302 3303

Definition An n by n matrix M over KhAi is full if, whenever M = M1 M2 for some matrices M1 ∈ KhAin×p and M2 ∈ KhAip×n , then p ≥ n.

3304

Remark Taking in the above definition a field instead of KhAi, one obtains exactly the definition of an invertible matrix over this field.

3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320

Corollary 3.2 (Cohn 1961) Let M be an n by n matrix over KhAi. If S1 , . . . , Sn in KhhAii are formal series, not all zero, such that (S1 , . . . , Sn )M = (0, . . . , 0), then M is not full. Proof. Let E be the set of vectors V ∈ KhAin×1 such that (S1 , . . . , Sn )V = 0. Then E is a right submodule of KhAin×1 . Let V = t (P P1 , . . . , Pn ) ∈ E be such that (V, 1) = 0. Then (Pi , 1) = 0Pfor any i. Moreover i Si Pi = 0, so that if a is a letter, by Eq. (3.1), one has i Si (Pi a−1 ) = 0. This means that V a−1 ∈ E; thus E is cancellative. By Theorem 3.1, the right KhAi-module E admits a basis consisting of p vectors V1 , . . . , Vp such that rank((V1 , 1), . . . , (Vp , 1)) = p and p ≤ n. Now suppose that p = n. Then the matrix N = ((V1 , 1), . . . , (Vn , 1)) ∈ K n×n is invertible. But N is the constant matrix of H = (V1 , . . . , Vn ) ∈ KhAin×n , that is N = (H, 1); this implies that H is invertible in KhhAiin×n . Now we have (S1 , . . . , Sn )H = 0 (because (S1 , . . . , Sn )Vi = 0 for all i), hence (S1 , . . . , Sn ) = 0 (multiply by H −1 ), a contradiction. So p < n. Let M = (C1 , . . . , Cn ), where Ck is the k-th column of M . Then, p P Vj Pj,k for some polynomials by hypothesis, Ck belongs to E, hence Ck = j=1

Pj,k . Thus

M = (V1 , . . . , Vp )(Pj,k )1≤j≤p, 1≤k≤n 3321

and M is not full.



165

3. Inertia 3322 3323 3324 3325

Corollary 3.3 (Cohn 1982) Let P1 , P2 , P3 , P4 be polynomials such that P2 is invertible as a formal series, that is (P2 , 1) 6= 0, and such that P1 P2−1 P3 = P4 holds in KhhAii. Then there exist polynomials Q1 , Q2 , Q3 , Q4 such that P1 = Q1 Q2 , P2 = Q3 Q2 , P3 = Q3 Q4 , P4 = Q1 Q4 . Proof. Consider the 2 by 2 matrix over KhAi:   P1 P4 M= P2 P3 By assumption, we have (1, −P1 P2−1 )M = 0 . Hence M is not full by Corollary 3.2, and M may be written as   Q1 M= (Q2 , Q4 ) Q3

3326

for some polynomials Qi . This proves the corollary.



The next result is the Inertia Theorem. It will not be used in Chapter XI. Let S1 , . . . , Sn , T1 , . . . , Tn be formal series. We say that X Sj T j j

is trivially a polynomial if, for each j, either Sj = 0, or Tj = 0, or both Sj and Tj are polynomials. Note that one has X j

 T1   Sj Tj = (S1 , . . . , Sn )  ...  . 

Tn

Corollary 3.4 (Inertia Theorem, Bergmann 1967, Cohn 1961) Let (Si,h )i∈I, 1≤h≤n and (Th,j )1≤h≤n, P j∈J be two families of formal series such that for each i ∈ I and j ∈ J, h Si,h Th,j is a polynomial. Then there exists an invertible matrix M over KhhAii such that for any i and j    T1,j     (Si,1 , . . . , Si,n )M M −1  ...  Tn,j

3327

is trivially a polynomial.

Proof. 1. We prove the theorem first in the case where each Th,j is a polynomial. Let E = {V ∈ KhAin×1 | ∀i ∈ I, (Si,1 , . . . , Si,n )V ∈ KhAi}. Then E is a cancellative right submodule of KhAin×1 as may be easily verified (cf. the proof of Corollary 3.2). By Theorem 3.1 there exist p vectors V1 , . . . , Vp in E which form a basis of E (as a right KhAi-module) and such that the constant matrix

166

Chapter X. Noncommutative Polynomials

of (V1 , . . . , Vp ) is of rank p ≤ n. By performing a permutation of coordinates, we may assume that   X (V1 , . . . , Vp ) = , Y where (X, 1) ∈ K p×p is invertible. Let   X 0 , M= Y In−p 3328 3329

where In−p is the identity matrix of order n − p. Then (M, 1) ∈ K n×n is invertible, hence M is invertible in KhhAiin×n . Note that the first p columns of M (that is the Vi ’s) are in E: this implies, by definition of E, that for any i ∈ I the first p components of (Si,1 , . . . , Si,n )M are polynomials. Moreover, let 1 ≤ h ≤ p: then M −1 Vh is equal to the hth column of M −1 M , that is to the hth canonical vector Eh ∈ K n×1 . PNow let j ∈ J. Then by assumption V = t (T1,j , . . . , Tn,j ) is in E. Hence V = 1≤h≤p Vh Ph for P some polynomials Ph . Thus M −1 V = h M −1 Vh Ph is equal, by the previous P remark, to h Eh Ph = t (P1 , . . . , Pp , 0, . . . , 0). This shows that the product    T1,j    −1  .  (Si,1 , . . . , Si,n )M M  ..  Tn,j

3330

is trivially a polynomial. 2. We come to the general case. Let

H = {h ∈ {1, . . . , n} | ∀j ∈ J, Th,j ∈ KhAi} . If H = {1, . . . , n}, then we are in case 1. Suppose |H| < n: we may suppose that H = {1, . . . , p} with 0 ≤ p < n (including the case H = ∅). Suppose that ∀i ∈ I, ∀h ∈ / H, Si,h = 0. Then n X

Si,h Th,j =

h=1

p X

Si,h Th,j

h=1

is a polynomial, so we are also in case 1 (with p instead of n). Otherwise, there / H such is some i0 ∈ I such that for some h0 ∈ / H, Si0 ,h0 6= 0. Choose h0 ∈ / H (for the definition of ω, see Section I.3). that ω(Si0 ,h0 ) ≤ ω(Si0 ,h ) for any h ∈ Choose polynomials R1 , . . . , Rp such that for 1 ≤ h ≤ p, ω(Si0 ,h + Rh ) ≥ ω(Si0 ,h0 ). Define Sh′ by Sh′ = Si0 ,h + Rh if 1 ≤ h ≤ p and Sh′ = Si0 ,h if p < h ≤ n. Then ω(Sh′ 0 ) ≤ ω(Sh′ ), Sh′ 0 = Si0 ,h0 6= 0 and X

Sh′ Th,j =

1≤h≤n

X

(Si0 ,h + Rh )Th,j +

h≤p

=

X

1≤h≤n

3331 3332

Si0 ,h Th,j +

X

X

Si0 ,h Th,j

h>p

Rh Th,j

h≤p

is a polynomial, by definition of H = {1, . . . , p}. Let w be a word of minimal length in the support of Sh′ 0 ; then w−1 Sh′ 0 is an invertible formal series,

4. Gauss’s lemma

167

3341 3342

and for any h, since ω(Sh′ ) ≥ |w|, one has w−1 (Sh′ Th,j ) = (w−1 Sh′ )Th,j . Hence P −1 ′ Sh )Th,j is a polynomial. Define the matrix N ∈ KhhAiin×n which coinh (w cides with the n × n identity matrix except in the h0 th row, where it is equal to (w−1 S1′ , . . . , w−1 Sn′ ); in particular the entry of the coordinate (h0 , h0 ) of N is the invertible series w−1 Sh′ 0 , so N is invertible in KhhAiin×n . Let M = N −1 . Then for any j, M −1 t (T1,j , . . . , Tn,j ) = N t (T1,j , . . . , Tn,j ) is to t (T1,j , . . . , Tn,j ) Pequal −1 ′ except in the h0 th component, where it is equal to (w Sh )Th,j : hence the first p and the h0 th components of M −1 t (T1,j , . . . , Tn,j ) are polynomials and we may conclude the proof by induction on n − p because we have increased |H|. 

3343

4

3344 3345

We consider in this section polynomials with integer or rational coefficients. Everything would work, however, with any factorial ring instead of Z.

3346

Definition A polynomial P ∈ QhAi is primitive if P 6= 0, P ∈ ZhAi and if its coefficients have no nontrivial common divisors in Z.

3333 3334 3335 3336 3337 3338 3339 3340

3347

Gauss’s lemma

3349

Definition The content of a nonzero polynomial P ∈ QhAi is the unique positive rational number c(P ) such that P/c(P ) is primitive.

3350

Notation P/c(P ) will be denoted by P .

3351 3352

Example 4.1 c(4/3+6a−2ab) = 2/3 because 3/2(4/3+6a−2ab) = 2+9a−3ab is primitive.

3348

Note that for P 6= 0 P primitive ⇐⇒ c(P ) = 1 P ∈ ZhAi ⇐⇒ c(P ) ∈ N . 3353 3354 3355 3356 3357 3358 3359 3360 3361 3362 3363 3364

3365 3366

(4.1) (4.2)

Theorem 4.1 (Gauss’s Lemma) (i) If P, Q are primitive, then so is P Q. (ii) If P, Q are nonzero polynomials, then c(P Q) = c(P )c(Q) and P Q = P Q. Proof (i) Suppose P Q is not primitive. Then there is some prime number n which divides each coefficient of P Q. This means that the canonical image φ(P Q) of P Q in (Z/nZ)hAi vanishes. But Z/nZ is a field, so (Z/nZ)hAi is an integral domain (Section I.1); moreover 0 = φ(P Q) = φ(P )φ(Q), so φ(P ) = 0 or φ(Q) = 0. This means that n divides all coefficients of P or of Q, and contradicts the fact that P and Q are primitive. (ii) By (i), P Q/c(P )c(Q) = (P/c(P ))(Q/c(Q)) is primitive. So, by definition of the content of P Q, c(P Q) = c(P )c(Q). Now, P Q = P Q/c(P Q) so that P Q = P Q/c(P )c(Q) = P Q.  Corollary 4.2 Let a1 , . . . , an be polynomials. Then the continuant polynomials p(a1 , . . . , an ) and p(an , . . . , a1 ) are both zero or have the same content.

168

Chapter X. Noncommutative Polynomials

Proof (Induction on n). The result is obvious for n = 0, 1. Let n ≥ 2. By Lemma 2.5, we may suppose that both polynomials are 6= 0. Now we have, by Proposition 2.1 p(a1 , . . . , an )p(an−1 , . . . , a1 ) = p(a1 , . . . , an−1 )p(an , . . . , a1 ) . 3367 3368 3369 3370 3371

By induction, either p(a1 , . . . , an−1 ) = p(an−1 , . . . , a1 ) = 0, in which case p(a1 , . . . , an ) = p(a1 , . . . , an−2 ) by Eq. (2.1) and p(an , . . . , a1 ) = p(an−2 , . . . , a1 ) and we conclude by induction; or c(p(an−1 , . . . , a1 )) = c(p(a1 , . . . , an−1 )), which implies by Eq. (2.4) and Theorem 4.1 that c(p(a1 , . . . , an )) = c(p(an , . . . , a1 )).  Corollary 4.3 Let P1 , P2 , P3 , P4 be nonzero polynomials in ZhAi such that P2 is invertible in QhhAii and such that P1 P2−1 P3 = P4 . Then there exist polynomials R1 , R2 , R3 , R4 ∈ ZhAi such that P1 = R1 R2 , P2 = R3 R2 , P3 = R3 R4 , P4 = R1 R4 . Proof. By Corollary 3.3 we have P1 = Q1 Q2 , P2 = Q3 Q2 , P3 = Q3 Q4 , P4 = Q1 Q4

3372

for some polynomials Q1 , Q2 , Q3 , Q4 ∈ QhAi. Let ci = c(Qi ), i = 1, 2, 3, 4. By Theorem 4.1 we have c(P1 ) = c1 c2 , c(P2 ) = c3 c2 , c(P3 ) = c3 c4 , c(P4 ) = c1 c4 .

3373

Thus c(P4 ) = c(P1 )c(P3 )/c(P2 ). As by hypothesis and Eq. (4.2) c(Pi ) ∈ N, there exist positive integers d1 , d2 , d3 , d4 such that c(P1 ) = d1 d2 , c(P2 ) = d3 d2 , c(P3 ) = d3 d4 , c(P4 ) = d1 d4 . Moreover, by Theorem 4.1, P 1 = Q1 Q2 , P 2 = Q3 Q2 , P 3 = Q3 Q4 , P 4 = Q1 Q4 . Put Ri = di Qi , i = 1, 2, 3, 4. Then Ri ∈ ZhAi. Moreover P1 = c(P1 )P 1 = d1 d2 Q1 Q2 = R1 R2 .

3374

Similarly P2 = R3 R2 , P3 = R3 R4 and P4 = R1 R4 .

3375 3376

Proposition 4.4 Let Y be a primitive polynomial of degree 1 which vanishes for some integer values of the variables. Let P, Q ∈ ZhAi and let α ∈ Z, α 6= 0 be such that P Q ≡ α mod Y ZhAi. Then P ≡ β, Q ≡ γ mod Y ZhAi for some β, γ ∈ Z such that α = βγ.

3377 3378 3379 3380 3381 3382 3383 3384 3385 3386 3387



Proof. We have P Q = Y Q2 + α for some polynomial Q2 . As α 6= 0, we have Q 6= 0 and we may apply Corollary 1.3. This shows that P = β + Y T for some β ∈ Q and T ∈ QhAi. Hence Y Q2 + α = βQ + Y T Q. Since α 6= 0 and deg(Y ) > 0, we obtain β 6= 0: indeed, otherwise P = Y T and Y T Q = Y Q2 + α, implying that Y divides α. This shows that Q = γ + Y S for some γ ∈ Q such that α = βγ. Now the assumption on Y and the fact that P, Q have integer coefficients imply that β, γ ∈ Z. Since Y T = P − β ∈ ZhAi, we obtain that c(Y )c(T ) ∈ N by Eq. (4.2) and Theorem 4.1 (ii). But Y is primitive, so c(Y ) = 1, which shows that c(T ) ∈ N and T ∈ ZhAi by (4.2). Similarly, S ∈ ZhAi. 

169

4. Gauss’s lemma 3388

Exercises for Chapter X n P

Pi Qi = 0 is P called trivial if for each i, either Pi = 0 or Qi = 0. Note that Pi Qi may be written   Q1   (P1 , . . . , Pn )  ...  .

