FREE PROFINITE SEMIGROUPS OVER SOME CLASSES OF SEMIGROUPS LOCALLY IN DG

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1 FREE PROFINITE SEMIGROUPS OVER SOME CLASSES OF SEMIGROUPS LOCALLY IN DG JOSÉ CARLOS COSTA1 Departamento de Matemática Universidade do Minho Camps de Galtar Braga - Portgal jcosta@math.minho.pt AMS Mathematics Sbject Classification: 20M07, 20M35, 68Q45 This paper is concerned with the strctre of semigrops of implicit operations on varios sbpsedovarieties V of DReG LDG, where DReG and DG are the psedovarieties of all semigrops S in which each reglar D-class is, respectively, a rectanglar grop and a grop, and where LDG is the psedovariety of semigrops locally in DG. As an application, we give a characterization of the variety of langages recognized by semigrops in V and derive some join decompositions of psedovarieties. 1 Introdction The theory of free profinite semigrops, which received its major impets with the pblication of Reiterman s paper [17] in the early eighties, has proven to be an important tool in the stdy of psedovarieties of semigrops and on the varieties of recognizable langages associated with them (via Eilenberg s Theorem on varieties [13]). The importance of Reiterman s theorem was immediately nderstood by Almeida [1, 2, etc] and Azevedo [10] who developed the theory. More recently, this approach has also received the attention of athors like Selmi, Trotter, Volkov, Weil, Zeiton and others [7, 18, 22, 23, 25]. For a psedovariety V, denote by LV the psedovariety of all finite semigrops S sch that ese V for each idempotent e of S, and by DV the psedovariety of all finite semigrops S in which each reglar D-class is a sbsemigrop of S which lies in V. Particlarly important in this work are the psedovarieties DReH, DRH and DLH, where, for a psedovariety H of grops, ReH, RH and LH denote, respectively, the psedovarieties of rectanglar grops, of right grops and of left grops, all of whose sbgrops lie in H. We recall that DReG is sally denoted by DO. This paper is devoted to the stdy of implicit operations on some sbpsedovarieties of DS, where S is the psedovariety of all finite semigrops, and consists of part of the athor s doctoral dissertation [11]. The sbpsedovarieties V of DS have a particlarly important property (proved by Azevedo [9, 10] extending a similar reslt of Almeida [2] on J, the psedovariety of J -trivial semigrops), which is the fact that the implicit operations on V can be factored as finite prodcts of words and reglar elements. For some sch psedovarieties V, a certain form of sch a factorization is known to be canonical for V. This is the case, for instance, of J (Almeida [2]), J LSl (Selmi [18]) DH ECom, DRH (Almeida and Weil [6, 8]) and R LSl (Costa [12]), where Sl, ECom and R are, respectively, the psedovarieties of semilattices (i.e. idempotent and 1 Research spported by INVOTAN, grant 4/A/94/PO and by Centro de Matemática da Universidade do Minho. The athor is member of the project AGC contract Praxis/2/2.1/MAT/63/94.

