the rst section are dened words, free monoids, and some terms about words,

Transcription

1 1 CHAPTER 1 Words 1.0. Introduction This chapter contains the main denitions used in the rest of the book. It also presents some basic results about words that are of constant use in the sequel. In the rst section are dened words, free monoids, and some terms about words, such as length and factors. Section 1.2 is devoted to submonoids and to morphism of free monoids, one of the basic tools for words. Many of the proofs of properties of words involve a substitution from the alphabet into words over another alphabet, which is just the denition of a morphism of free monoids. A nontrivial result called the defect theorem is proved. The theorem asserts that if a relation exists among words in a set, those words can be written on a smaller alphabet. This is a weak counterpart for free monoids of the Nielsen-Schreier theorem for subgroups of a free group. In Section 1.3 the denition of conjugate words is given, together with some equivalent characterizations. Also dened are primitive words, or words that are not a repetition of another word. A very useful result, due to Fine and Wilf, is proved that concerns the possibility of multiple repetitions. The last section introduces the notation of formal series that deal with linear combinations of words, which will be used in Chapters??-?? and??. A list of problems, some of them dicult, is collected at the end. Two of them (1.1.2 and 1.2.1) deal with free groups their object is to point out the existence of a combinatorial theory of words in free groups, although the theory is not developed in the present book (see Lyndon and Schupp 1977). Two others (1.1.3 and 1.3.5) deal with the analysis of algorithms on words Free Monoids and Words Let A be a set that we shall call an alphabet. Its elements will be called letters. (In the development of this book, it will often be necessary to suppose that the alphabet A is nite. Because this assumption is not always necessary, however, it will be mentioned explicitly whenever it is used.)

2 2 Words 1.1 Aword over the alphabet A is a nite sequence of elements of A: (a 1 a 2 ::: a n ) a i 2 A: The set of all words over alphabet A is denoted by A. It is equipped with a binary operation obtained by concatenating two sequences. (a 1 a 2 ::: a n )(b 1 b 2 ::: b m )=(a 1 a 2 ::: a n b 1 b 2 ::: b m ): This binary operation is obviously associative, which allows writing a word as instead of a 1 a 2 a n (a 1 a 2 ::: a n ) by identifying a letter a 2 A with the sequence (a). The empty sequence, called the empty word, is a neutral element for the operation of concatenation. It is denoted by 1 hence, for any word w 1w = w1 =w: A monoid is a set M with a binary operation that is associative and has a neutral element denoted by 1 M. Hence, what has been dened on the set A is a monoid structure. A morphism of a monoid M into a monoid N is a mapping ' of M into N compatible with operations of M and N: '(mm 0 )='(m)'(m 0 ) m m 0 2 M and such that '(1 M )=1 N : Proposition For any mapping of A into a monoid M, thereexistsa unique morphism ' of monoids from A into M such that the following diagram is commutative: i A - A Z ZZ ' Z Z~? M where i is the natural injection of A into A. Proof. Left to the reader. Because of this property (called a universal property), the set A of all words over the alphabet A is called the free monoid over the set A. The set of all nonempty words over A will be denoted by A + : A + = A ; 1:

3 1.1. Free Monoids and Words 3 It is called the free semigroup over A (recall that a semigroup is a set with an associative binary operation). It may be readily veried that Proposition can be stated for A + instead of A by replacing the term \monoids" by \semigroups". As for any monoid the binary operation of A may be extended to the subsets of A by dening for X Y A. XY = fxy j x 2 X y 2 Y g : We shall come back to this extension in Section 1.4. Consider now some terminology about words. The length of the word w = a 1 a 2 a n, a i 2 A is the number n of the letters w is a product of. It will be denoted by jwj: jwj = n: The length of the empty word is 0 and the mapping w 7! jwj is a morphism of the free monoid A onto the additive monoid N of positive integers. For a subset B of the alphabet A, we denote by jwj B the number of letters of w that belong to B. Therefore, X jwj = jwj a : a2a Denoted by alph(w) is the subset of the alphabet formed by the letters actually occurring in w. Therefore a 2 A belongs to alph(w) i jwj a 1 : Aword v 2 A is said to be a factor of a word x 2 A if there exist words u w 2 A such that x = uvw : The relation \v is factor of x" isanorderona. A factor v of x 2 A is said to be proper if v 6= x. Aword v issaidtobealeft factor of x 2 A if there exists a word w 2 A such that x = vw anditissaidtobeaproper left factor if v 6= x. The relation \v is a left factor of x" is again an order on A it will be denoted by v x: This order has the fundamental property that if v x v 0 x then v and v 0 are comparable: v v 0 or v 0 v.