1.1 Let P1 , . . . , Pn , Q1 , . . . , Qn be polynomials. A relation

i=1

Qn

Show that if

n P

Pi Qi = 0, then there exists an invertible n by n matrix

i=1

M with coefficients in KhAi such that the relation    Q1    −1  .  (P1 , . . . , Pn )M M  ..  = 0 Qn

3389 3390 3391 3392

is trivial (cf. Cohn 1961). 1.2 a) Let X, Y X ′ , Y ′ be nonzero formal series such that XY ′ = Y X ′ , with ω(X) ≥ ω(Y ) (cf Chapter I). Show that there exists a formal series U such that X = Y U , X ′ = U Y ′ . b) Let S be a formal series and let C be its centralizer, that is C = {T ∈ KhhAii | ST = T S}. Show that if T1 , T2 ∈ C and ω(T2 ) ≥ ω(T1 ), then there exists T ∈ C such that T2 = T1 T . (Hint : one may suppose ω(S) ≥ 1; let n be such that ω(S n ) ≥ ω(T1 ), ω(T2 ): use a) three times.) Let T ∈ C such that ω(T ) ≥ 1 is minimum. Show that C = K[[T ]], that is nX o C= an T n | an ∈ K n∈N

3393

( (see Cohn 1961). 2.1 Show that for n ≥ k ≥ 1 the continuant polynomials satisfy the identities p(a1 , . . . , an )p(an−1 , . . . , ak ) − p(a1 , . . . , an−1 )p(an , . . . , ak ) = (−1)n+k p(a1 , . . . , ak−2 )

3394 3395 3396 3397

with the conventions: p(a1 , . . . , ak−2 ) = 0 if k = 1, = 1 if k = 2, and p(an−1 , . . . , ak ) = 1 if k = n. Show that the number of words in the support of p(a1 , . . . , an ) is the nth Fibonacci number Fn (F0 = F1 = 1, Fn+2 = Fn+1 + Fn ). 2.2 Show that if a1 , . . . , an are commutative polynomials, then 1

a1 +

=

1

a2 + a3 +

1 ···+

1 an

p(a1 , . . . , an ) . p(a2 , . . . , an )

170 2.3 Show that the entries   a1 1 a2 1 0 1

Chapter X. Noncommutative Polynomials of the matrix    1 a 1 ··· n 0 1 0

3401

may be expressed by means of continuant polynomials. 3.1 Let M be an n by n polynomial matrix such that M = M1 M2 with M1 ∈ KhhAiin×p and M2 ∈ KhhAiip×n . Show that then one may choose M1 , M2 to be polynomial matrices (use the inertia theorem; see Cohn 1985).

3402

Notes to Chapter X

3398 3399 3400

3403 3404 3405 3406 3407 3408 3409

Most of the results of this chapter are due to P. M. Cohn. We have already seen a result concerning noncommutative polynomials in Chapter II (Corollary II.3.3): in P. M. Cohn’s terminology, it means that KhAi is a fir (“free ideal ring”). The terminology “continuant” stems from its relation to continuous fractions (see Exercises 2.2 and 2.3). Corollary 3.2 is a special case of a more general result, stating that every polynomial matrix which is singular over the free field is not full (see Cohn 1961).

3410

Chapter XI

3411

Codes and Formal Series

3421 3422

The aim of this chapter is to present an application of formal series to the theory of (variable-length) codes. The main result (Theorem 4.1) states that every finite complete code admits a factorization into three polynomials which reflect its combinatorial structure. The first section contains some basic facts on codes and prefix codes. These are easily expressed by means of formal power series. Section 2 is devoted to complete codes and their relations to Bernoulli morphisms (Theorem 2.4). Concerning the degree of a code, we give in Section 3 only the very basic results needed in Section 4. This last section is devoted to the proof of the main result. It uses the material of the previous section and from Chapter X.

3423

1

3412 3413 3414 3415 3416 3417 3418 3419 3420

Codes

Definition A code is a subset C of A∗ such that whenever u1 , . . . , un , v1 , . . . , vp in C satisfy u1 · · · un = v1 · · · vp , 3424 3425 3426

(1.1)

then n = p and ui = vi for i = 1, . . . , n. In this case, any word in C ∗ (= the submonoid generated by C) is called a message. Note that if C is a code, then C ⊂ X + (= X ∗ \ 1). Example 1.1 The set {a, ab, ba} is not a code, because the word aba has two factorizations in it: aba = a(ba) = (ab)a .

3429

Example 1.2 The set {a, ab, bb} is a code; indeed, no word in it is a prefix of another, so in each relation of the form (1.1), either u1 is a prefix of v1 or vice versa, so one has u1 = v1 and one concludes by induction on n.

3430 3431

Example 1.3 The set {b, ab, a2 b, a3 b, . . . , an b, . . .} = a∗ b is a code, for the same reason as in Example 1.2.

3427 3428

171

172 3432 3433

Chapter XI. Codes and Formal Series

Example 1.4 The set {a3 , a2 ba, a2 b2 , ab, ba2 , baba, bab2, b2 a, b3 } is a code, for the same reason; note that in this case, moreover no word is a suffix of another. Example 1.5 The set C = {a2 , ab, a2 b, ab2 , b2 } is a code. Indeed, let C denote its characteristic polynomial; then we have 1 − C = 1 − a2 − ab − a2 b − ab2 − b2

= (1 − b − a2 − ab) + (b − b2 − a2 b − ab2 )

= (1 − b − a2 − ab)(1 + b)

= ((1 − a − b) + (a − a2 − ab))(1 + b)

= (1 + a)(1 − a − b)(1 + b) . Thus, in ZhhAii, we have

(1 − C)−1 = (1 + b)−1 (1 − a − b)−1 (1 + a)−1 . By the results of Section I.4, for any proper formal series S, (1 − S)−1 = P n ∗ −1 = A∗ = A∗ is the sum of all words on A n≥0 S = S and (1 − a − b) (and hence, its nonzero coefficients are all equal to 1). Hence  X C n (1 + a) . A∗ = (1 + b) n≥0

P This shows that the series n≥0 C n has no coefficient ≥ 2, since otherwise A∗ would have such a coefficient. From X X X Cn = u1 · · · un n≥0

n≥0 u1 ,...,un ∈C

3434 3435

we obtain that no word has two distinct factorizations of the form u1 · · · un (ui ∈ C), so C is a code.

3436

Recall that for any language X, X denotes its characteristic series (considered as an element of QhhAii in the present chapter). One of the arguments of the last example may be generalized as follows.

3437 3438

Proposition 1.1 Let C be a subset of A+ and let C be its characteristic series. Then C is a code if and only if one has in ZhhAii (1 − C)−1 = C ∗ = C ∗ .

(1.2)

Proof. The first equality is always true, as shown in Section I.4. We have X X X u1 · · · un = Cn = C∗ . n≥0 u1 ,...,un ∈C

n≥0

If C is a code, then the words u1 · · · un 3439 3440

(n ≥ 0, ui ∈ C)

are all distinct, so the left-hand side is equal to C ∗ . If C is not a code, then two of these words are equal, so the left-hand side is a series with at least one

173

1. Codes 3441 3442

coefficient ≥ 2: it cannot be equal to C ∗ , because the latter has only 0, 1 as coefficients. 

3443 3444

The previous result provides an effective algorithm for testing whether a given rational subset of C of A+ is a code. Indeed, one has merely to check if the rational power series C ∗ − C ∗ is equal to 0; for this, apply Corollary II.3.4. However, there is a more direct algorithm. We give below, without proof, the algorithm of Sardinas and Patterson (see Lallement 1979, Berstel and Perrin 1985). Recall that for any language X and any word w, we denote by w−1 X the language

3445

w−1 X = {u ∈ A∗ | wu ∈ X} . More generally, if Y is a language, we denote by Y −1 X the language [ Y −1 X = w−1 X . w∈Y

Now let C be a subset of A+ . Define a sequence of languages Cn by C0 = C −1 C \ 1

Cn+1 = Cn−1 C ∪ C −1 Cn

(n ≥ 0) .

3453 3454

Then C is a code if and only if no Cn contains the empty word. If C is finite, the sequence (Cn ) is periodic (because each word in Cn is a factor of some word in C). The same is true if C is rational (see Berstel and Perrin 1985, Prop. I.3.3). Hence in these cases, we obtain an effective algorithm. Another way to express the fact that a set of words is a code is by means of the so-called unambiguous operations. Let X, Y be languages. We say that their union is unambiguous if they are disjoint languages. We say that their product is unambiguous if x, x′ ∈ X, y, y ′ ∈ Y , and xy = x′ y ′ implies x = x′ , y = y ′ . We say that the star X ∗ is unambiguous if X is a code.

3455

Proposition 1.2 Let X, Y be languages.

3446 3447 3448 3449 3450 3451 3452

3456 3457 3458

(i) The union of X and Y is unambiguous if and only if X ∪ Y = X + Y . (ii) The product XY is unambiguous if and only if XY = X Y . (ii) If 1 ∈ / X, then the star X ∗ is unambiguous if and only if X ∗ = X ∗ .

3459 3460

Proof. The first two assertions are a direct consequence of their definition. The last one is merely a reformulation of Proposition 1.1. 

3461

We have already met a family of codes in Section II.3: the prefix codes. A set is prefix if no word in it is a prefix of another word in it. A prefix set which is not reduced to the empty word is easily seen to be a code, called a prefix code. Symmetrically, one defines suffix codes. A code is called bifix if it is both prefix and suffix.

3462 3463 3464 3465

3466 3467

Proposition 1.3 Let C be a code such that for any word v in C ∗ , one has v −1 C ∗ ⊂ C ∗ . Then C is a prefix code.

174 3468 3469 3470 3471 3472 3473

Chapter XI. Codes and Formal Series

Note the converse: for any set C and for any word v in C ∗ , one has C ∗ ⊂ v C∗. Proof. Suppose u = vw, with u, v in C and w ∈ A∗ . We have to show that w = 1. Now w = v −1 u ∈ v −1 C ∗ ⊂ C ∗ , hence w ∈ C ∗ . Therefore w = c1 · · · cn (ci ∈ C) and u = vc1 · · · cn ∈ C. The only possibility for C to be a code is n = 0, that is w = 1, and C is a prefix code.  −1

Proposition 1.4 Let C be a prefix code such that CA∗ ∩ wA∗ is nonempty for any word w. Let P be the set of proper prefixes of the words in C. Then one has in ZhhAii C − 1 = P (A − 1) . 3474 3475 3476 3477 3478 3479 3480 3481 3482 3483 3484 3485

Proof. Let P ′ = A∗ \ CA∗ . Then, by Proposition II.3.1, we have A∗ = C ∗ P ′ . But, because C is a prefix code, the conditions u1 · · · un q = v1 · · · vp r, ui , vj ∈ C, q, r ∈ P ′ imply n = p, ui = vi for i = 1, . . . , n, hence also q = r. This shows that the product C ∗ P ′ is unambiguous, hence by Proposition 1.2, we have A∗ = C ∗ P ′ . Now, by Proposition 1.1, A∗ = (1 − A)−1 and C ∗ = (1 − C)−1 . Moreover, the empty word is in P ′ , so P ′ is invertible in ZhhAii. Hence 1−A = P ′−1 (1−C), which implies C − 1 = P ′ (A − 1). It remains to show that P = P ′ . Let w be in P ; then w is a proper prefix of some word in C and so has no prefix in C, C being a prefix code; hence w∈ / CA∗ =⇒ w ∈ P ′ . Let w be in P ′ . By assumption, there are words c ∈ C, u, v ∈ A∗ such that cu = wv; as w ∈ / CA∗ , w must be a proper prefix of c, so w ∈ P .  Let C be a code. Define, for any word u, the series Su inductively by S1 = 1 Su = a−1 Sv + (Sv , 1)a−1 C ,

3486 3487 3488 3489 3490

for u = va (a ∈ A)

Note that, obviously, Su has nonnegative coefficients. The reader may verify that the support of Su consists of proper suffixes of C (cf. Exercise 1.3). Lemma 1.5 Let C be a code. Then for any word u, u−1 (C ∗ ) = Su C ∗ . In particular, Su is a characteristic series. If C is finite, then Su is a polynomial. Proof. We shall use the formulas of Lemma I.7.2. We prove u−1 (C ∗ ) = Su C ∗ by induction on |u|. If u = 1, it is clearly true. Let u = va, (a ∈ A). Then by induction v −1 (C ∗ ) = Sv C ∗ . Thus, by Lemma I.7.2, u−1 (C ∗ ) = a−1 v −1 (C ∗ ) = (a−1 Sv )C ∗ + (Sv , 1)(a−1 C ∗ ) = (a−1 Sv )C ∗ + (Sv , 1)(a−1 C)C ∗ = Su C ∗ .

3491 3492

Now, since u−1 (C ∗ ) is obviously a characteristic series, the same holds for Su . It is easily verified by induction that Su is a polynomial if C is finite.  One defines symmetrically the series Pu ∈ ZhhAii by P1 = 1 Pav = Pv a−1 + (Pv , 1)Ca−1 ,

for a ∈ A and v ∈ A∗

175

2. Completeness

Now we define, for a couple (u, v) of words another series in the following way: Fu,1 = 0 Fu,av = (Pv , 1)Su a−1 + Fu,v a−1 . 3493 3494 3495 3496

As above, the series Fu,v clearly has nonnegative coefficients. Proposition 1.6 Let C be a code. Then for any words u and v, u−1 (C ∗ )v −1 = Su C ∗ Pv + Fu,v . In particular, Fu,v is a characteristic series. If C is finite, then Fu,v is a polynomial. Proof (Induction on |v|). The result is obvious if v = 1 by Lemma 1.5. Let a ∈ A. Then u−1 (C ∗ )(av)−1 = [u−1 (C ∗ )v −1 ]a−1 is equal, by induction and Lemma I.7.2, to (Su C ∗ Pv )a−1 + Fu,v a−1 = Su C ∗ (Pv a−1 ) + (Pv , 1)Su (C ∗ a−1 ) + (Pv , 1)Su a−1 + Fu,v a−1 = Su C ∗ (Pv a−1 ) + (Pv , 1)Su C ∗ (Ca−1 ) + Fu,av = Su C ∗ Pav + Fu,av .

3499 3500

This proves the formula. Now, since Su C ∗ Pv has nonnegative coefficients and since u−1 (C ∗ )v −1 is a characteristic series, the same holds for Fu,v . If C is finite, it is easily seen by induction on the definition that Fu,v is a polynomial. 

3501

2

3497 3498

3502 3503

Completeness

Definition A language C ⊂ A∗ is complete if, for any word w, the set C ∗ ∩ A∗ wA∗ is nonempty. Lemma 2.1 If C is complete, then any word w is either a factor of a word in C or may be written as w = smp ,

3504

with m ∈ C ∗ and where s (p) is a suffix (prefix) of a word of C. Proof. We have xwy ∈ C ∗ for some words x, y. Let us represent a word in C ∗ schematically by

3505

Then we have two cases: 1)

w

176

Chapter XI. Codes and Formal Series

2)

w 3506 3507

3508 3509 3510 3511 3512

In the first case, w is a factor of a word in C. In the second case, w = smp as in the lemma.  Definition A Bernoulli morphism is a mapping π : A∗ → R such that (i) (ii) (iii) (iv)

π(w) > 0 for any word w, π(1) = 1, π(uv) P = π(u)π(v) for any words u, v, π(a) = 1. a∈A

It is called uniform if π(a) = 1/|A| for any letter a. We define for any language X the measure of X by X π(X) = π(w) w∈X

(it may be infinite). We shall frequently use the following inequalities: X π(Xi ) π(∪Xi ) ≤ π(XY ) ≤ π(X)π(Y ) .