2 2 j. costa commtative semigrops), of semigrops in which the idempotents commte and of R- trivial semigrops. However, the general problem of describing canonical factorizations for all sbpsedovarieties of DS (or even for DS itself) is very far from being achieved. A crcial reslt in this paper is the characterization of the reglar implicit operations on psedovarieties V in the interval [Sl LI, DReG LDG], where I is the trivial psedovariety. We prove in Corollary 3.2 that they are characterized by their restrictions to Sl, LI and V G. We also show that DReG LDG is the greatest sbpsedovariety of DReG with this property. Note that V is sch that V B = NB, where B and NB are, respectively, the psedovarieties of bands and of normal bands. Trotter and Weil [22] proved that the greatest sbpsedovariety of DA, the psedovariety of semigrops in which all reglar elements are idempotents, having intersection NB with B is DA LJ(= DA LDG). Using their reslts, one can show that DReG LDG is the greatest sbpsedovariety of DReG whose intersection with B is NB. So Corollary 3.2 is somehow related with the reslt of Trotter and Weil. This paper is a contribtion to the stdy of the psedovarieties in the interval [Sl, DReG LDG], i.e., the sbpsedovarieties of DReG whose intersection with B is in the interval [Sl, NB]. More precisely, we stdy the strctre of the semigrops of implicit operations on the psedovarieties DA LJ, R LJ, V W and V W ECom, with V {DReH, DRH, DH} and W {LECom, LZE, L(Sl G), Com D}, where denotes the operation of semidirect prodct of psedovarieties of semigrops and Com, D and ZE are, respectively, the psedovarieties of commtative semigrops, of semigrops S in which es = S for each idempotent e S and of semigrops in which idempotents are central (i.e., commte with every element). The techniqes that we se are in close connection with the ones sed by Almeida and Weil [6] in the stdy of the psedovarieties of the form DH ECom. As a conseqence of this work, we are able to give combinatorial descriptions of the classes of langages recognized by each of these psedovarieties U. More precisely, for each finite alphabet A, we describe a set of generators for the Boolean algebra of the langages of A + that are recognized by semigrops in U. Excepting the cases U = DA LJ and U = R LJ, the generators are very simple langages. Depending on the psedovariety U considered, they are of the form 0 A 1 A l 1 l 1L l l A l+1 A n n or of the form 0 A + 1 A+ l 1 l 1L l l A + l+1 A+ n n, where n 0, the i are words over A, L l is a grop langage over A l (if U is aperiodic, then L l = A l or L l = A + l, respectively), the A i are non-empty sbsets of A, and where the i and the A i satisfy some conditions depending on the psedovariety involved. Note that several varieties of langages have been described as Boolean combinations of langages of one of the above forms (e.g. piecewise testable langages (Simon [19]), R-trivial langages (Eilenberg [13]), level 2 langages in the Strabing hierarchy (Pin and Strabing [16]), etc). The previos reslts also permit s to compte some joins of psedovarieties. Recall that the join V W is the least psedovariety containing both the psedovarieties V and W. Among several eqalities we prove that, in the case W = Com D for instance, if H is a psedovariety of abelian grops, then DReH (Com D) = (DA (Com D)) H = (DRH DLH) (Com D) (DRH (Com D)) (DLH (Com D)). This paper is organized as follows. In section 2 we briefly recall some definitions

3 free profinite semigrops over some classes 3 and properties that we shall need in the seqel. Sections 3 to 8 are dedicated to the description of the strctre of the semigrops of implicit operations on the varios psedovarieties mentioned above. Finally, section 9 is devoted to the characterization of the corresponding varieties of langages. 2 Preliminaries We assme the reader is familiar with the basic notions of finite semigrop theory and its relationships with the theory of rational langages and finite atomata. For a comprehensive treatment of the theory and for ndefined notions and notation, the reader is referred to the books of Almeida [3], Eilenberg [13] and Pin [15], and to the srvey [7]. 2.1 Generalities By an alphabet, we mean a finite non-empty set A. We denote by A N (resp. A N ) the set of all words over A that are infinite to the right (resp. infinite to the left ), that is, the set of seqences of letters of A indexed by N (resp. N). We denote by + (resp. ) the infinite word to the right (resp. left) obtained by repeating infinitely often the word A +. The set of all letters appearing in a word (finite or infinite) is denoted by c() and is called the content of. A word A is a prefix (resp. sffix, factor) of a word x (finite or infinite) if there exist words y and z sch that x = y (resp. x = y, x = yz). For each integer k we denote by p k (x) (resp. s k (x)) the prefix (resp. sffix) of x of length k, if it exists. It is well known that every finite semigrop S admits an integer k sch that s k is idempotent for every element s S. Sch an integer will be called an exponent of S. Notice that if k is an exponent of a finite semigrop S, then every mltiple of k is also an exponent of S. Let V be a psedovariety and let A be an alphabet. We denote by ˆF A (V) the free pro-v semigrop over A. The semigrop ˆF A (V) can be viewed as the completion of a certain niform strctre on the free semigrop A + or as the semigrop of A-ary implicit operations on V. For this reason, the elements of ˆFA (V) are sally called (A-ary) implicit operations (on V). It is well known that, for instance, ˆF A (Sl) is the semigrop 2 A of non-empty sbsets of A nder nion. The following important properties of ˆF A (V), will be sed freely in this paper. There exists a natral injective mapping ι : A ˆF A (V) sch that ι(a) generates a dense sbsemigrop of ˆF A (V). Any mapping from A into a semigrop S of V can be niqely extended to a continos morphism from ˆF A (V) into S. In particlar, if W is a sbpsedovariety of V, the identity of A indces a continos onto homomorphism π : ˆFA (V) ˆF A (W), called the canonical projection of ˆFA (V) onto ˆF A (W). The image π(x) of an element x ˆF A (V) is called the restriction of x to W. In particlar, when V is a psedovariety containing Sl, the canonical projection c : ˆF A (V) ˆF A (Sl) = 2 A is called the content homomorphism on V. As one can easily show, c extends to the elements of ˆF A (V) the notion of content for words of A +.