4 4 Words 1.2 More precisely, if vw = v 0 w 0 either there exists s 2 A such that v = v 0 s (and then sw = w 0 )orthereexists t 2 A such thatv 0 = vt (and then w = tw 0 ). This will be referred to as the property of equidivisibility of the free monoid. The denition of a right factor is symmetrical to that of a left factor. The reversal of a word w = a 1 a 2 a n, a i 2 A, istheword ~w = a n a 2 a 1 : Hence v is a left factor of x i ~v is a right factor of ~x. We shall also use the notation w instead of ~w wemay then write or all u v 2 A +, (uv) =~v~u: Aword w is palindrome if w =~w. Aword v 2 A is said to be a subword of a word x 2 A if v = a 1 a 2 a n a i 2 A n 0 and there exist y 0 y 1 ::: y n 2 A such that x = y 0 a 1 y 1 a 2 a n y n : Therefore v is a subword of x if it is a sub-sequence of x Submonoids and Morphisms A submonoid of a monoid M is a subset N of M containing the neutral element of M and closed under the operation of M : NN N. Given a subset X of the free monoid A,we denote by X the submonoid of A generated by X. Conversely, given a submonoid P of A, there exists a unique set X that generates P and is minimal for set-inclusion. In fact, X is the set X =(P ; 1) ; (P ; 1) 2 of the nonempty words of P that cannot be written as the product of two nonempty words of P. It is a straightforward verication that X generates P and that it is contained in any sety A generating P. The set X will be referred to as the minimal generating set of P. A monoid M is said to be free if there exist an alphabet B and an isomorphism of the free monoid B onto M. For instance, for any word w 2 A + the submonoid generated by w, writtenw instead of fwg, is free. It is very important to observe that not all the submonoids of a free monoid are themselves free (see Example 1.2.2).

5 1.2. Submonoids and Morphisms 5 Let P be a submonoid of A and X be its minimal gen- Proposition erating set. Then P is free i any equality x 1 x 2 x n = y 1 y 2 y m n m 0 x i y j 2 X implies n = m and x i = y i, 1 i n. The proof is again left to the reader. The minimal generating set of a free submonoid P of A is called a code it is referred to as the basis of P. A set X A is called a prex if for x y 2 X x y implies x = y it can easily be veried that any prexx A + is a code. Example Let A = fa bg the set X = fa b abg is not a code since it is not the minimal generating set of X. The set Y = fa ab bag is the minimal generating set of Y yet it is not a code because a(ba) =(ab)a is a nontrivial equality between products of elements of Y. The set Z = faa ba, baa, bb bbag can be veriedtobeacode. The following characterization of free submonoids of A is useful: Proposition A submonoid P of A is free i for any word w 2 A, one has w 2 P whenever there exist p q 2 P such that pw wq 2 P: Proof. Let P be a submonoid of A and denote by X its minimal generating set. First suppose that the preceding condition holds for P. Then if x 1 x 2 x n = y 1 y 2 y m x i 2 X y j 2 X (1.2.1) we may suppose that x 1 y 1 and let y 1 = x 1 w, w 2 A.Then x 2 x n = wy 2 y m and therefore x 1 w, wy 2 y m 2 P this implies by hypothesis w 2 P. Since X is the minimal generating set of P,wehave w = 1, and this proves that eq. (1.2.1) is trivial by induction on n + m. Therefore P is free. Conversely, ifp is free, let ' be an isomorphism of a free monoid B onto P, with X = '(B). Then if for p q 2 P,onehaspw wq 2 P,let'(x) =p, '(y) =wq, '(z) =pw, '(t) =q. Since '(xy) ='(zt) wehave xy = zt, and this implies that z = xu, u 2 B. Therefore w = '(u) 2 P. Corollary An intersection of free submonoids of A is free.