3513

Note that, for any n, one has π(An ) = 1.

3514

Lemma 2.2 Let C be a code. Then π(C) ≤ 1. Proof. Since C is the limit of its finite subsets, it is enough to show the lemma in the case where C is finite. Let p be the maximal length of words in C. Then C n ⊂ A ∪ A2 ∪ · · · ∪ Apn . Thus π(C n ) ≤ pn. Now, as C is a code, each word in C n has only one factorization of the form u1 · · · un (ui ∈ C). As π is multiplicative, we obtain π(C n ) = π(C)n . Hence π(C)n ≤ pn .

3515

This shows that π(C) ≤ 1.

3516

Lemma 2.3 Let C be a finite complete language. Then π(C) ≥ 1. Proof. By Lemma 2.1, we may write A∗ = SC ∗ P ∪ F , where S, P, F are finite languages. Thus ∞ = π(A∗ ) ≤ π(S)π(C ∗ )π(P ) + π(F ) .



177

2. Completeness This shows that π(C ∗ ) = ∞. Now [ C∗ = Cn n≥0

3517 3518

3519 3520 3521 3522 3523 3524 3525 3526 3527 3528

P so that π(C ∗ )P ≤ n≥0 π(C n ). Moreover, π(C n ) ≤ π(C)n , π being multiplica tive. So ∞ ≤ n≥0 π(C)n , which shows that π(C) ≥ 1.

Theorem 2.4 (Sch¨ utzenberger and Marcus 1959, Bo¨e et al. 1980) Let C be a finite subset of A∗ and let π be a Bernoulli morphism. Then any two of the following assertions imply the third one: (i) C is a code, (ii) C is complete, (iii) π(C) = 1 . Note that this gives an algorithm for testing whether a given finite code is complete (see Exercise 2.3). We need another lemma. Lemma 2.5 Let X be a language and let w be a word such that X ∩ A∗ wA∗ is empty. Then π(X) < ∞. Proof. Let ℓ = |w| and for i = 0, . . . , ℓ − 1 Xi = {v ∈ X | |v| ≡ i mod ℓ} .

3529 3530 3531

Then Xi ⊂ Ai (Aℓ \ w)∗ . Indeed v ∈ Xi implies v = uv1 · · · vn with |u| = i and for any j, |vj | = ℓ; by assumption, w is not factor of v, hence w is none of the vj ’s: thus vj ∈ Aℓ \ w, which proves the claim. Now π(Aℓ \ w) = π(Aℓ ) − π(w) = 1 − π(w) < 1 and π[(Aℓ \ w)∗ ] = π ≤

X

ℓ

[

 X (Aℓ \ w)n ≤ π[(Aℓ \ w)n ]

n≥0

n

n≥0

[π(A \ w)] < ∞ .

n≥0

3532 3533

Thus π(Xi ) = π[Ai (Aℓ \ w)∗ ] ≤ π(Ai )π[(Aℓ \ w)∗ ] < ∞ and since X = ∪0≤i≤ℓ−1 Xi , we obtain π(X) < ∞. 

3534

Proof of Theorem 2.4. Lemma 2.2 and 2.3 show that (i) and (ii) imply (iii). Let C be a code with π(C) = 1. Suppose C is not complete. Then for some word w, C ∗ ∩ A∗ wA∗ is empty. P Hence, by Lemma 2.5, π(C ∗ ) < ∞. As C is a ∗ code, π(C ) is equal to the sum n≥0 π(C)n . The latter being finite, we deduce that π(C) < 1, a contradiction. Let C be complete and π(C) = 1. Then C n is complete for any n; indeed, for any word w, there are words u, v, c1 , . . . , cp (ci ∈ C) such that uwv = c1 · · · cp (C being complete). Let r be such that p + r is a multiple of n; then uwvcr1 = c1 · · · cp cr1 ∈ (C n )∗ , which shows that (C n )∗ ∩ A∗ wA∗ is not empty. Hence

3535 3536 3537 3538 3539 3540 3541 3542

178 3543 3544

Chapter XI. Codes and Formal Series

C n is complete. Thus, by Lemma 2.3, π(C n ) ≥ 1 for any n. But as usually π(C n ) ≤ π(C)n = 1, thus π(C n ) = π(C)n for any n. Suppose C is not a code. Then for some words u1 , . . . , un , v1 . . . , vp in C we have u1 · · · un = v1 · · · vp and u1 6= v1 . Hence u1 · · · un v1 · · · vp = v1 · · · vp u1 · · · un , and we have obtained a word in C n+p which has two distinct factorizations. Hence
π(C n+p ) = π {w1 · · · wn+p | wi ∈ C} X < π(w1 · · · wn+p ) = π(C n+p ) w1 ,...,wn+p ∈C

3545

which is a contradiction.



Let π be a Bernoulli morphism. Since π is multiplicative, it may be extended to an algebra morphism, still denoted by π, π : ZhAi → R by the formula X  X π (P, w)w = (P, w)π(w) . w

w

Note that, because the measure of A is 1, one has π(A − 1) = 0 . 3546 3547

Theorem 2.6 (Sch¨ utzenberger 1965) Let C be a finite code such that for any word w, the set C ∗ ∩ wA∗ is nonempty. Then C is a prefix code. Proof. Let C ′ be the set of words in C having no proper prefix in C, that is C ′ = C \ CA+ . Clearly C ′ is a prefix code. Moreover, if w is a word, then for some words c1 , . . . , cn ∈ C, u ∈ A∗ , one has by assumption c1 · · · cn = wu .

3548 3549 3550 3551 3552 3553

3554 3555 3556 3557

Then either c1 ∈ C ′ , or c1 has a prefix in C ′ . Thus C ′ A∗ ∩ wA∗ is nonempty. Let P be the set of proper prefixes of the words in C ′ . Then by Proposition 1.4, C ′ − 1 = P (A − 1). Apply the morphism π : ZhAi → R, obtaining π(C ′ − 1) = 0 because π(A − 1) = 0. Thus π(C ′ ) = 1. As C is a code, we have by Lemma 2.2, π(C) ≤ 1. But C ′ ⊂ C and π is positive. Hence C = C ′ is prefix.  Theorem 2.7 (Reutenauer 1985) Let P in NhAi be without constant term such that P − 1 = X(A − 1)Y for some polynomials X, Y in RhhAii. Then P = C for some finite complete code C. Furthermore, if Y ∈ R (X ∈ R), then C is a prefix (suffix) code. Proof. 1. Note that if S, T are formal series, then supp(ST ) ⊂ supp(S) supp(T ) .

3. The degree of a code

179

Moreover, if S is proper, then supp(S ∗ ) ⊂ supp(S)∗ . 2. We have 1 − P = X(1 − A)Y . By assumption, 1 − P is invertible in RhhAii. The same holds for 1 − A since its inverse is A∗ = A∗ . This shows that X and Y are also invertible. So we obtain (1 − P )−1 = Y −1 (1 − A)−1 X −1 which implies (1 − A)−1 = Y (1 − P )−1 X . Thus A∗ = Y P ∗ X .

(2.1)

By 1, this implies that each word w may be written as w = ymx, with y ∈ supp(Y ), m ∈ supp(P )∗ and x ∈ supp(X). Let C = supp(P ) and let u be a word such that |u| > deg(X), deg(Y ). Let v be any word. Then w = uvu may be written uvu = ymx as above, which shows, by the choice of u, that m = v1 vv2 . Hence C ∗ ∩ A∗ vA∗ is nonempty: we have shown that C is complete. Thus, by Lemma 2.3, π(C) ≥ 1 (where π is some Bernoulli morphism). Now, as P − 1 = X(A − 1)Y , we obtain π(P ) = 1. Hence 1 ≤ π(C) ≤ π(P ) = 1

3564

because P has nonnegative integer coefficients. This shows, π being positive, that P = C and that π(C) = 1. Hence, by Theorem 2.4, C is a code, and thus a finite complete code. Suppose now that Y ∈ R. Then, as above, Eq. (2.1) shows that for any word v, one has vu = mx for some words m ∈ C ∗ , x ∈ supp(X) (u being chosen as before). Then, as |u| > |x|, we obtain m = vv1 which shows that C ∗ ∩ vA∗ is nonempty. We conclude by Theorem 2.6. 

3565

3

3558 3559 3560 3561 3562 3563

3566 3567 3568 3569 3570 3571 3572 3573 3574 3575 3576

The degree of a code

Given a monoid M , recall that an ideal in M is a nonempty subset J which is closed for left and right multiplication by elements of M . Moreover, an idempotent is an element e which is equal to its square, that is e2 = e. Theorem 3.1 (Suschkewitsch 1928) Let M be a finite monoid. There exists in M an ideal J which is contained in any ideal of M . Let e be an idempotent in J. Then eM e is a finite group whose neutral element is e. This ideal will be called the minimal ideal of M Proof. 1. Let J be the intersection of all ideals in M . Clearly J is closed for multiplication by elements of M . We have only to verify that it is not empty. But let m be the product of all elements of M , in some order. Then m is in each ideal of M , and hence in J.

180 3577 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588 3589 3590 3591 3592 3593

3594 3595 3596 3597

3598 3599 3600

3601 3602

3603 3604 3605 3606 3607 3608 3609 3610 3611 3612 3613 3614 3615 3616 3617

Chapter XI. Codes and Formal Series

2. We use the following classical fact: if a ∈ M , then some positive power of a is an idempotent. Indeed, chose i, j ≥ 1 such that j ≥ i and that ai = ai+j (this is possible because the set {a, a2 , . . . , an , . . .} is finite). Let k = j − i. Then ai+k is idempotent because ai+k ai+k = ak ai+i+k = ak ai+j = ak ai = ak+i . 3. Clearly, eeme = eme = emee and emeem′ e = e(mem′ )e, hence eM e is a (finite) monoid whose neutral element is M . 4. Let a = eme be in eM e. We show the existence of b ∈ eM e such that ab = e. We have a = et for some t ∈ M . Now M aM is an ideal of M contained in J (because M aM = M etM , e ∈ J and J is an ideal), hence M aM = J (J being minimal). Thus e = uav for some elements u, v of M . Next, e = uetv = uuetvtv = un e(tv)n for any n ≥ 1. Choose n such that (tv)n is idempotent. Then e = un e(tv)n = un e(tv)n (tv)n = e(tv)n = etv(tv)n−1 = aw (recall that et = a). But a = eme implies ae = eme2 = eme = a, whence e = aw = aew and e = e2 = aewe. Let b = ewe ∈ eM e. Then e = ab. 5. Symmetrically, we have e = ca for some c in eM e. Then, classically c = ce = cab = eb = b. This shows that each element of eM e has an inverse in eM e, that is, eM e is a group. 

Theorem 3.2 Let C be a finite complete code. There exist a finite monoid M and a surjective morphism φ : A∗ → M such that C ∗ = φ−1 φ(C ∗ ). Let J be the minimal ideal of M . There exists an idempotent e in J ∩ φ(C ∗ ); further φ(C ∗ ) ∩ eM e is a subgroup of the group eM e. It will not be shown here that the index of φ(C ∗ ) ∩ eM e in eM e depends only on C; for this, we refer the reader to the book by Berstel and Perrin (1985). This being admitted, we introduce the following definition.

Definition With the notation of Theorem 3.2, the index of eM e ∩ φ(C ∗ ) in eM e is called the degree of C. Proof of Theorem 3.2. Clearly, C ∗ is a rational subset of A∗ (cf. Section III.1). Hence, by Kleene’s theorem (Theorem III.1.1), it is recognizable. This shows that there exist a finite monoid M , a monoid morphism φ : A∗ → M , and a subset N of M such that C ∗ = φ−1 (N ). Clearly, we may assume that φ is surjective; then N = φ(C ∗ ) and C ∗ = φ−1 φ(C ∗ ). Let J be the minimal ideal of M and w a word in φ−1 (J). Then C ∗ ∩A∗ wA∗ is nonempty (because C is complete), hence there exist words u, v such that uwv is in C ∗ . Now m = φ(uwv) is in φ(C ∗ ) and also in J (because m = φ(u)φ(w)φ(v), φ(w) ∈ J, and J is an ideal). Some power e = mn with n ≥ 1 of m is idempotent and still lies in φ(C ∗ ) ∩ J. Now, φ(C ∗ ) is clearly a submonoid of M . Hence, the product of any two elements of eM e ∩ φ(C ∗ ) lies in eM e ∩ φ(C ∗ ). Take a ∈ eM e ∩ φ(C ∗ ). Then for some n ≥ 2, an = e (eM e being a finite group). Then an−1 is the inverse of a in eM e, and belongs to eM e ∩ φ(C ∗ ). Thus, the latter is a subgroup of eM e. 

4. Factorization 3618

4

181

Factorization

Theorem 4.1 (Reutenauer 1985) Let C be a finite complete code. Then there exist polynomials X, Y, Z in ZhAi such that C − 1 = X(d(A − 1) + (A − 1)Z(A − 1))Y 3619 3620 3621

(4.1)

and (i) d is the degree of C, (ii) C is prefix (suffix) if and only if Y = 1 (X = 1). Example 4.1 We have a2 + a2 b + ab + ab2 + b2 − 1 = (1 + a)(a + b − 1)(1 + b) .

3622 3623

The corresponding code is neither prefix nor suffix, but synchronizing (that is of degree 1). Example 4.2 Let C be the square of the code of Example 4.1. Then C is of degree 2 and C − 1 = (1 + a)(2(a + b − 1) + (a + b − 1)(1 + b)(1 + a)(a + b − 1))(1 + b) . Example 4.3 We have a3 + a2 ba + a2 b2 + ab + ba2 + baba + bab2 + b2 a + b3 − 1 = 3(a + b − 1) + (a + b − 1)(2 + a + b + ab)(a + b − 1) .

3624

The corresponding code is a bifix code and has degree 3.

3625 3626

The following corollary (which also uses Theorem 2.7) characterizes completely finite complete codes.

3627 3628

Corollary 4.2 (Reutenauer 1985) Let C be a language not containing the empty word. Then the following conditions are equivalent:

3629

(i) C is a complete finite code. (ii) There exist polynomials P, S in ZhAi such that C − 1 = P (A − 1)S .