4 4 j. costa For each x ˆF A (V), the seqence (x n! ) n converges in ˆF A (V). Its limit, denoted by x ω, is the only idempotent in the topological closre of the sbsemigrop generated by x. Let V be a psedovariety and let A be an alphabet. A V-psedoidentity on A is a pair (x, y) of elements of ˆFA (V), and is sally denoted x = y. We say that a semigrop S V satisfies x = y, written S = x = y, if, for any continos morphism µ : ˆFA (V) S, we have µ(x) = µ(y). We say that a sbclass C of V satisfies a set Σ of V-psedoidentities, written C = Σ, if each element of C satisfies each element of Σ. The class of all finite semigrops which satisfy Σ is said to be defined by Σ and is denoted [[Σ] V. By a psedoidentity we will mean an S-psedoidentity, and we will also set [[Σ] = [[Σ] S. For instance, adopting the convention of replacing in a psedoidentity expressions of the form x ω, y ω and z ω by symbols e, f and g if, respectively, x, y and z do not appear elsewhere in the psedoidentity, we have the following eqalities: A = [[x ω+1 = x ω ], B = [[x 2 = x]] Com = [[xy = yx], Com D = [[exfyezf = ezfyexf]] D = [[xe = e]], ECom = [[ef = fe]] J = [[(xy) ω = (yx) ω ]] A, K = [[ex = e]] L = [[y(xy) ω = (xy) ω ]], LG = [[ex = x]] LNB = [[xyz = xzy]] B, NB = [[xyzx = xzyx]] B R = [[(xy) ω x = (xy) ω ]], ReG = [[x = x ω+1, efe = e] RG = [[xe = x], RNB = [[xyz = yxz]] B Sl = [[xy = yx] B, ZE = [[ey = ye]]. As far as the D operator is concerned, the following eqalities are well known. DA = [[(xy) ω (yx) ω (xy) ω = (xy) ω ] A, DG = [[(xy) ω = (yx) ω ]] DLG = [[(xy) ω (yx) ω = (yx) ω ], DReG = [[(xy) ω (yx) ω (xy) ω = (xy) ω ] DRG = [[(xy) ω (yx) ω = (xy) ω ]. Let Σ be a set of psedoidentities defining a psedovariety V. Then LV is defined by the set of all psedoidentities which are obtained from Σ by sbstitting each variable x by y ω xy ω where y is a variable that does not occr in Σ. For instance, we have that LDG = [[(exeye) ω = (eyexe) ω ], LECom = [[(exe) ω (eye) ω = (eye) ω (exe) ω ]] LI = [[exe = e]], LSl = [[exexe = exe, exeye = eyexe]] LZE = [[(exe) ω eye = eye(exe) ω ]]. The following fndamental theorem is de to Reiterman [17]. Theorem 2.1 Let V be a psedovariety of semigrops and let W be a sbclass of V. Then W is a psedovariety if and only if there exists a set Σ of V-psedoidentities sch that W = [[Σ] V.