6 6 Words 1.2 Proof. If the submonoids P i, i 2 I are free, and if there exists p q 2 P = \ i2i P i such that pw wq 2 P, then by Proposition 1.2.3, w 2 P i for each i 2 I and therefore w 2 P. By 1.2.2, this shows that P is free. If X is any subset of A, the set F of free submonoids of A containing X is not empty (it contains A )and,by Corollary 1.2.4, it is closed under intersection. Therefore the intersection of all elements of F is the smallest free submonoid containing X the code generating this submonoid is called the free hull of X. Theorem (Defect theorem). The free hull Y of a nite subset X A, which is not a code, satises the inequality Card(Y ) Card(X) ; 1: Proof. Consider the mapping of X into Y associating to x 2 X the word y 2 Y such that x 2 yy sincey is a code, the mapping is well dened. As X is not a code, there exists an equality x 1 x 2 x n = y 1 y 2 y n with x i y j 2 X and x 1 6= y 1. Therefore, (x 1 )= (y 1 )andcannot be injective. The following shows that is surjective: if it were not, let z 2 Y be such that z=2 (X) consider the set Z =(Y ; z)z : The set Z is a code since an equality can be rewritten z 1 z 2 z n = z 0 1z 0 2 z 0 n 0 z i z 0 j 2 Z (1.2.2) y 1 z k1 y 2 z k2 y n z kn = y 0 1z k0 1y 0 2z k0 2 y 0 n 0zk0 n0 (1.2.3) with z i = y i z ki, zj 0 = y0 j zk0 j, y i yj 0 2 Y ; Z, k i kj 0 0. Since Y is a code Eq. (1.2.3) is trivial. This implies y 1 = y 0 1, k 1 = k 0, y 1 2 = y 0 2 ::: and nally n = n0 and z i = zi 0. But we have X Z and Z Y,whichcontradicts the minimalityofthe submonoid Y. Hence is surjective, which implies that Y has fewer elements than X. As an immediate consequence, of Theorem 1.2.5, there is the following corollary.

7 1.3. Conjugacy 7 Corollary and y are powers of a single word z 2 A. Each pair of words fx yg (x y 2 A ) is a code unless x Morphisms of free monoids play an essential role in the sequel. Let ' : B! A be a morphism of free monoids. Clearly it is completely characterized by the images '(b) 2 A of the letters b 2 B. It is an isomorphism of B into A i its restriction to B is injective and if the submonoid '(B ) is a free submonoid of A. A morphism ' : B! A is called nonerasing if '(B + ) A +. If ' is nonerasing, then for all w 2 B, 1.3. Conjugacy j'(w)j jwj: Aword x 2 A is said to be primitive if it is not a power of another word that is, if x 6= 1andx 2 z for z 2 A implies x = z. Proposition If x n = y m x y 2 A n m 0 there exists a word z such that x y 2 z. In particular, for each word w 2 A +, there exists a unique primitive word x such that w 2 x. Proof. Ifw = x n = y m with x 6= y the set fx yg is not a code and, by the defect theorem (1.2.5) there exists a word z 2 A such that x y 2 z.ifw = x n = y m with x and y primitive, then there exists a word z 2 A such that x = z i, y = z j, i j 0. This implies x = y = z. Proposition Two words x y 2 A + commute i they are powers of the same word. More precisely the set of words commuting with a word x 2 A + is a monoid generated by a single primitive word. Proof. Letz be the unique primitiveword such thatx 2 z. Then if xy = yx for y 2 A +, the set fx yg is not a code and there exists t 2 A + such that x y 2 t. Then by Proposition 1.3.1, t 2 z. Therefore the set of words commuting with x is generated by z. Two words x and y are said to be conjugate if there exist words u v 2 A such that x = uv y = vu: (1.3.1)