3630



In order to prove Theorem 4.1, we need the following lemma. Lemma 4.3 Let C be a finite complete code of degree d. Then there exist words u1 , . . . ud , v1 , . . . , vd , with u1 , v1 ∈ C ∗ , such that for any i, 1 ≤ i ≤ d: X ∗ −1 A∗ = u−1 i (C )vj 1≤j≤d

and for any j, 1 ≤ j ≤ d: X ∗ −1 u−1 A∗ = i (C )vj . 1≤i≤d

182 3631 3632 3633 3634

Chapter XI. Codes and Formal Series

Proof. By Theorem 3.2 there exist a finite monoid M and a surjective morphism φ : A∗ → M such that C ∗ = φ−1 φ(C ∗ ); moreover, there exists an idempotent e in J ∩ φ(C ∗ ), where J is the minimal ideal of M , G = eM e is a finite group and H = eM e ∩ φ(C ∗ ) is a subgroup of G of index d. Let u1 , . . . ud , v1 , . . . , vd be words in φ−1 (G) such that G=

[

φ(vi )H

[

Hφ(uj )

(4.2)

1≤i≤d

and G=

1≤j≤d

3635 3636 3637 3638 3639 3640 3641 3642 3643 3644

(disjoint unions). By elementary group theory, we may assume that φ(u1 ) = φ(v1 ) = e (hence u1 , v1 ∈ φ−1 (e) ⊂ φ−1 φ(C ∗ ) = C ∗ ) and that φ(ui ) is the inverse of φ(vi ) in G. Let 1 ≤ j ≤ d and w be a word. Then there exists one and only one i, ∗ −1 ∗ 1 ≤ i ≤ d, such that w ∈ u−1 i (C )vj , that is ui wvj ∈ C . Indeed, the element eφ(wvj ) of G is in some φ(vi )H by Eq. (4.2). Hence, φ(ui wvj ) = φ(ui )eφ(wvj ) ∈ φ(ui )φ(vi )H = eH = H, which implies that ui wvj ∈ φ−1 (H) ⊂ φ−1 φ(C ∗ ) = C ∗ . Conversely, ui wvj ∈ C ∗ implies φ(ui wvj ) ∈ eM e ∩ φ(C ∗ ) = H, because φ(ui wvj ) = eφ(ui wvj )e is already in eM e. Hence eφ(wvj ) = φ(vi )φ(ui wvj ) ∈ φ(vi )H, and i is completely determined by j and w. We have shown that one has the disjoint union, for any j, 1 ≤ j ≤ d: [ ∗ −1 A∗ = u−1 i (C )vj . 1≤i≤d

3645 3646 3647

3648 3649 3650 3651 3652 3653 3654 3655 3656 3657 3658

But this is equivalent to the last relation of the lemma. By symmetry, we have also the first.  We easily derive the following lemma Lemma 4.4 Let C be a finite complete code of degree d. Then there exist polynomials P, P1 , S, S1 , Q, G1 , D1 with coefficients 0, 1 such that (i) (ii) (iii) (iv) (v) (vi)

dA∗ − Q = SC ∗ P . A∗ − G1 = SC ∗ P1 . A ∗ − D 1 = S1 C ∗ P . P1 , S1 have constant term 1. G1 , D1 have constant term 0. If C is a prefix (suffix) code, then S1 = 1 (P1 = 1).

Proof. We use Lemma 4.3 and the notation of Section 1. We have, by Proposi∗ −1 tion 1.6, u−1 = Sui C ∗ Pvj + Fui ,vj ; moreover, by Lemma 1.5 and Propoi (C )vj sition 1.6, Sui , Pvj and Fui ,vj are polynomials with nonnegative coefficients. Now, by Lemma 4.3, for any i X X Fui ,vj Sui C ∗ Pvj + A∗ = 1≤j≤d

1≤j≤d

183

4. Factorization and for any j X

A∗ =

Sui C ∗ Pvj +

X

Fui ,vj

1≤i≤d

1≤i≤d

Let P =

X

Pvj ,

S=

1≤j≤d

G1 =

X

X

Sui ,

P1 = Pv1 ,

1≤i≤d

Fui ,v1 ,

D1 =

X

Fu1 ,vj

Q=

j

i

X

S1 = Su1 Fui ,vj .

i,j

Then we obtain dA∗ = SC ∗ P + Q, 3659 3660 3661 3662 3663

A∗ = SC ∗ P1 + G1 ,

A∗ = S1 C ∗ P + D1 ,

(4.3)

which proves (i), (ii) and (iii). ∗ −1 ∗ As u1 ∈ C ∗ by Lemma 4.3, u−1 1 (C ) contains 1, hence u1 (C ) has constant ∗ ∗ −1 term 1. As u1 (C ) = Su1 C by Lemma 1.5, S1 = Su1 must have constant term 1. TheP same holds for P1 by symmetry, and proves (iv). As S = Sui , the Sui ’s are nonnegative and as Su1 has constant term 1, i

3664 3665 3666 3667 3668 3669 3670

S has nonnegative constant term. Moreover, P1 has constant term 1. Hence, because A∗ has constant term 1 and by Eq. (4.3), G1 has constant term 0. Similarly, D1 has constant term 0. This proves (v). ∗ ∗ Suppose now that C is prefix. Then, by Proposition 1.3, u−1 1 (C ) = C ∗ ∗ ∗ ∗ −1 −1 ∗ (because u1 ∈ C ). Hence u1 (C ) = C . As by Lemma 1.5, u1 (C ) = Su1 C , we obtain S1 = Su1 = 1. Similarly, if C is suffix, then P1 = 1. This proves (vi).  Given a Bernoulli morphism π, define a mapping λ for each word w by λ(w) = π(w) |w| . For each language X, define λ(X) by λ(X) =

X

w∈X

λ(w) ∈ R+ ∪ ∞ .

This is called the average length of X. On the other hand λ extends to a linear mapping ZhAi → R by λ(P ) =

X

(P, w)λ(w) .

w

3671

Lemma 4.5 Let P1 , . . . , Pn be polynomials. Then X λ(P1 · · · Pn ) = π(P1 ) · · · π(Pi−1 )λ(Pi )π(Pi+1 ) · · · π(Pn ) . 1≤i≤n

184

Chapter XI. Codes and Formal Series

Proof. For n = 2, it is enough, by linearity, to prove the lemma when P1 = u, P2 = v are words. But in this case λ(uv) = π(uv) |uv| = π(u)π(v)(|u| + |v|) = π(u)|u|π(v) + π(u)π(v)|v| = λ(u)π(v) + π(u)λ(v) . 3672

The general case is easily proved by induction.



Proof of Theorem 4.1. 1. First, note that the “if” part of (ii) is a consequence of Theorem 2.7. We use the notation of Lemma 4.4. We have A∗ − G1 = (1 − A)−1 − G1 = (1 − A)−1 (1 − (1 − A)G1 ). As A∗ − G1 = SC ∗ P1 and P1 has constant term 1 (Lemma 4.4), P1 is invertible in ZhAi and we obtain from SC ∗ P1 = (1 − A)−1 (1 − (1 − A)G1 ) , by multiplying by 1 − A on the left and by P1−1 on the right, (1 − A)SC ∗ = (1 − (1 − A)G1 )P1−1 .

(4.4)

Multiply the relation (i) of Lemma 4.4 by 1 − A on the left. This yields d − (1 − A)Q = (1 − A)SC ∗ P . Hence, by Eq. (4.4), d − (1 − A)Q = (1 − (1 − A)G1 )P1−1 P . Note that, because G1 has no constant term, 1−(1−A)G1 is invertible in ZhhAii, so that we obtain, by multiplying the previous relation by P1 (1 − (1 − A)G1 )−1 on the left P = P1 (1 − (1 − A)G1 )−1 (d − (1 − A)Q) . 2. We apply Corollary X.4.3 to the last equality: there exist E, F, G, H in ZhAi such that P1 = EF,

1 − (1 − A)G1 = GF

d − (1 − A)Q = GH,

P = EH

(4.5)

By Proposition X.4.4 applied to the second equality (with 1 − A instead of Y ), we obtain G ≡ ±1 mod (1 − A)ZhAi . Replacing if necessary E, F, G, H by their opposites, we may suppose that G ≡ +1, and hence we obtain, again by Proposition X.4.4, and by the third equality in Eq. (4.5), that H ≡ d mod (1 − A)ZhAi, which implies P = E(d + (A − 1)R) ,

R ∈ ZhAi .

(4.6)

3. We have A∗ − D1 = (1 − A)−1 (1 − (1 − A)D1 ) so that by Lemma 4.4 (iii), S1 C ∗ P = (1 − A)−1 (1 − (1 − A)D1 ) .

185

4. Factorization

As D1 has constant term 0, (1 − (1 − A)D1 ) is invertible in ZhhAii; moreover S1 is also invertible because it has constant term 1. So we obtain, by multiplying by (1 − C)S1−1 on the left and by (1 − (1 − A)D1 )−1 (1 − A) on the right, (1 − C)S1−1 = P (1 − (1 − A)D1 )−1 (1 − A) . Now we use Eq. (4.6) and multiply by −S1 on the right, thus obtaining C − 1 = E(d + (A − 1)R)(1 − (1 − A)D1 )−1 (A − 1)S1 . 4. By Corollary X.4.3, there exist E ′ , F ′ , G′ , H ′ ∈ ZhAi such that E(d + (A − 1)R) = E ′ F ′ ,

(A − 1)S1 = G′ H ′ ,

1 − (1 − A)D1 = G′ F ′

C − 1 = E′H ′ .

(4.7)

Let π be any Bernoulli morphism. Replacing if necessary E ′ , F ′ , G′ , H ′ by their opposites, we may assume that π(F ′ ) ≥ 0 . So, by Eq. (4.7) and Proposition X.4.4, we obtain (since π(A − 1) = 0) G′ = 1 + (A − 1)G′′ ,

F ′ = 1 + (A − 1)F ′′

(4.8)

for some G′′ , F ′′ ∈ ZhAi. This and Eq. (4.7) imply that (A − 1)S1 = (1 + (A − 1)G′′ )H ′ = H ′ + (A − 1)G′′ H ′ . Thus, we have H ′ = (A − 1)H ′′ ,

H ′′ ∈ ZhAi .

(4.9)

Now, Eqs. (4.7) and (4.8) imply also E(d + (A − 1)R) = E ′ (1 + (A − 1)F ′′ ) . 5. We now apply Theorem X.2.2 to this equality and denote by pi the continuant polynomial p(a1 , . . . , ai ) and p˜i = p(ai , . . . , a1 ). Thus there exist polynomials U, V ∈ QhAi such that E = U pn , d + (A − 1)R = p˜n−1 V, E ′ = U pn−1 , 1 + (A − 1)F ′′ = p˜n V .

(4.10)

Applying Corollary X.1.3 to the second and the last equalities (with X → p˜n−1 or p˜n , Y → A − 1, Q1 → 0, P → V , R → d or 1), we obtain that the left Euclidean division of p˜n−1 and p˜n by A− 1 is possible, that is p˜n−1 and p˜n are both congruent to a scalar mod (A−1)QhAi. This implies, by Proposition X.2.3, that pn−1 and p˜n−1 (pn and p˜n ) are congruent to the same scalar mod (A − 1)QhAi. lary X.4.2, they have the same content c(pn−1 ) = c(˜ pn−1 ),

c(pn ) = c(˜ pn ) .

(4.11) Moreover, by Corol(4.12)

186

Chapter XI. Codes and Formal Series

6. As D1 has coefficients 0, 1, the polynomial 1 − (A − 1)D1 is primitive. Hence, by Eq. (4.7) and by Gauss’s Lemma, G′ and F ′ are primitive. As by Eqs. (4.10) and (4.8) p˜n V = 1 + (A − 1)F ′′ = F ′ , we obtain by Gauss’s Lemma c(˜ pn )c(V ) = 1 and p¯ ˜n V = F ′ . Hence, by Proposition X.4.4 and Eq. (4.8), V = ε + (A − 1)V ′ ,

ε = ±1, V ′ ∈ ZhAi .

(4.13)

Furthermore, C − 1 is primitive, hence so is E ′ by Eq. (4.7). As E ′ F ′ = E(d + (A − 1)R) by Eq. (4.7) and E ′ , F ′ are primitive, we obtain by Gauss’s Lemma that d + (A − 1)R is primitive. Thus by Eq. (4.10) and Gauss’s Lemma again d + (A − 1)R = p¯˜n−1 V . This implies, by Proposition X.4.4 and Eq. (4.13), p¯ ˜n−1 = εd + (A − 1)L,

L ∈ ZhAi .

By Eqs. (4.11) and (4.12), we obtain that p¯n−1 and p¯˜n−1 are congruent to the same scalar mod(A − 1)QhAi. Hence p¯n−1 = εd + (A − 1)M 3673 3674

with M ∈ QhAi. But p¯n−1 − εd = (A − 1)M and A − 1 is primitive, so that c(M ) = c(¯ pn−1 − εd) ∈ N and M ∈ ZhAi, by Eq. (4.2) in Chapter X. We have seen that E ′ is primitive, so that by Gauss’s Lemma and Eq. (4.10), we have E ′ = U p¯n−1 which implies E ′ = U (εd + (A − 1)M ) . Hence, by Eqs. (4.7) and (4.9), C − 1 = U (εd + (A − 1)M )(A − 1)H ′′ , where all polynomials are in ZhAi and where ε = ±1. This shows that we have a relation of the form C − 1 = X(ε′ d + (A − 1)D)(A − 1)Y , where X = ±U , Y = ±H ′′ , ε′ d + (A − 1)D = ±(εd + (A − 1)M )

4. Factorization

187

are chosen in such a way that, for some Bernoulli morphism π, one has π(X) ≥ 0, π(Y ) ≥ 0 . 7. Apply Lemma 4.5 to this relation, using the fact that π(A−1) = 0; we obtain λ(C − 1) = π(X)ε′ dλ(A − 1)π(Y ) . Now λ(1) = 0, λ(C) > 0, λ(A) > 0, and we obtain ε′ dπ(X)π(Y ) > 0 .

3682

This shows that ε′ = 1 and proves Eq. (4.1) and (i). 8. Now, if C is a prefix code, we have by Lemma 4.4 (vi) that S1 = 1. Hence, by Eq. (4.7), A − 1 = G′ H ′ , which implies by Eq. (4.9) A − 1 = G′ (A − 1)H ′′ . Hence H ′′ = ∓1, and we obtain Y = ±1. But π(Y ) ≥ 0, so Y = 1. On the other hand, if C is suffix, then P1 = 1 by Lemma 4.4 (vi). Then, by Eq. (4.5), E = ±1 which implies by Eq. (4.10) and Gauss’s Lemma that U = ±1. Thus X = ±1. As π(X) ≥ 0, we obtain X = 1. This proves the theorem. 

3683

Exercises for Chapter XI

3675 3676 3677 3678 3679 3680 3681

1.1 Show that a submonoid of A∗ is of the form C ∗ , C a code, if and only if it is free (that is isomorphic to some free monoid). Show that a submonoid M of A∗ is free if and only if for any words u, v, w u, uv, vw, w ∈ M =⇒ v ∈ M . 3684 3685 3686 3687 3688 3689 3690 3691 3692 3693 3694 3695 3696 3697

1.2 Show that, given rational languages K, L, it is decidable whether their union (their product, the star of K) is unambiguous. 1.3 Show that Su (Pu , Fu,v ) as defined in Section 1 is a sum of proper suffixes (prefixes, factors) of words of C. 2.1 Show that for a finite code C the three following conditions are equivalent: (i) C is a complete and prefix code. (ii) For any word w, wA∗ ∩ CA∗ is not empty. (iii) For any word w, wA∗ ∩ C ∗ is not empty. 2.2 Let C be a finite complete language. Show that for any word w, there exists some power of a conjugate of w which is in C ∗ (two words w, w′ are conjugate if w = uv, w′ = vu for some words u, v). 2.3 Deduce from Theorem 2.4 an algorithm to show that a finite set C is a code (hint: it is decidable whether C is complete, since the set of factors of a rational language is rational). 3.1 Show that if e, e′ are idempotents in the minimal ideal J of a finite monoid M , then there exists an idempotent e1 in J which is a right multiple of E and a left multiple of e′ . Show that the mapping a 7→ ae1

3698 3699

defines a group isomorphism eM e → e1 M e1 . Deduce that all the maximal groups in J are isomorphic.