5 free profinite semigrops over some classes Langages recognized by a psedovariety V Let A be an alphabet and let V be a psedovariety. A sbset L of A + is called a langage. It is said to be recognizable (resp. V-recognizable) if there exists a finite semigrop S (resp. in V) and a morphism µ : A + S sch that L = µ 1 (µ(l)). In that case, we say that S recognizes L. The syntactic congrence of a langage L is the congrence L over A + given by L v if and only if xy L xvy L for all x, y A. The syntactic semigrop of L, denoted by S(L), is the qotient of A + by L. We know that L is recognizable (resp. V-recognizable) if and only if S(L) is finite (resp. S(L) V). Frthermore, a semigrop S recognizes a langage L if and only if S(L) divides S (that is, if S(L) is a homomorphic image of a sbsemigrop of S). For more details on recognizable langages, the reader is referred to [15, 13]. A class of (recognizable) langages is a correspondence C associating with each alphabet A a set A + C of (recognizable) langages of A +. A variety of langages is a class V of recognizable langages sch that (1) for every alphabet A, A + V is closed nder finite nion, finite intersection and complement; (2) for every morphism φ : A + B +, L B + V implies φ 1 (L) A + V; (3) if L A + V and a A, then a 1 L = { A + a L} and La 1 = { A + a L} are in A + V. Let V be a psedovariety and let V be the class of recognizable langages which associates with each alphabet A the set A + V of V-recognizable langages of A +. One can show that V is a variety of langages. Moreover, Eilenberg [13] proved the following fndamental reslt. Theorem 2.2 The correspondence V V defines a bijective correspondence between psedovarieties of semigrops and varieties of langages. We say that a family X of sbsets of ˆFA (V) separates the points of ˆFA (V) if, for each pair of distinct elements x and y in ˆF A (V), there exists an element X of X sch that either x X and y X, or x X and y X. The next reslt, de to Almeida [3, 7], will be very sefl. Proposition 2.3 Let A be an alphabet, let V be a psedovariety satisfying no nontrivial identity, and let V be the corresponding variety of langages. Let L be a sbset of A + V and let L be the set of the topological closres in ˆF A (V) of the elements of L. The Boolean algebra A + V is generated by L if and only if the points of ˆFA (V) are separated by L. 2.3 Sbpsedovarieties of DS In this paper we will be particlarly interested in some sbpsedovarieties of DS. Almeida and Azevedo [5] gave a nmber of factorization and reglarity reslts for the implicit operations on sbpsedovarieties of DS, which will prove fndamental in this paper. Some of these reslts are smmarized in the following propositions.

6 6 j. costa Proposition 2.4 Let V be a sbpsedovariety of DS containing Sl and let x, y ˆF A (V). (1) x can be written as a prodct of the form x = 0 x 1 1 x n n where the i are words and the x i are reglar implicit operations on V. (2) If x and y are reglar, then x J y if and only if c(x) = c(y). (3) If w ˆF A (V), c(w) c(y), x = wy (resp. x = yw) and y is reglar, then x is reglar and x L y (resp. x R y). Proposition 2.5 Let V be a sbpsedovariety of DReG. Two reglar elements x and y of ˆF A (V) are eqal if and only if x ω = y ω and V G satisfies x = y. We will need also the following reslt (see [3, Corollary 5.6.2]). Proposition 2.6 Let V be a psedovariety of semigrops and let x ˆF A (V) \ A +. Then x = yz ω w for some y, z, w ˆF A (V). We now consider the psedovariety of nilpotent semigrops N = K D. It is well known that N satisfies no non-trivial identity. This means that the natral morphism ι : A + ˆF A (N) is injective for each alphabet A. In particlar, we may identify the free semigrop A + with a sbsemigrop of ˆF A (N). Since N is contained in K, D and LI, the same is tre for each of these psedovarieties. Frthermore, it is known (see [3]) that: ˆFA (N) is obtained from A + by adding a zero 0; ˆFA (K) = A + A N and the prodct in ˆF A (K) is extended from the prodct in A + by letting ww = w if w A N (dally ˆF A (D) = A + A N and the prodct in ˆF A (D) is extended from the prodct in A + by letting w w = w if w A N ); ˆFA (LI) = A + (A N A N ) where A N A N is a rectanglar band