8 8 Words 1.3 This is an equivalence relation on A since x is conjugate to y i y can be obtained by a cyclic permutation of the letters of x. More precisely, let be the permutation of A + dened by (ax) =xa a 2 A x 2 A then the classes of conjugate elements are the orbits of. Let x y 2 A and z t be the primitive words such that Proposition x 2 z, y 2 t. Then x and y are conjugate i z and t are also conjugate in this case, there exists a unique pair (u v) 2 A A + such thatz = uv, t = vu. Proof. Letx = z k.ifx = rs, there exists u v 2 A such thatz = uv, r = z k1 u, s = vz k2 and k 1 + k 2 +1=k. Then the conjugate y = sr of x can be written y = t k with t = vu. Moreover the pair (u v) suchthatz = uv, t = vu is unique since by Proposition z has jzj distinct conjugates. Proposition Two words x y 2 A + are conjugate i there exists a z 2 A such that xz = zy: (1.3.2) More precisely, equality (1.3.2) holds i there exist u v 2 A such that x = uv y = vu z 2 u(vu) : (1.3.3) Proof. If Eq. (1.3.3) holds, then (1.3.2) also holds. Conversely, if xz = zy, for x y 2 A +, z 2 A,wehave for each n 1 x n z = zy n (1.3.4) Let n be such that njxj jzj (n ; 1)jxj. Then we deduce from Eq. (1.3.4) that z = x n;1 u x = uv vz = y n : (1.3.5) Finally y n = vz = vx n;1 u is also equal to (vu) n and since jyj = jxj, we obtain y = vu, proving that Eq. (1.3.3) holds. It may be observed that, in accordance with the defect theorem, the equality xz = zy implies x y z 2fu vg, a submonoid with two generators. The properties of conjugacy in A proved thus far can be viewed as particular cases of the properties of conjugacy in the free group on A (see Problem 1.3.1). If Card(A) =k is nite, let us denote by k (n) thenumber of classes of conjugates of primitive words of length n on the alphabet A. If w is a word of length n and if w = z q with z primitive and n = qd, then the number of conjugates of w is exactly d. Hence k n = X djn d k (d) (1.3.6)

9 1.3. Conjugacy 9 the sum running over the divisors of n. Problem 1.3.2) this is equivalent to: k(n) = 1 n X where is the Mobius function dened on N ; 0 as follows: djn (1) = 1 (n) =(;1) i if n is the product of i distinct primes and if n is divisible by a square. (n) =0 By Mobius inversion formula (see (d)k n=d (1.3.7) Proposition admits the following renement (Fine and Wilf 1965): Proposition Let x y 2 A, n = jxj, m = jyj, d = gcd(n m). If two powers x p and y q of x and y have a common left factor of length at least equal to n + m ; d, then x and y are powers of the same word. Proof. Letu be the common left factor of length n + m ; d of x p y q.werst suppose that d = 1 and show thatx and y are powers of a single letter. We may assume that n m ; 1. It will be enough to show that the rst n ; 1 letters of u are equal. Denote by u(i) the ith letter of u. By hypothesis, we have u(i) =u(i + n), 1 i m ; 1 (1.3.8) u(j) =u(j + m), 1 j n ; 1: (1.3.9) Let 1 i j m;1 andj i+n mod m. Then either j = i+n or j = i+n;m. In the rst case u(i) =u(j) by (1.3.8). In the second case u(j) =u(j + m) by (1.3.9) since j = i + n ; m n ; 1. Therefore u(i) =u(i + n) =u(j + m) =u(j) : Hence u(i) =u(j) whenever 1 i j m ; 1 and j ; i n mod m. But since m n are supposed to be relatively prime, any element of the set f1 2 ::: m;1g is equal modulo m to a multiple of n. This shows that the rst m ; 1 letters of u are equal. In the general case, we consider the alphabet B = A d and, by the foregoing argument, x and y are powers of a single word of length d. Example Consider the sequence of words on A = fa bg dened as follows: f 1 = b, f 2 = a and f n+1 = f n f n;1 n 2:

10 10 Words 1.4 The sequence of the lengths n =j f n j is the Fibonacci sequence. Two consecutive elements n and n+1 for n 3 are relatively prime. Let g n be the left factor of f n of length n ; 2 for n 3. Then g n+1 = f 2 n;1 g n;2 for n 5, as it may beveried by induction. We thenhave simultaneously f n+1 f 2 n g n+1 f 3 n;1 : Therefore, for each n 5, fn 2 and f 3 n;1 have a common left factor of length n + n;1 ;2. This shows that the bound given by Proposition is optimal. For instance, 1.4. Formal Series g 7 = f 6 z } { {z } f 5 abaab abaaba {z } f 5 Enumeration problems on words often lead to considering mappings of the free monoid into a ring. Such mappings may be viewed (and usefully handled) as nite or innite linear combinations of words (see for instance Problem 1.4.2). This is the motivation for introducing the concept of a formal series. Let K be a ring with unit in the sequel K will be generally be the ring Z of all integers. A formal series (or series) with coecients in K and variables in A is just a mapping of the free monoid A into K. The set of these series is denoted by KhhAii. For a series 2 KhhAii and a word w 2 A,thevalue of on w is denoted by h wi and called the coecient of w in itisanelementofk. For a set X A,we denote by X the characteristic series of X, dened by hx xi =1 if x 2 X, hx xi =0 if x=2 X. The operations of sum and product of two series 2 KhhAii are dened by: h + wi = h wi X + h wi h wi = h uih vi w=uv for any w 2 A. These operations turn the set KhhAii into a ring. This ring has a unit that is the series 1, where 1 is the empty word. A formal series 2 KhhAii such that all but a nite number of its coecients are zero is called a polynomial. ThesetKhAi of these polynomials is a subring of the ring KhhAii. It is called the free (associative) K-algebra over A (see Problem 1.4.1). For each 2 KhhAii and 2 KhAi, we dene h i = X w h wih wi :

11 1.4. Formal Series 11 This is a bilinear map of KhhAiiKhAi in K. The sum may be extended to an innite number of elements with the following restriction: A family ( i ) i2i of series is said to be locally nite if for each w 2 A, all but nitely many of the coecients h i wi are zero. If ( i ) i2i is a locally nite family of series, the sum = X i2i i is well dened since for each w 2 A, the coecient h wi is the sum of a nite set of nonzero coecients h i wi. In particular, the family (w) w2a is locally nite, and this allows to write for any 2 KhhAii X = h wiw w2a or, by identifying w with w, X = h wiw w2a This is the usual notation for formal series X in one variable: = n a n n0 with n = h a n i. Let be a series such that h 1i = 0 the family ( i ) i0 is then locally nite since h i wi = 0 for i jwj + 1. This allows us to dene the new series = which is called the star of. It is easy to verify the following: Proposition Let 2 KhhAii be such thath 1i =0. The series is the unique series such that: (1 ; ) =(1; ) =1: Following is a list of statements relating the operations in KhhAii with the operations on the subsets of A when K is assumed to be of characteristic zero. Proposition For two subsets X Y of A,onehas (i) let Z = X [ Y. Then Z = X + Y i X \ Y =, (ii) let Z = XY.ThenZ = XY i xy = x 0 y 0 ) x = x 0 y = y 0, for x x 0 2 X, y y 0 2 Y, (iii) let X A +,andp = X.ThenP = X i X is a code. The proof is left to the reader as an exercise.