188 3700 3701 3702 3703 3704 3705 3706 3707

Chapter XI. Codes and Formal Series

3.2 Let C be a finite complete code. Show that C is synchronizing (that is of degree 1) if and only if for some word w, one has wA∗ w ⊂ C ∗ . 4.1 Let C be a finite complete code which is bifix. Let n be such that an ∈ C for some letter a. a) Show that for any i, 1 ≤ i ≤ n, Ci = a−i C is a prefix set such that Ci A∗ ∩ wA∗ is not empty for any word w. b) Show that the set of proper suffixes of C is the disjoint union of the Ci ’s. c) Deduce that C i − 1 = Pi (A − 1) and that n  X Pi (A − 1) . C − 1 = n(A − 1) + (A − 1) i=1

3708 3709

3710 3711 3712 3713 3714 3715 3716 3717 3718 3719 3720 3721 3722

Show that n is the degree of C. Show that it is also equal to the average length of C (cf. Perrin 1977).

Notes to Chapter XI Theorem 4.1 is a non commutative generalization of a theorem due Sch¨ utzenberger (1965). Corollary 4.2 is a partial answer to the main conjecture in the theory of finite codes, the factorization conjecture which states that P and S may be chosen to have nonnegative coefficients (or equivalently coefficients 0 and 1). Finite complete codes are maximal codes, and conversely, every maximal code is complete. Most of the general results on codes are stated here in the finite case. However, they hold for rational and even for thin codes. For a general exposition of the theory of codes, see the book by Berstel and Perrin (1985). Another illustration of the close relation between codes and formal series is the following result (roughly): a thin code is bifix if and only if its syntactic algebra is semisimple (Reutenauer 1981, Berstel and Perrin 1985).

3723

3724

3725

Chapter XII

Semisimple Syntactic Algebras

3728

This chapter has two appendices, one on semisimple algebras and another on simple semigroups, where we have collected the results which are needed and which are not proved here. We use the symbols A1 and A2 to refer to them.

3729

1

3726 3727

3730 3731 3732 3733 3734 3735 3736 3737 3738 3739

Bifix codes

Let E be a set of endomorphisms of a finite dimensional vector space V . Recall that E is called irreducible if there is no subspace of V other than 0 and V itself which is invariant under all endomorphisms in E. Similarly, we say that E is completely reducible if V is a direct sum V = V1 ⊕ · · · ⊕ Vk of subspaces such that for each i, the set of induced endomorphisms e|Vi , for e ∈ E, of Vi is irreducible. A set of matrices in K n×n (K being a field) is irreducible (resp. completely reducible) if it is so, viewed as a set of endomorphisms acting at the right on K 1×n , or equivalently at the left on K n×1 (for this equivalence, see Exercises 1.1 and 1.2). A linear representation (λ, µ, γ) of a series S ∈ KhhAii is irreducible (resp. completely reducible) if the set of matrices {µa | a ∈ A} (or equivalently µA∗ or µ(KhAi)) is so. By a change of basis, we see that (λ, µ, γ) is completely reducible if and only if it is similar to a linear representation which has a block diagonal form

λ = (λ1 , . . . , λk ),

3740 3741 3742



µ1 0   µ =  ...  0 0

0 ··· µ2 0 .. .. .. . . . 0 µk−1 ··· 0

 0 0  ..  , .   0 µk



γ1



      γ =  ...      γk

where each representation (λi , µi , γi ) is irreducible. Recall that codes, bifix codes and complete codes have been defined in Section XI.1 and XI.2. We assume that K is a field of characteristic 0. 189

190 3743 3744 3745 3746

Chapter XII. Semisimple Syntactic Algebras

Theorem 1.1 Let C be a rational code and let S be the characteristic series of C ∗ . Let ρ = (λ, µ, γ) be a minimal representation of S. (i) If C is bifix, then ρ is completely reducible. (ii) If C is complete and ρ is completely reducible, then C is bifix.

3748

An equivalent formulation of this result is the following. For semisimple algebras, see A2.

3749 3750

Corollary 1.2 Let C and S be as in the theorem and let A be the syntactic algebra of S.

3747

3751 3752 3753 3754 3755 3756 3757 3758 3759 3760 3761 3762 3763 3764 3765 3766 3767 3768 3769 3770 3771 3772 3773 3774 3775 3776 3777 3778 3779 3780 3781

(i) If C is bifix, then A is semisimple. (ii) If C is complete and A is semisimple, then C is bifix. We thus obtain that a complete rational code C is bifix if and only if the syntactic algebra of C ∗ is semisimple. Proof. Let ρ = (λ, µ, γ) be as in the theorem. Then A = µ(KhAi) is isomorphic to the syntactic algebra of S by Corollary II.2.2. Evidently, A acts on K 1×n , and it acts faithfully. Thus statement (i) follows from Theorem 1.1(i) and from A1.5. For (ii), we use A1.6.  For the proof of Theorem 1.1 we need a lemma. Lemma 1.3 Let C, S, ρ be as in the theorem. Then in the finite monoid M = µ(A∗ ), there is a finite group G, with neutral element e, such that e ∈ µ(C ∗ ) and that • if M has no zero, then eM e = G; • if M has a zero, then e 6= 0 and eM e = G ∪ 0. Proof. By Propositions III.3.1 and III.3.2, M is the syntactic monoid of C ∗ and is finite. If M has no zero, let J be its minimal ideal. If M has a zero, let J be a 0minimal ideal. For these notions, see A2.1 and A2.2. In both cases, Card J ≥ 2. Hence µ(C ∗ ) intersects J since otherwise we obtain a coarser congruence than the syntactic congruence by taking µ−1 (J) as a single equivalence class. If M has a zero, µ(C ∗ ) does not contain it. Indeed, if 0 = µ(w) for some w ∈ C ∗ , then for any letter a, one has 0 = µ(aw) = µ(wa), hence w, wa, aw ∈ C ∗ and by Exercise 1.4, a ∈ C ∗ . Thus C = A and M = {1} which would yield 1 = 0, a contradiction. We conclude that in both cases (zero or not) some element and its powers are in µ(C ∗ ) ∩ J and are nonzero. Hence, there is some nonzero idempotent e in µ(C ∗ ) ∩ J and the lemma follows from the Rees matrix representation of J, see A2.4.  Proof of Theorem 1.1. (i) Let the algebra A = µ(KhAi) act on the right on V = K 1×n . In view of Exercise 1.3, it is enough to show that each subspace W of V which is invariant under A has a supplementary space W ′ which is also invariant. With the notations of Lemma 1.3, in particular M = µ(A∗ ), define the subspace E = {ve | v ∈ V } of V . Set F = W ∩ E. If g ∈ G, then W g ⊂ W (W being invariant under A) and g = ge, hence Eg = Ege ⊂ E. This implies that

1. Bifix codes

191

F is invariant under G. By Maschke’s theorem A1.7, there exists a G-invariant subspace F ′ of E such that E is the direct sum over K of F and F ′ . Let W ′ = {v ∈ V | vM e ⊂ F ′ } . 3782 3783 3784 3785 3786 3787 3788 3789 3790 3791 3792 3793 3794 3795 3796 3797 3798 3799 3800 3801

We show that W ′ is a subspace of V , supplementary of W and invariant under A. First, it is invariant, since for m in M , the inclusion vM e ⊂ F ′ implies vmM e ⊂ F ′ . We claim that λ ∈ E. This will imply that λ = t + t′ for some t ∈ F, t′ ∈ F ′ . Since F ⊂ W and F ′ ⊂ W ′ (indeed, t′ ∈ F ′ implies t′ ∈ E, and therefore t′ = t′ e from which t′ M e = t′ eM e ⊂ F ′ G ⊂ F ′ , thus t′ ∈ W ′ ), we obtain λ ∈ W + W ′ . Since these two subspaces are invariant and since λA = V (Proposition II.2.1), we obtain that V = W + W ′ . In order to prove the claim, it suffices to show that λ = λe. We know that e = µ(w) for some w ∈ C ∗ . Since C is a prefix code, we have u ∈ C ∗ ⇐⇒ wu ∈ C ∗ for any word u ∈ A∗ (see Exercise 1.5). Thus (S, u) = (S, wu) and therefore (S, (1 − w)u) = 0. This implies that for any P in KhAi, one has 0 = (S, (1−w)P ) = (S ◦(1−w), P ). We obtain that 1−w is in the right syntactic ideal of S (Proposition II.1.4) and therefore λµ(1 − w) = 0 (Proposition II.2.1), and finally λ = λe. It remains to show that W ∩ W ′ = 0. For this, consider a vector in W ∩ W ′ . By Proposition II.2.1, it is of the form λµP for some P in KhAi. If m ∈ M , then λµP me ∈ E ∩ W = F since W is stable and by definition of E. Moreover, λµP me ∈ F ′ since λµP ∈ W ′ and by definition of W ′ . Thus λµP me ∈ F ∩F ′ = 0. Finally, since C is a suffix code, we have (S, u) = (S, uw) for any word u, and w as above. Thus, for Q ∈ KhAi, we have (S, Q) = (S, Qw) or equivalently λµQγ = λµQµwγ. We deduce that for any word u, λµP µuγ = λµP µuµwγ = λµP meγ = 0

3802 3803 3804 3805 3806 3807

by the preceding argument and with m = µu. Since the µuγ span K n×1 , we conclude by Proposition II.2.1 that λµP = 0. (ii) It is enough, by left-right symmetry, to show that C is prefix. By Lemma III.1.3, we know that M = µ(A∗ ) is a finite monoid. Since C is complete, C ∗ intersects each ideal in A∗ , hence µ(C ∗ ) intersects the minimal ideal L of M . Let V = K 1×n , with its right action of A = µ(KhAi). Let W be the subspace of V composed of the elements v in V such that vHγ = vKγ for any maximal subgroups H,P K in L contained in the same minimal left ideal of M , where we write H for m∈H m. The subspace W is invariant under M , hence under A. Indeed, if v ∈ W and m ∈ M , then for any H, K as above, mH and mK are maximal subgroups of the same minimal left ideal contained in L, by A2.4 and A2.5, and the mapping h 7→ mh is a bijection H → mH. Consequently vmHγ = vmHγ = vmKγ = vmKγ ,

3808 3809 3810 3811

which implies that vm ∈ W . Observe that for any m in L and v in V , one has vm ∈ W . This is because for any maximal subgroups H, K contained in the same minimal left ideal of M , one has mH = mK (see A2.4 and A2.5).

192

Chapter XII. Semisimple Syntactic Algebras

Since V is completely reducible, we know by A1.3 that V = W ⊕ W ′ for some stable subspace W ′ . Let λ = v + v ′ with v ∈ W, v ′ ∈ W ′ . Let H, K be as before. Then λHγ − λKγ = vHγ − vKγ + v ′ Hγ − v ′ Kγ = v ′ Hγ − v ′ Kγ 3812 3813 3814 3815 3816 3817 3818 3819 3820

since v is in W . By our previous observation, v ′ H and v ′ K are in W . Since they are also in W ′ , they vanish, hence λHγ = λKγ. This shows that if µ(C ∗ ) intersects some maximal subgroup of a minimal left ideal, then it intersects each such maximal subgroup. In other words, µ(C ∗ ) intersects each minimal right ideal of M (see A2.3, A2.4 and A2.5). Applying A2.4, we have L = I × G × J and by Exercise 1.6, L ∩ µ(C ∗ ) = I1 × H × J1 , where H is a subgroup of G and I1 ⊂ I, J1 ⊂ J. In fact, by what we have just said, we must have I = I1 . Moreover, pj,i ∈ H for j ∈ J1 , i ∈ I1 . By Exercise 1.5, C is a prefix code, if we establish that for any words u, v, u, uv ∈ C ∗ implies v ∈ C ∗ . Since the syntactic congruence of C ∗ saturates C ∗ , and in view of Proposition III.3.2, it suffices to show that for any m, n in M , m, mn ∈ µ(C ∗ ) =⇒ n ∈ µ(C ∗ ). By multiplying m on the left by some element in L ∩ µ(C ∗ ), we may assume that m ∈ L. We may write m = (i, h, j) for some i ∈ I, h ∈ H, j ∈ J1 and mn ∈ L ∩ µ(C ∗ ). Now nm ∈ L and is a left multiple of m; hence it is in the same minimal left ideal as m and therefore, by A2.5, nm = (i′ , g, j) with i′ ∈ I, g ∈ G. Thus (i, hpj,i′ g, j) = (i, h, j)(i′ , g, j) = mnm ∈ L ∩ µ(C ∗ ) .

3821 3822

Thus hpj,i′ g ∈ H, which implies g ∈ H. We conclude that m, mn and nm are all in µ(C ∗ ) and therefore n ∈ µ(C ∗ ) by Exercise 1.4. 

3823

2

3824

A language L ⊂ A∗ is called cyclic if it has the following two properties:

3825 3826

Cyclic languages

(i) for any words u, v ∈ A∗ , uv ∈ L ⇐⇒ vu ∈ L. (ii) for any nonempty word w and any integer n ≥ 1, w ∈ L ⇐⇒ wn ∈ L.

Given a finite deterministic automaton A over A, we call character of A, denoted by χA , the formal series X χA = αw w , w∈A∗

3827 3828 3829 3830 3831 3832 3833 3834 3835

where αw is the number of closed paths labeled w in A. Recall that a 0, 1-matrix is a matrix with entries equal to 0 or 1, and that a row-monomial matrix is a matrix having at most one nonzero entry in each row. A series is the character of some finite deterministic automaton if and only if there is a representation µ of A∗ by row-monomial 0, 1-matrices such that this P series is equal to w∈A∗ tr(µw)w. This follows from the equivalence between automata and linear representations, see Section I.7. Theorem 2.1 The characteristic series of a rational cyclic language is a Zlinear combination of characters of finite deterministic automata.

193

2. Cyclic languages 3836 3837 3838

Corollary 2.2 The syntactic algebra over a field K of a rational cyclic language is semisimple. This will follow from the theorem and the next lemma. Lemma 2.3 Let µ1 , . . . , µk be linear representations of A∗ , let α1 , . . . , αk ∈ K and let S be the series defined by X S= αi tr(µi w) . 1≤i≤k

3839

Then the syntactic algebra of S is semisimple.

3840

Proof. We may assume that each representation is irreducible. Indeed, if µi is reducible, we put it, by an appropriate change of basis, into block-triangular form with each block irreducible, and then, keeping only the diagonal blocks, into block-diagonal form. These transformations do not change the trace. Since the trace of a diagonal sum is the sum of the traces of the blocks, we obtain the desired form. Consider now the algebra

3841 3842 3843 3844 3845

A = {(µ1 P, . . . , µk P ) | P ∈ KhAi} . 3846 3847 3848 3849 3850 3851 3852 3853 3854

It acts faithfully on the right on K 1×n , where n is of the appropriate size; moreover K 1×n is completely reducible under this action. Hence A is semisimple by A1.5. There is a surjective algebra morphism µ : KhAi → A, namely µ = (µ1 , . . . , µk ), and a linear mapping ϕ : A → K such that (S, w) = ϕ(µw). Hence, by Lemma II.1.1, the syntactic algebra of S is a quotient of A, hence is semisimple by A1.1.  Corollary 2.2 follows from Lemma 2.3 because of the trace form of the character of an automaton seen above. Let L be a language and let an be the number of words of length n in L. The zeta function of L is the series  X xn  an ζL = exp . n n≥1

3855 3856

Corollary 2.4 Let L be a rational cyclic language. Then its zeta function is rational. Proof. Let A be a finite deterministic automaton with associated representation µ : A∗ → Zn×n , see the remark before Theorem 2.1. Then the character of A is X tr(µw) w . w∈A∗

P n Thus, setting an = |w|=n tr(µw), we obtain an = tr(M ), where M =
P a∈A µa . It follows that k  X xn   X tr(M n )  X X λni n  ζA = exp = exp xn = exp x an n n n i=1 n≥1

n≥1

n≥1

194

Chapter XII. Semisimple Syntactic Algebras

where λ1 , . . . , λk are the eigenvalues of M with multiplicities. Thus this series is equal to k  X λn xn  Y  1  i exp exp log = n 1 − λi x i=1 i=1 k Y

n≥1

= 3857 3858 3859 3860 3861 3862 3863 3864 3865 3866 3867 3868 3869 3870 3871 3872 3873 3874

k Y

1 = det(1 − M x)−1 . 1 − λ x i i=1

Since by Theorem 2.1 L is a Z-linear combination P of characters of finite deterministic automata Aj for j ∈ J, we have L = j∈J αj χAj for some αj in Z. Q α Then it is easily verified that ζL = j∈J ζAjj , which concludes the proof. 

In view of the proof of Theorem 2.1 we establish two lemmas. For this, we call permutation character of a group G a function χ : G → N, where χ(g) is the number of fixpoints of g in some action of G on a finite set. Equivalently, χ(g) = tr(µ(g)), where µ : G → Zn×n is a representation of G such that each matrix µ(g) is a permutation matrix. Lemma 2.5 Let G be a group and let θ : G → Zn×n be a multiplicative morphism such that each matrix θ(g) is a row-monomial 0, 1-matrix. Then g 7→ tr(θ(g)) is a permutation character. Proof. The row vector ei of the canonical basis of Z1×n is mapped by each g in G onto some ej or onto 0. Thus each g ∈ G induces a partial function from {1, . . . , n} into itself. These partial functions have all the same image E. The restriction of g to E is a bijection and the number of fixpoints of this bijection is tr(θ(g)).  Recall that two elements in a semigroup S are conjugate if, for some x, y in S, they may be written xy and yx.

3875 3876

Lemma 2.6 Let S be a 0-simple semigroup and let G be a maximal subgroup in S \ 0. Any element x ∈ S with x2 6= 0 is conjugate to some element in G.

3877

Proof. We use the Rees matrix semigroup form for S, see A2.4. We may therefore assume that the maximal subgroup is {(i, g, j) | g ∈ G} and that x = (i′ , g ′ , j ′ ). Since x2 6= 0, we have pj ′ ,i′ 6= 0. Similarly pj,i 6= 0. Let u = (i′ , g ′ , j) −1 ′ −1 ′ ′ and v = (i, pj,i , j ). Then uv = (i′ , g ′ pj,i p−1 j,i , j ) = x and vu = (i, pj,i pj ′ ,i′ g , j) which proves the lemma. 

3878 3879 3880 3881 3882 3883 3884 3885 3886 3887 3888 3889 3890

P We call a formal series S = w∈A∗ (S, w) cyclic if it has the following properties: (i) There is a finite monoid M , a surjective monoid morphism µ : A∗ → M and a function ϕ : M → Z such that for any word w, (S, w) = ϕ(µw). Moreover, for any group G in M , the restriction of ϕ to G is a Z-linear combination of permutation characters of G. (ii) For any words u and v, one has (S, uv) = (S, vu). (iii) For any word w, the sequence un = (S, wn+1 ) satisfies a proper linear recurrence relation (see Section VI.1).

2. Cyclic languages 3891 3892 3893 3894 3895 3896 3897 3898 3899 3900 3901 3902 3903 3904 3905 3906 3907 3908 3909 3910 3911 3912 3913 3914 3915 3916 3917 3918 3919 3920

195

Observe that a Z-linear combination of cyclic series is a cyclic series (take the product monoid) and that the character of a finite deterministic automaton is a cyclic series (use Lemma 2.5). Proof of Theorem 2.1. The proof is in two parts. First, we show that the characteristic series of a rational cyclic language is a cyclic series. Next, we prove that each cyclic series satisfies the conclusion of the theorem. This implies the theorem. 1. Let S be the characteristic series of a rational cyclic language L. Since L is recognizable by Theorem III.1.1, there is some monoid morphism µ : A∗ → M , where M is a finite monoid, and a subset P of M such that L = µ−1 (P ). We may assume that µ is surjective. Define ϕ : M → Z by ϕ(m) = 1 if m ∈ P , and ϕ(m) = 0 otherwise. Then (S, w) = ϕ(µw). If G is a group in M , then either ϕ(G) = 1 or ϕ(G) = 0. Indeed, any two elements in G have a positive power in common, namely the neutral element e of G, and we conclude according to e ∈ P or not, since L is cyclic and µ is surjective. Hence condition (i) is satisfied for S. Moreover, condition (ii) is satisfied since L is cyclic, and (iii) follows also, since un is constant, for the same reason. This proves that S is cyclic. 2. It remains to prove that each cyclic series S is a Z-linear combination of characters of finite deterministic automata. We take the notations of conditions (i),(ii) and (iii) above and prove the claim by induction on the cardinality of M . If M has a 0, we may assume that ϕ(0) = 0 by replacing ϕ by ϕ − ϕ(0) and S by S − ϕ(0)A∗ , since A∗ is evidently the character of some finite deterministic automaton. Now, let J be some 0-minimal ideal of M if M has a zero, and the minimal ideal of M if M has no zero. Note that Card J ≥ 2. Suppose that no element of J is idempotent. Then x2 = 0 for each element of J by A2.4. Hence the sequence ϕ(xn+1 ) is ϕ(x), 0, 0, . . ., and therefore by (iii) we have ϕ(x) = 0. Hence ϕ vanishes on J and we may replace M by the quotient M/J and conclude by induction. Thus we may suppose that J contains an idempotent, hence some maximal group G. By A2.6 there exists a monoid representation θ : M → (G0 )r×r where G0 is G with a zero adjoined, where each matrix is row-monomial, and where the restriction of θ to G is of the form   g 0 ··· 0 ∗ 0 · · · 0   θ(g) =  . . ..   .. .. . ∗ 0 ··· 0

3921

and moreover θ(0) = 0. Let β : G → Zd×d be a representation of G by permutation matrices. Replacing in each matrix θ(m), for m ∈ M , each nonzero entry g ∈ G by β(g), we obtain a representation ψ : M → Zdr×dr by row-monomial 0, 1-matrices. Hence X tr(ψ(µw)) w w∈A∗

3922 3923 3924

is the character of some finite deterministic automaton. If H is a group in M , then the function H → Z, h → 7 tr(ψ(h)) is a permutation character of H by Lemma 2.5.

196

Chapter XII. Semisimple Syntactic Algebras

3930 3931

Since ϕ|G is a Z-linear combination of permutation characters of G, the previous construction shows that for some Z-linear combination T of characters of finite deterministic automata, the series S ′ = S − T vanishes on G. Moreover S ′ is a cyclic series. By Lemma 2.6 it vanishes on J. Indeed, let x ∈ J. If x2 6= 0, we use this lemma and the cyclicity of S ′ . On the contrary, if x2 = 0, we use property (ii) of cyclic series together with the fact that θ(0) = 0. Thus we may replace M by the quotient M/J and conclude by induction. 

3932

Appendix 1: Semisimple algebras

3933 3934

Here, all algebras are finite dimensional over the field K. Likewise the modules over the algebras that we consider will be finite dimensional over K.

3935 3936 3937 3938

A1.1 An algebra is called simple if it has no two-sided ideal other than 0 and itself. An algebra is called semisimple if it is a finite direct product of simple algebras. It follows that a quotient of a semisimple algebra is semisimple (see Exercise 1.1).

3939 3940

A1.2 A right-module M over an algebra A is faithful if, whenever M a = 0 for some a in A, then a = 0. Similarly for left modules.

3941 3943 3944

A1.3 A module is irreducible, or simple, if it has no submodules other than 0 and itself. It is completely reducible if it is a finite direct sum of irreducible modules. A module is completely reducible if and only if each submodule has a supplementary submodule.

3945

A1.4 If an algebra has a faithful irreducible module, then this algebra is simple.

3946 3947

A1.5 If an algebra has a faithful completely reducible module, then this algebra is semisimple.

3948

A1.6 Each module over a semisimple algebra is completely reducible and this property characterizes semisimple algebras.

3925 3926 3927 3928 3929

3942

3949

3950 3951 3952

A1.7 If K is a field of characteristic 0 and G is a finite group, then the group algebra KG is semisimple. In other words, a finite group of endomorphisms of a vector space is completely reducible (Maschke’s theorem).

3955 3956

A1.8 Each simple algebra is isomorphic to a matrix algebra Dn×n , where D is a skew field containing K in its center and finite dimensional over K. In particular, if K is algebraically closed, then each simple algebra is a matrix algebra K n×n .

3957

Appendix 2: Simple semigroups

3958

All semigroups considered here are finite.

3953 3954

2. Cyclic languages 3959 3960 3961 3962 3963 3964 3965 3966 3967 3968

197

A2.1 An ideal in a semigroup S is a subset I of S such that for all s ∈ S, t ∈ I, the elements st and ts are in I. A zero in S is an element 0 such that S 6= {0} and such that {0} is an ideal. It is necessarily unique. Note that if S is a monoid, that is, has a neutral element, then the latter is 6= 0. A2.2 The minimal ideal of a semigroup S is the smallest ideal in S. It always exists. If S has a zero, a 0-minimal ideal of a semigroup S is is an ideal in S strictly containing 0, and minimal for this property. A2.3 A semigroup S is simple if it has no ideal except itself. A semigroup with zero is 0-simple if it has no ideal except {0} and itself. The minimal (resp. a 0-minimal) ideal of a semigroup is a simple (resp. a 0-simple) semigroup. A2.4 Each simple or 0-simple semigroup is isomorphic to a Rees matrix semigroup S. Such a semigroup is given by a group G, two sets of indices I and J, and a matrix P ∈ GI×J , where G0 is G with a 0 added. The matrix P is called 0 the sandwich matrix , and the elements of S are the triples (i, g, j) with i ∈ I, g ∈ G, j ∈ J together with 0 if S has a zero. The product is ( (i, gpj,i′ g ′ , j ′ ) if pj,i′ 6= 0 ′ ′ ′ (i, g, j)(i , g , j ) = 0 otherwise.

3969 3970 3971 3972 3973 3974 3975 3976

The nonzero idempotents in S are the elements e = (i, p−1 j,i , j) for any i, j with pj,i 6= 0. In this case, eSe = G′ or G′ ∪ {0}, according to 0 ∈ S or 0 ∈ / S, and G′ = {(i, g, j) | g ∈ G} is a subgroup of S isomorphic to G. It is a maximal subgroup of S, and all nonzero maximal subgroups of S are of this form. A2.5 Let M be a monoid and take a Rees matrix representation of its minimal ideal L (the latter is a simple semigroup). Then, for fixed i, the set {(i, g, j) | g ∈ G, j ∈ J} is a minimal right ideal of M , and all minimal right ideals of M are of this form. Similarly for minimal left ideals of M .

3983

A2.6 Let M be a monoid and let S be its minimal ideal if M has no zero, and a 0-minimal ideal if M has a zero. Suppose that S contains an idempotent e. Then M has a maximal subgroup G containing e, which is the neutral element of G. There exists a representation of M by square row-monomial matrices over G ∪ {0} such that the image of each g in G has nonzero coefficients only in the first column, and such that the image of 0 is the zero matrix.

3984

Exercises for Chapter XII

3977 3978 3979 3980 3981 3982

3985 3986 3987 3988 3989

1.1 Show that a set M of square matrices of order n is reducible (that is, not irreducible) if and only if for some invertible matrix g and some i, j ≥ 1 with i + j = n, the matrices gmg −1 , for m ∈ M , have all the block a b triangular form , where a (resp. b) is square of order i (resp. j). 0 c   a 0 Show that equivalently the form may be . b c

198

Chapter XII. Semisimple Syntactic Algebras

1.2 Show that a set M of square matrices is completely reducible if and only if for some invertible matrix g, the matrices gmg −1 have all the block diagonal matrix form of the same size   a1 0 . . 0  0 a2 . .   . . . . .   . . 0 0 . . 0 ak 3990 3991

1.3

3992 3993 3994 3995 3996

1.4 1.5

3997

1.6

where for each i = 1, . . . , k the induced set of matrices ai is irreducible. Show that a set of endomorphisms of a finite dimensional vector space is completely reducible if and only if for each subspace which is invariant under these endomorphisms, there is a supplementary subspace which is also invariant. Hint: use A1.3. Let C be a code. Show that if u, uv, vu ∈ C ∗ , then v ∈ C ∗ (consider uvu). Let C be a code. Show that C is prefix if and only if for any words u and v, u, uv ∈ C ∗ implies v ∈ C ∗ . Let S be the Rees matrix semigroup as in A2.4. Let T be a subsemigroup of S not containing 0. Show that for some subgroup H of G, some subsets I1 of I and J1 of J, one has T = {(i, h, j) | i ∈ I1 , h ∈ H, j ∈ J1 } ,

3998 3999 4000 4001 4002 4003 4004 4005 4006 4007 4008 4009 4010 4011 4012 4013 4014 4015 4016 4017 4018 4019 4020 4021 4022

together with the condition pj,i ∈ H for all i ∈ I1 , j ∈ J1 . 1.7 Let G be a finite group and take as alphabet A = G. Let µ : A∗ → G be the natural monoid morphism which is the identity on G. Show that µ−1 (1) = C ∗ for some rational bifix code C. Show that the syntactic algebra of C ∗ is isomorphic to the group algebra KG. 2.1 Let L be a rational language such that for any w in L, one has wn ∈ L for all n ≥ 1. Show that the cyclic closure of L (that is the smallest cyclic language containing L) is rational. A1.1 Let A, B be two algebras with A simple. Show that if I is a two-sided ideal of A × B, then either I = A × J or I = 0 × J for some ideal J of B. Deduce that each quotient of A × B is either a quotient of B or of the form A × (a quotient of B). Deduce that a quotient of a semisimple algebra is semisimple. A1.2 Let A be a subalgebra of K n×n . Show that it acts faithfully at the right on K 1×n . A1.3 Let A1 , . . . , An be simple algebras and let A be a subalgebra of A1 × · · · × An such that the projections A → Ai are surjective. Show that A is semisimple. Hint: let B be the projection of A onto A1 × · · · × An−1 . It is semisimple by induction. If A → B is not injective, then (0, . . . , 0, a) ∈ A for some a 6= 0 in An . Then 0 × · · · × 0 × An ⊂ A and finally A = B × An . A1.4 Let A act faithfully on a completely reducible module M . Using A1.4 and the previous exercise, prove A1.5. A2.1 Let L be the minimal ideal of some finite semigroup S. (i) Show that if m ∈ L and H is a maximal subgroup of L, then h 7→ mh is a bijection from H onto the maximal subgroup of L which is the

2. Cyclic languages 4023 4024 4025 4026 4027

4028 4029 4030 4031 4032

199

intersection of the minimal right ideal containing m and the minimal left ideal containing H. (ii) Show that if s ∈ S and H is as before, then sH is a maximal subgroup of L contained in the same minimal left ideal as H. Hint: sH = seH, where e is the neutral element of H, and use (i).