12 12 Problems Notes The terminology used for words presents some variations in the literature. Some authors call subword what is called here factor the term subword is reserved for another use (see Chapter 6). Some call prex or initial segment what we call a left factor. Also, the empty word is often denoted by instead of 1. General references concerning free submonoids are Eilenberg 1974 and Lallement 1979 Proposition was known to Schutzenberger (1956) and to Cohn (1962). The defect theorem (Theorem 1.2.5) is virtually folklore it has been proved under various forms by several authors (see Lentin 1972 Makanin 1976 Ehrenfeucht and Rozenberg 1978). The proof given here is from Berstel et al. 1979, where some generalizations are discussed. The results of Section 1.3 are also mainly common knowledge. For further references see Chapters 8 and 9. The standard reference for Section 1.4 is Eilenberg Problems Section (Levi's lemma). A monoid M is free i there exists a morphism of M into the monoid N of additive integers such that ;1 (0) = 1 M and if for any x y z t 2 M xy = zt implies the existence of a u 2 M such that either x = zu, uy = t or xu = z, y = ut Let A be an alphabet and A = fa j a 2 Ag be a copy ofa. Consider in the free monoid over the set A [ A the congruence generated by the relations aa =aa =1 a 2 A: a. Show that each word has a unique representative of minimal length, called a reduced word. b. Show that the quotient of(a [ A) by this congruence is a group F the inverse of the reduced word w is denoted by w. c. Show that for any mapping of A into a group G, there exists a unique morphism ' of F onto G making the following diagram commutative: A - F Z ZZ ' Z Z~? G F is called the free group over A (see Magnus, Karass, and Solitar 1976 or Hall 1959 or Lyndon and Schupp 1977). Henceforth in problems

13 Problems about free groups, : F! (A [ A) denotes the mapping associating to each elementoff the unique reduced word representing it. Let ' : A! A be the mapping assigning to each word w 2 A the longest word that is both a proper left and a proper right factor of w. a. Let w = a 1 a 2 a n and denote '(i) =j instead of '(a 1 a i )= a 1 a j show that the following algorithm allows computation of ': 1. '(1) 0 2. for i 2 until n do begin 3. j '(i ; 1) 4. while j>0 and a i 6= a j+1 do j '(j) 5. if j =0and a i 6= a j=1 then '(i) 0 6. else '(i) j +1 end (For the notations concerning algorithms, see Aho, Hopcroft, and Ullman 1974.) b. Show that the number of successive comparisons of two letters of the word w in performing the foregoing algorithm does not exceed 2n. (Hint : Note that the variable j can be increased at most n times by one unit.) c. Show that the foregoing algorithm can be used to test whether a word u 2 A + is a factor of a word v 2 A +. (Hint : Apply the algorithm of (1) to the word w = uv.) This is called a string-matching algorithm (see Knuth, Morris, and Pratt 1977). Section Let F be the free group over the set A and H be a subgroup of F. a. Show that it is possible to choose a set Q of representatives of the right cosets of H in F such that the set (Q) of reduced words representing Q contains all its left factors. Such a set Q is called a Schreier system for H. b. Let Q be a Schreier system for H and X = fpaq j p q 2 Q a 2 A pa 2 (H ; 1)qg Show that X generates H. c. Show that each paq 2 X is reduced as written and that, in the product of two elements of X [ X the letters a in the triple (p a q) never cancel unless the whole product does.

14 14 Problems d. Deduce from (a), (b), and (c) that any subgroup H of a free group F is free and that if H is of nite index d in F, then it is isomorphic with a free group on r generators with r ; 1=d(k ; 1) k = Card(A) (Schreier's formula see the references of Problem 1.1.2) A submonoid N of A is generated by a prex i it satises: m mn 2 N ) n 2 N for all m n 2 A. Such a submonoid is called (right) unitary Let P be the set of words P = fw ~w j w 2 A g: Then P is the set of palindromes (i.e., u = ~u) of even length. Show that the submonoid P is right and left unitary. (Hint : Let be the basis of P show that is prex.) (See Knuth, Morris and Pratt 1977.) Let : A! B be a morphism and P B be a free submonoid of B.Showthat ;1 (P ) is a free submonoid of A. Section Show thattwo words x y 2 A are conjugate i they are conjugate in the free group F over A that is, i there exists an element g of F such that x = gyg ;1 (Identify A to a subset of F.) (Mobius inversion formula) Letbe the Mobius function show that X 1 if n =1, (d) = 0 if n 2. djn Deduce from this that two functions ' of N ; 0inZare related by i X djn X djn (d) ='(n) (d)'(n=d) = (n) : Show directly (without using the defect theorem, that is) that if fx yg is not a code, then x and y are powers of a single word.