Notes to Chapter XII Corollary 1.2 is from (Reutenauer 1981). For the proof of the equivalent Theorem 1.1, we have followed (Berstel and Perrin 1985), Section VIII.7. Theorem 2.1 and Corollary 2.4 are from (Berstel and Reutenauer 1990). For Appendix 1, see (Lang 1984) and for Appendix 2, see (Lallement 1979).

200

References

4033

References

4034 4035

Allouche, J.-P., Shallit, J. O. The ring of k-regular sequences. Theoret. Comput. Sci., 98:163–197, 1992.

4036 4037

Allouche, J.-P., Shallit, J. O. Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, 2003.

4038 4039

Amice, Y. Les nombres p-adiques. Presses Universitaires de France, Paris, 1975. Pr´eface de Ch. Pisot, Collection SUP: Le Math´ematicien, No. 14.

4040 4041

Barcucci, E., Del Lungo, A., Frosini, A.,, Rinaldi, S. A technology for reverseengineering a combinatorial problem from a rational generating function. Adv. in Appl. Math., 26(2):129–153, 2001.

4042 4043 4044 4045 4046 4047 4048 4049 4050 4051 4052 4053 4054 4055 4056 4057 4058 4059 4060 4061 4062

Benzaghou, B. Alg`ebres de Hadamard. Bull. Soc. Math. France, 98:209–252, 1970. Bergmann, G. M. Commuting elements in free algebras and related topics in ring theory. Thesis, Harvard University, 1967. Berstel, J. Sur les pˆ oles et le quotient de Hadamard de s´eries N-rationnelles. C. R. Acad. Sci. Paris S´er. A-B, 272:A1079–A1081, 1971. Berstel, J., Mignotte, M. Deux propri´et´es d´ecidables des suites r´ecurrentes lin´eaires. Bull. Soc. Math. France, 104(2):175–184, 1976. Berstel, J., Perrin, D. Theory of codes, volume 117 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1985. Berstel, J., Reutenauer, C. Recognizable formal power series on trees. Theoret. Comput. Sci., 18:115–148, 1982. Berstel, J., Reutenauer, C. Zeta functions of formal languages. Trans. American Math. Soc., 321:533–546, 1990. Berstel, J., Reutenauer, C. Another proof of Soittola’s theorem. Theoret. Comput. Sci., 2007. to appear. B´ezivin, J.-P. Factorisation de suites r´ecurrentes lin´eaires et applications. Bull. Soc. Math. France, 112(3):365–376, 1984. Bo¨e, J. M., de Luca, A.,, Restivo, A. Minimal complete sets of words. Theoret. Comput. Sci., 12(3):325–332, 1980. 201

202

References

4065

´ ements de math´ematique. Fasc. XXX. Alg`ebre commutative. Bourbaki, N. El´ Chapitre 5: Entiers. Chapitre 6: Valuations. Actualit´es Scientifiques et Industrielles, No. 1308. Hermann, Paris, 1964.

4066 4067

Brown, T. C. An interesting combinatorial method in the theory of locally finite semigroups. Pacific J. Math., 36:285–289, 1971. ISSN 0030-8730.

4068

Brzozowski, J. A. Derivatives of regular expressions. J. Assoc. Comput. Mach., 11:481–494, 1964.

4063 4064

4069

4071

´ ements quasi-entiers et extensions de Fatou. J. Cahen, P.-J., Chabert, J.-L. El´ Algebra, 36(2):185–192, 1975.

4072 4073

Carlyle, J. W., Paz, A. Realizations by stochastic finite automata. J. Comput. System Sci., 5:26–40, 1971.

4074

Chabert, J. L. Anneaux de Fatou. Enseignement Math., 18:141–144, 1972.

4075 4076

Christol, G. Ensembles presque p´eriodiques k-reconnaissables. Theoret. Comput. Sci., 9:141–145, 1979.

4077

Christol, G., Kamae, T., Mend`es France, M.,, Rauzy, G. Suites alg´ebriques, automates et substitutions. Bull. Soc. Math. France, 108:401–419, 1980.

4070

4078

4080

Cobham, A. On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Th., 3:186–192, 1969.

4081

Cobham, A. Uniform tag sequences. Math. Systems Th., 6:164–192, 1972.

4082 4083

Cobham, A. Representation of a word function as the sum of two functions. Math. Systems Th., 12:373–377, 1978.

4084 4085

Cohen, R. S. Star height of certain families of regular events. J. Comput. System Sci., 4:281–297, 1970.

4086 4087

Cohn, P. M. On a generalization of the Euclidean algorithm. Proc. Cambridge Philos. Soc., 57:18–30, 1961.

4088

Cohn, P. M. Free associative algebras. Bull. London Math. Soc., 1:1–39, 1969.

4089

Cohn, P. M. The universal field of fractions of a semifir. I. Numerators and denominators. Proc. London Math. Soc. (3), 44(1):1–32, 1982.

4079

4090

4092 4093

Cohn, P. M. Free rings and their relations, volume 19 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1985.

4094

Connes, A. Noncommutative geometry. Academic Press Inc., 1994.

4095

Conway, J. H. Regular algebra and finite machines. Chapman and Hall, 1971.

4096 4097 4098

Cori, R. Un code pour les graphes planaires et ses applications. Soci´et´e Math´ematique de France, Paris, 1975. With an English abstract, Ast´erisque, No. 27.

4099 4100

Drensky, V. Free algebras and PI-algebras. Springer-Verlag Singapore, Singapore, 2000. ISBN 981-4021-48-2. Graduate course in algebra.

4091

References

203

4101 4102

Dubou´e, M. Une suite r´ecurrente remarquable. European J. Combin., 4(3): 205–214, 1983.

4103

Duchamp, G., Reutenauer, C. Un crit`ere de rationalit´e provenant de la g´eom´etrie non commutative. Invent. Math., 128(3):613–622, 1997.

4104 4105 4106

Eggan, L. C. Transition graphs and the star-height of regular events. Michigan Math. J., 10:385–397, 1963. ISSN 0026-2285.

4108

Ehrenfeucht, A., Parikh, R.,, Rozenberg, G. Pumping lemmas for regular sets. SIAM J. Comput., 10(3):536–541, 1981.

4109 4110

Eilenberg, S. Automata, languages, and machines. Vol. A. Academic Press, New York, 1974.

4111

Eilenberg, S., Sch¨ utzenberger, M.-P. Rational sets in commutative monoids. J. Algebra, 13:173–191, 1969.

4107

4112 4113 4114

Fatou, P. Sur les s´eries enti`eres ` a coefficients entiers. Comptes Rendus Acad. Sci. Paris, 138:342–344, 1904.

4116

Fliess, M. Formal languages and formal power series. In IRIA., editor, S´eminaire Logique et Automates, pages 77–85, Le Chesnay, 1971.

4117

Fliess, M. Matrices de Hankel. J. Math. Pures Appl. (9), 53:197–222, 1974a.

4118 4119

Fliess, M. Sur divers produits de s´eries formelles. Bull. Soc. Math. France, 102: 181–191, 1974b.

4120 4121

Fliess, M. S´eries rationnelles positives et processus stochastiques. Ann. Inst. H. Poincar´e Sect. B (N.S.), 11:1–21, 1975.

4122

Fliess, M. Fonctionnelles causales non lin´eaires et ind´etermin´ees non commutatives. Bull. Soc. Math. France, 109(1):3–40, 1981.

4115

4123

4126

Gessel, I. Rational functions with nonnegative integer coefficients. In The 50th s´eminaire Lotharingien de Combinatoire, page Domaine Saint-Jacques, march 2003. unpublished, available at Gessel’s homepage.

4127 4128

Halava, V., Harju, T.,, Hirvensalo, M. Positivity of second order linear recurrent sequences. Discrete Applied Math., 154(447-451), 2006.

4129

Hansel, G. Une d´emonstration simple du th´eor`eme de Skolem-Mahler-Lech. Theoret. Comput. Sci., 43(1):91–98, 1986.

4124 4125

4130 4131 4132

Harrison, M. A. Introduction to formal language theory. Addison-Wesley Publishing Co., Reading, Mass., 1978.

4134

Hashiguchi, K. Limitedness theorem on finite automata with distance functions. J. Comput. System Sci., 24(2):233–244, 1982. ISSN 0022-0000.

4135 4136

Herstein, I. N. Noncommutative rings. The Carus Mathematical Monographs, No. 15. Published by The Mathematical Association of America, 1968.

4137 4138

Isidori, A. Nonlinear control systems: an introduction, volume 72 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1985.

4133

204

References

4139 4140

Jacob, G. Repr´esentations et substitutions matricielles dans la th´eorie alg´ebrique des transductions. Thesis, University of Paris, 1975.

4141

Jacob, G. La finitude des repr´esentations lin´eaires des semi-groupes est d´ecidable. J. Algebra, 52(2):437–459, 1978.

4142 4143 4144 4145 4146 4147

Jacob, G. Un th´eor`eme de factorisation des produits d’endomorphismes de k n . J. Algebra, 63:389–412, 1980. Katayama, T., Okamoto, M.,, Enomoto, H. Characterization of the structuregenerating functions of regular sets and the DOL growth functions. Information and Control, 36(1):85–101, 1978. ISSN 0890-5401.

4151

Kleene, S. C. Representation of events in nerve nets and finite automata. In Shannon, C. E., McCarthy, J., editors, Automata Studies, Annals of mathematics studies, no. 34, pages 3–41. Princeton University Press, Princeton, N. J., 1956.

4152 4153

Koblitz, N. p-adic numbers, p-adic analysis, and zeta-functions, volume 58 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1984.

4154

Koutschan, C. Regular languages and their generating functions: the inverse problem. Diplomarbeit informatik, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, 2005.

4148 4149 4150

4155 4156 4157 4158 4159 4160 4161

Koutschan, C. Regular languages and their generating functions: the inverse problem. Technical report, Universit¨ at Linz, 2006. 10 pages. Krob, D. Expressions rationnelles sur un anneau. In Topics in invariant theory (Paris, 1989/1990), volume 1478 of Lecture Notes in Math., pages 215–243. Springer-Verlag, 1991.

4164

Krob, D. The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Internat. J. Algebra Comput., 4(3):405– 425, 1994. ISSN 0218-1967.

4165 4166

Kuich, W., Salomaa, A. Semirings, automata, languages, volume 5 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1986.

4167 4168

Lallement, G. Semigroups and combinatorial applications. John Wiley & Sons, New York-Chichester-Brisbane, 1979.

4169 4170

Lang, S. Algebra. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, second edition, 1984. first edition in 1965.

4171 4172

Lascoux, A. Suites r´ecurrentes lin´eaires. Adv. in Appl. Math., 7(2):228–235, 1986. ISSN 0196-8858.

4173 4174

Lascoux, A., Sch¨ utzenberger, M.-P. Formulaire raisonn´e de fonctions sym´etriques. Technical report, LITP, Universit´e Paris VII, 1985.

4175

Lech, C. A note on recurring series. Ark. Mat., 2:417–421, 1953. ISSN 0004-2080.

4176

Leung, H. On the topological structure of a finitely generated semigroup of matrices. Semigroup Forum, 37(3):273–287, 1988. ISSN 0037-1912.

4162 4163

4177

References

205

4178 4179

Lewin, J. Free modules over free algebras and free group algebras: The Schreier technique. Trans. Amer. Math. Soc., 145:455–465, 1969.

4180

Lothaire, M. Combinatorics on words, volume 17 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., 1983. ISBN 0-201-13516-7.

4181 4182

4184

Lyndon, R. C., Schupp, P. E. Combinatorial group theory. Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89.

4185 4186

Mahler, K. Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen. Akad. Wetensch. Amsterdam Proc., 38:50–60, 1935.

4187 4188

Mandel, A., Simon, J. On finite semi-groups of matrices. Theoret. Comput. Sci., 5:101–111, 1977.

4189

Manin, Y. I. A course in mathematical logic. Springer-Verlag, New York, 1977.

4190 4191

McNaughton, R., Zalcstein, Y. The Burnside problem for semigroups. J. Algebra, 34:292–299, 1975.

4192

Perrin, D. Codes asynchrones. Bull. Soc. Math. France, 105(4):385–404, 1977.

4193

Perrin, D. On positive matrices. Theoret. Comput. Sci., 94(2):357–366, 1992. Discrete mathematics and applications to computer science (Marseille, 1989).

4183

4194 4195

4197

P´ olya, G. Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen. J. reine angew. Math., 151:1–31, 1921.

4198 4199

Procesi, C. Rings with polynomial identities. Marcel Dekker Inc., New York, 1973. Pure and Applied Mathematics, 17.

4200

Restivo, A., Reutenauer, C. On cancellation properties of languages which are supports of rational power series. J. Comput. System Sci., 29(2):153–159, 1984.

4196

4201 4202 4203 4204 4205 4206 4207 4208 4209 4210 4211 4212 4213 4214 4215 4216

Reutenauer, C. Une caract´erisation de la finitude de l’ensemble des coefficients d’une s´erie rationnelle en plusieurs variables non commutatives. C. R. Acad. Sci. Paris S´er. A-B, 284(18):A1159–A1162, 1977a. Reutenauer, C. On a question of S. Eilenberg (automata, languages, and machines, vol. a, Academic Press, New York, 1974). Theoret. Comput. Sci., 5 (2):219, 1977b. Reutenauer, C. Vari´et´es d’alg`ebres et de s´eries rationnelles. In 1er Congr`es Math. Appl. AFCET-SMF, volume 2, pages 93–102. AFCET, 1978. Reutenauer, C. S´eries formelles et alg`ebres syntactiques. J. Algebra, 66(2): 448–483, 1980a. Reutenauer, C. S´eries rationnelles et alg`ebres syntactiques. Thesis, University of Paris, 1980b. Reutenauer, C. An Ogden-like iteration lemma for rational power series. Acta Inform., 13(2):189–197, 1980c.

206

References

4217 4218

Reutenauer, C. Semisimplicity of the algebra associated to a biprefix code. Semigroup Forum, 23(4):327–342, 1981.

4219

Reutenauer, C. Sur les ´el´ements inversibles de l’alg`ebre de Hadamard des s´eries rationnelles. Bull. Soc. Math. France, 110(3):225–232, 1982.

4220 4221 4222 4223 4224

Reutenauer, C. Noncommutative factorization of variable-length codes. J. Pure Appl. Algebra, 36(2):167–186, 1985. Reutenauer, C. Inversion height in free fields. Selecta Math. (N.S.), 2(1):93–109, 1996.

4226 4227

Rowen, L. H. Polynomial identities in ring theory, volume 84 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. ISBN 0-12-599850-3.