15 Problems (Problem 1.1.3) Show that '(w) = u i w =(st) k+1 s u =(st) k s k 0 s t2 A with jsj minimal. Deduce that the algorithm of Problem allows computation of the primitive word such thatw = v n, n 1(Hint: Use Proposition 1.3.5) Let w = a 1 a 2 a n, n 1, a i 2 A. For 1 i n, let (i) be the greatest integer j i ; 1suchthat a 1 a 2 a j;1 = a i;j+1 a i;1 a i 6= a j with (i) = 0ifnosuchinteger j exists. a. Show that the following algorithm computes : 1. (i) 0 2. i 1 j 0 3. while i<ndo begin 4. while j>0 and a i 6= a j do j (j) 5. i i +1 j j if a i = a j then (i) (j) else (i) (j) end (Hint: Show that the value of the variable j at line 6 is '(i ; 1) + 1 wheres ' is as in Problem ) b. Show that the algorithm of problem part (a) can be used to test whether a word u is a factor of a word v. c. Show that the numberofconsecutive times the while loop of line 4 may be executed does not exceed the integer r such that r+3 n< r+4 where r is the rth term of the Fibonacci sequence. Show, using the sequence of Example that this bound can be reached. (See Knuth, Morris and Pratt 1978 Duval 1981.) Section For any mapping of A into an associative K-algebra R, there exists a unique morphism ' of KhAi into R such that the following diagram is commutative: A - ; ;; R Let W A + and P = A ; A WA

16 16 Problems be the set of words having no factor in W. Let for each u 2 W, X u = A u ; A WA + be the set of words having u as a right factor but no other factor in W. For each u v 2 W, let R u v be the nite set R u v = ft 2 A + ; A v j ut 2 A vg: a. Show that the following equalities hold in ZhhAii: 1+PA = P + X u2w X u (a.1) and for each u 2 W, Pu = Xu + X v2w XR v u (a.2) b. Show that the system of equalities (a.1) and (a.2), for u 2 W, allows computation of P. c. Show that the formal series = X n0 n z n with n = Card(A n \ P ), is rational. (Hint: Use the morphism of ZhhAii onto Zhhzii sending a 2 A on z.) d. Apply the foregoing method to show that for W = fabag one has n =2 n;1 ; n;2 + n;3 n 3: (See Schutzenberger 1964 for a general reference concerning linear equations in the ring ZhhAii, see Eilenberg 1974.)

17 BIBLIOGRAPHY 17 Bibliography Berstel, J., Perrin, D., Perrrot, J.-F., and Restivo, A. (1979). Sur le theoreme du defaut, J. Algebra, 60, 169{180. Cohn, P. M. (1962). On subsemigroups of free semigroups, Proc. Amer. Math. Soc., 13, 347{351. Ehrenfeucht, A. and Rozenberg, G. (1978). Elementary homomorphisms and a solution of the DOL sequence equivanlence problem, Theoret. Comput. Sci., 7, 169{183. Eilenberg, S. (1974). Automata, Languages and Machines, Vol. A. Academic Press. Lallement, G. (1979). Semigroups and Combinatorial Applications. J. Wiley and Sons. Lentin, A. (1972). Equations dans les mono ides libres. Gauthier{Villars, Paris. Makanin, G. S. (1976). On the rank of equations in four unknownsinafree semigroup, Mat. Sb., 100, 285{311. (English trans. in Math USSR Sb., 32, 257{280). Schutzenberger, M. P. (1956). Une theorie algebrique du codage, Seminaire Dubreil{Pisot, annee 55{56.