4228 4229

Ryser, H. J. Combinatorial mathematics. The Carus Mathematical Monographs, No. 14. Published by The Mathematical Association of America, 1963.

4230

Sakarovitch, J. Elements of Automata Theory. Cambridge University Press, 2007. to appear.

4225

4231 4232 4233 4234 4235

Salomaa, A., Soittola, M. Automata-theoretic aspects of formal power series. Springer-Verlag, New York, 1978. Sch¨ utzenberger, M.-P. Sur certains sous-mono¨ıdes libres. Bull. Soc. Math. France, 93:209–223, 1965.

4237

Sch¨ utzenberger, M. P. On the definition of a family of automata. Information and Control, 4:245–270, 1961a.

4238 4239

Sch¨ utzenberger, M. P. On a special class of recurrent events. Ann. Math. Statist., 32:1201–1213, 1961b.

4240 4241

Sch¨ utzenberger, M.-P. On a theorem of R. Jungen. Proc. Amer. Math. Soc., 13:885–890, 1962a. ISSN 0002-9939.

4242 4243

Sch¨ utzenberger, M.-P. Finite counting automata. Information and Control, 5: 91–107, 1962b. ISSN 0890-5401.

4244

Sch¨ utzenberger, M.-P. On a theorem of R. Jungen. Proc. Amer. Math. Soc., 13:885–890, 1962c. ISSN 0002-9939.

4236

4245 4246 4247 4248 4249 4250 4251 4252 4253 4254 4255 4256

Sch¨ utzenberger, M.-P. Parties rationnelles d’un mono¨ıde libre. In Proc. Intern. Math. Conf., volume 3, pages 281–282, 1970. Sch¨ utzenberger, M.-P., Marcus, R. S. Full decodable code-word sets. IRE Trans., IT-5:12–15, 1959. Simon, I. Limited subsets of a free monoid. In 19th Annual Symposium on Foundations of Computer Science (Ann Arbor, Mich., 1978), pages 143– 150. IEEE, Long Beach, Calif., 1978. Simon, I. Recognizable sets with multiplicities in the tropical semiring. In Mathematical foundations of computer science, 1988 (Carlsbad, 1988), volume 324 of Lecture Notes in Comput. Sci., pages 107–120. Springer, Berlin, 1988.

References

207

4257 4258

Simon, I. On semigroups of matrices over the tropical semiring. RAIRO Inform. Th´eor. Appl., 28(3-4):277–294, 1994. ISSN 0988-3754.

4259 4261

Skolem, T. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen. C. R. 8e Congr. Scand. Stockholm, pages 163–188, 1934.

4262 4263

Soittola, M. Positive rational sequences. Theoret. Comput. Sci., 2(3):317–322, 1976. ISSN 0304-3975.

4264

Sontag, E. D. On some questions of rationality and decidability. J. Comput. System Sci., 11(3):375–381, 1975. ISSN 0022-0000.

4260

4265

4267

Sontag, E. D., Rouchaleau, Y. Sur les anneaux de Fatou forts. C. R. Acad. Sci. Paris S´er. A-B, 284(5):A331–A333, 1977.

4268 4269

Steyaert, J.-M., Flajolet, P. Patterns and pattern-matching in trees: an analysis. Inform. and Control, 58(1-3):19–58, 1983. ISSN 0019-9958.

4266

4270 4271

¨ Suschkewitsch, A. K. Uber die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit. Math. Ann., 99:30–50, 1928.

4273

Turakainen, P. A note on test sets for ⋉-rational languages. Bull Europ. Assoc. Theor. Comput. Sci., 25:40–42, 1985.

4274 4275

Wedderburn, H. M. Noncommutative domains of integrity. J. reine angew. Math., 167:129–141, 1932.

4272

208

Index

Index 4276

0–minimal ideal, 197

4277 4278 4279 4280 4281 4282 4283 4284 4285 4286 4287 4288 4289 4290 4291 4292 4293 4294 4295 4296 4297 4298 4299 4300 4301 4302

absolute star height, 67 adjoint morphism, 28 admissible – polynomial identity, 116 algebra group –, 90 Hadamard –, 54, 90 monoid –, 46 polynomial identity, 116 semisimple, 196 simple –, 196 syntactic –, 26 syntactic –, 38, 46 algebraic series, 78 algorithm – of Sardinas and Patterson, 173 reduction –, 37 alphabet, 2 annihilator, 99 automaton, 28 associated representation, 15 character of an –, 192 deterministic –, 14 weighted –, 13 average length, 183

4303 4304 4305

Bernoulli morphism, 176 bifix code, 173 Boolean semiring, 2

4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316 4317

cancellative – right module, 162 Cauchy product, 74 character permutation, 194 character of an automaton, 192 characteristic – series, 6 – zero, 90 code, 171 codimension, 29 coefficient, 2

4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340 4341 4342 4343 4344 4345 4346

commutative semiring, 1 companion matrix, 106 complete – language, 175 – topological space, 4 completely integrally closed, 110 completely reducible module, 196 representation, 189 set of endomorphisms, 189 set of matrices, 189 composition, 152 congruence monoid –, 41 saturating, 41 semiring, 18 syntactic, 46 conjecture, 54, 107, 188 conjugate, 187 conjugate elements, 194 constant term, 4, 57, 162 content of a polynomial, 167 continuant polynomial, 158 continuous fraction, 170 cycle complexity, 62–64 cyclic language, 192 cyclic series, 194 cyclic closure – of a language, 198

4347 4348 4349 4350 4351 4352 4353 4354 4355 4356 4357 4358 4359 4360

degree – of a code, 180 – of a polynomial, 3, 155 – of a polynomial identity, 116 – of growth, 140 matrix polynomial, 146 denominator minimal –, 86 dense, 4 dependent, 156 deterministic automaton, 14 digit, 69 dimension of a linear representation, 8

209

Index 4361 4362 4363 4364 4365 4366 4367

discrete topology, 3 distance ultrametric –, 3 divisible – module, 144 dominating coefficient, 128 dominating eigenvalue, 127

4368 4369 4370 4371 4372 4373 4374 4375 4376 4377 4378 4379 4380 4381 4382

eigenvalue dominating –, 127 eigenvalues of a rational series, 86 Eisenstein’s criterion, 21 endomorphisms completely reducible set of, 189 irreducible set of, 189 equality set, 55 Euclidean, 155 – algorithm, 155 – division, 157 exponential polynomial, 90 extension Fatou –, 113

4383 4384 4385 4386 4387 4388 4389 4390 4391 4392 4393 4394 4395 4396 4397 4398 4399 4400 4401 4402 4403 4404 4405

factorization conjecture, 188 faithful module, 196 family locally finite –, 4 summable –, 4 Fatou – extension, 113 – lemma, 110 strong – ring, 121 weak – ring, 118 finitely generated – Abelian group, 93 – module, 8 fir, 170 formal series, 2 free – ideal ring, 170 – monoid, 2 full matrix, 164 function identity –, 70 k-regular – , 70 sum of digits –, 70

4406 4407 4408 4409 4410 4411

Gauss’s lemma, 167 generating function, 44 generic matrix, 64 geometric series, 30, 92 group algebra, 90 group-like series, 39

4412 4413 4414

growth degree of –, 140 polynomial –, 125, 140

4415 4416 4417 4418 4419 4420 4421

Hadamard – algebra, 90 – product, 12 Hankel – -like property, 23 – matrix, 29, 88 height function, 62

4422 4423 4424 4425 4426 4427 4428 4429 4430 4431 4432 4433 4434 4435 4436 4437 4438 4439 4440 4441 4442 4443 4444

ideal, 197 – in a monoid, 179 0-minimal –, 197 minimal –, 179, 197 syntactic –, 26 syntactic right –, 28 ideal of rational identities, 60 idempotent, 54, 179 identity function, 70 image of a series, 45 inertia theorem, 165 integral – element in an algebra, 111 integral domain, 155 integral part of a rational fraction, 89 invertible series, 5 irreducible – module, 196 representation, 189 set of endomorphisms, 189 set of matrices, 189 irreducible set of matrices, 137

4445 4446 4447 4448 4449 4450 4451 4452

k-automatic sequence, 76 k-kernel, 76 k-recognizable set, 70 k-regular – function, 70 – sequence, 70 kernel, 76 Kimberling function, 81

4453 4454 4455 4456 4457 4458 4459 4460 4461 4462

language, 2, 41 cyclic, 192 cyclic closure of –, 198 limited –, 147 proper –, 43 rational –, 41 recognizable –, 41 syntactic algebral of a –, 46 leap-frog construction, 159 length

210 4463 4464 4465 4466 4467 4468 4469 4470 4471

– of a word, 2 average –, 183 letter, 2 limited language, 147 linear recurrence relation, 36, 86 linear representation, 8 locally finite – semigroup, 152 locally finite family, 4

4472 4473 4474 4475 4476 4477 4478 4479 4480 4481 4482 4483 4484 4485 4486 4487 4488 4489 4490 4491 4492 4493 4494 4495 4496 4497 4498 4499 4500 4501 4502 4503 4504 4505

matrix proper –, 16 Hankel –, 29 proper, 58 row-monomial, 192 star of a –, 16 0, 1, 192 matrix polynomial, 145 measure, 176 merge, 92 message, 171 minimal – automaton, 28 – denominator, 86 – polynomial, 86 minimal ideal, 197 module, 8 completely reducible –, 196 divisible –, 144 faithful –, 196 finitely generated –, 8 irreducible –, 196 simple –, 196 monoid, 1 – algebra, 46 free –, 2 syntactic –, 46 morphism – of formal series, 19 – of semiring, 2 morphic –, 82 purely morphic –, 82 uniform –, 82 multiplicity of an eigenvalue, 86

4506 4507 4508

next function, 62 Noetherian ring, 18 normalized, 86

4509 4510

open peoblem, 107 open problem, 56, 121

4511 4512 4513

p-adic valuation, 94 palindrome, 38, 48 paper-folding sequence, 77

Index 4514 4515 4516 4517 4518 4519 4520 4521 4522 4523 4524 4525 4526 4527 4528 4529 4530 4531 4532 4533 4534 4535 4536 4537 4538 4539 4540 4541 4542 4543 4544 4545 4546 4547 4548 4549 4550 4551 4552 4553

pattern – of a ray, 116 periodic purely –, 99 quasi –, 99 permutation character, 194 Perrin companion matrix, 130 poles, 86 P´ olya series, 106 polynomial, 3 – growth, 125, 140 exponential –, 90 matrix –, 145 minimal –, 86 support of an exponential –, 91 polynomial identity, 116 admissible –, 116 degree of a –, 116 polynomially bounded series, 125, 140 Post correspondence problem, 55 prefix – -closed, 33 – code, 173 – set, 33 prime factors of a series, 94 prime subsemiring, 18 primitive polynomial, 167 product – of languages, 41 – of series, 3 Hadamard –, 12 proper – language, 43 – linear recurrence relation, 88 – matrix, 16 – series, 4 purely periodic, 99

4554 4555 4556 4557 4558

quasi-integral, 110 quasi-periodic, 99 quasi-power, 50 quasi-regular, 20 quotient of a semiring, 18

4559 4560 4561 4562 4563 4564 4565

rank – of a series, 29 of a matrix, 29 rational – closure, 5 – language, 41 – operations, 5

211

Index 4566 4567 4568 4569 4570 4571 4572 4573 4574 4575 4576 4577 4578 4579 4580 4581 4582 4583 4584 4585 4586 4587 4588 4589 4590 4591 4592 4593 4594 4595 4596 4597 4598 4599 4600 4601 4602 4603 4604 4605 4606 4607 4608 4609 4610

– series, 5 R+ - – function, 123 unambiguous – operations, 106 rational expression constant term, 57 star height, 57 rational identity, 58 rationally – closed, 5 – separated, 54 ray, 116, 140 pattern of a –, 116 reciprocal polynomial, 87 recognizable – language, 41 – series, 8 reduced linear representation, 30 reduction algorithm, 37 Rees matrix semigroup, 197 regular – linear representation, 88 – rational series, 88 – semiring, 19 representation – of an integer, 69 associated automaton, 15 canonical –, 69 completely reducible, 189 dimension of a linear –, 8 irreducible, 189 linear –, 8 reduced linear –, 30 regular – of a monoid, 42 reverse –, 82 tree –, 34 representations similar –, 143 reversal, 38 reverse representation, 82 right complete, 33 ring Noetherian, 18 weak Fatou –, 118 row-monomial matrix, 192

4611 4612 4613 4614 4615 4616 4617 4618

sandwich matrix, 197 saturating congruence, 41 Schreier’s formula, 25 semigroup, 1 locally finite –, 152 simple –, 197 torsion –, 139 semiring, 1

4619 4620 4621 4622 4623 4624 4625 4626 4627 4628 4629 4630 4631 4632 4633 4634 4635 4636 4637 4638 4639 4640 4641 4642 4643 4644 4645 4646 4647 4648 4649 4650 4651 4652 4653 4654 4655 4656 4657 4658 4659 4660 4661 4662 4663 4664 4665 4666 4667 4668 4669 4670 4671 4672

– morphism, 2 Boolean –, 2 prime –, 18 regular –, 19 simplifiable –, 19 topological –, 4 tropical–, 148 semisimple, 188 semisimple algebra, 196 separated rationally –, 54 sequence k-automatic, 76 k-regular –, 70 Thue-Morse, 76 series – recognized, 8 algebraic –, 78 characteristic – of a language, 6 cyclic, 194 formal –, 2 morphism of formal –, 19 polynomially bounded –, 125, 140 proper –, 4 rational –, 5 recognizable –, 8 shuffle product, 21 similar – representations, 143 similar linear representations, 31 simple – elements, 91 – module, 196 – semigroup, 197 – set of recognizable series, 94 simple algebra, 196 simplifiable semiring, 19 Soittola denominator, 129 stable, 9 – submodule, 71 star – height, 5 – of a matrix, 16 – of a series, 4 star height – of a rational expression, 57 absolute –, 67 sub-invertible, 54 submodule, 8 stable –, 71 subsemiring, 1 suffix

212 4673 4674 4675 4676 4677 4678 4679 4680 4681 4682 4683 4684 4685 4686 4687 4688 4689 4690

– -closed, 36 – set, 37 suffix code, 173 sum of digits function, 70 summable family, 4 support – of a series, 2 – of an exponential polynomial, 91 synchronizing, 181 syntactic – algebra, 26, 38, 46 – ideal, 26 – monoid, 46 – right ideal, 28 syntactic algebra – of a language, 46 syntactic congruence, 46

4691 4692 4693 4694 4695 4696 4697 4698 4699 4700

thin, 188 Thue-Morse sequence, 76, 82 topological semiring, 4 torsion element, 139 torsion semigroup, 139 torsion-free, 110 tree representation, 34 trivial relation, 169 trivially a polynomial, 165 tropical semiring, 148

4701 4702 4703 4704 4705

ultrametric distance, 3 unambiguous rational operations, 106, 173 uniform Bernoulli morphism, 176 uniform morphism, 82

4706 4707 4708 4709 4710

weak algorithm, 155 weak Fatou ring, 118 weighted automaton, 13 word, 2 empty –, 2

4711 4712 4713

zero, 197 0, 1-matrix, 192 zeta function, 193

Index