and if A + and (v, w) A N A N, then (v, w) = (v, w) and (v, w) = (v, w). Note that if x = (v, w) is an element of ˆF A (LI)\A +, then v (resp. w) is the restriction of x to K (resp. D). In particlar, LI satisfies a psedoidentity x = y if and only if K and D satisfy x = y. This is another way of stating the well known eqality LI = K D. 3 Reglar elements of ˆF A (DReG LDG) In this section, we give a characterization of the reglar elements of the semigrops ˆF A (V) of implicit operations on sbpsedovarieties V of DReG LDG and derive some important properties of them. Proposition 3.1 Let V be a sbpsedovariety of DReG LDG containing Sl and K (resp. D). Two reglar elements x and y of ˆFA (V) are R-(resp. L-)eqivalent if and only if they have the same content and the same restriction to K (resp. D). Proof. Sppose first that x R y. In particlar, x J y and so by Proposition 2.4, c(x) = c(y). Moreover, x = yz for some z ˆF A (V). Since y (and x) is not in A +, this clearly implies that the restrictions of x and y to K are eqal. Sppose now that c(x) = c(y) and that K satisfies x = y. We claim that the second condition implies that x = z and y = w for some, z, w ˆF A (V) sch that A +. Indeed, if (x n ) n and (y n ) n are seqences of A + converging, respectively, to x and y in

7 free profinite semigrops over some classes 7 ˆF A (V), then we can choose sbseqences (x n) n and (y n) n of (x n ) n and (y n ) n, sch that x n = n z n and y n = n w n for some n, z n, w n A +. We may choose n sch that n > n and, by compactness of ˆFA (V), we may sppose that the seqences ( n ) n, (z n ) n and (w n ) n are convergent in ˆF A (V) proving the claim. Moreover, Proposition 2.6 says that = 1 ω 2 3 for some 1, 2, 3 ˆF A (V). Now since c(x) = c(y), we dedce from Proposition 2.4 that x, y, xy and yx are J -eqivalent reglar elements, and that xy R x. In particlar, xy is a grop element becase V DS and so xy = (xy) ω+1. Frthermore, we dedce sccessively xy = (xy) ω+1 = ( 1 ω 2 3z 1 ω 2 3w) ω+1 = 1 ( ω 2 3z 1 ω 2 3w 1 ω 2 )ω 3 z 1 ω 2 3w = 1 ( ω 2 3w 1 ω 2 3z 1 ω 2 )ω 3 z 1 ω 2 3w since V LDG = ( 1 ω 2 3w 1 ω 2 3z) ω 1 ω 2 3z 1 ω 2 3w = (yx) ω xy. This means that xy R y and, conseqently, that x R y. Corollary 3.2 Let V be a sbpsedovariety of LDG and DReG containing Sl and LI. Two reglar elements of ˆF A (V) are eqal if and only if they have the same content and the same restriction to LI and to V G. Frthermore, DReG LDG is the greatest sbpsedovariety of DReG with this property. Proof. We only need to prove the sfficient condition. Since c(x) = c(y) and LI satisfies x = y, we have x H y from Proposition 3.1. So as the H-class of x is a grop (say becase x is reglar and V is a sbpsedovariety of DS) we dedce x ω = y ω. Now the eqality x = y follows from Proposition 2.5. Now sppose that W is a sbpsedovariety of DReG not contained in LDG. Then there are two distinct idempotents of ˆF A (W) of the form x ω yx ω and x ω zx ω, respectively, in the same J -class. These elements have clearly the same restriction to LI and W G. Moreover, since they are J -eqivalent, they have the same content by Proposition 2.5. Let V be a psedovariety in the interval [Sl LI, DReG LDG] and let x be a reglar element of ˆFA (V). The previos reslt shows that x is characterized by its content, say B A, and by its restrictions to LI and to V G, say (w, w ) B N B N and g ˆF A (V G), respectively. So we will denote x by [w, B, g, w ]. In particlar, when x is idempotent it will be denoted by [w, B, 1, w ]. Frthermore, if V is an aperiodic psedovariety (i.e., it is sch that V G = I), then V is a sbpsedovariety of DA LDG. In particlar, every reglar element of ˆF A (V) is idempotent and it is characterized by its restrictions to Sl and LI. In this case we simplify the notation and denote it simply by (w, B, w ). Remark. We notice that one can show, as above, that for a psedovariety V in the interval [Sl K, DRG LDG] (resp. [Sl D, DLG LDG]), a reglar element x ˆF A (V)

8 8 j. costa is characterized by its content, say B A, and by its restrictions to K (resp. D) and to V G, say w B N (resp. w B N ) and g ˆF A (V G), respectively. Ths, x will be denoted by [w, B, g] (resp. [B, g, w ]). When V is an aperiodic psedovariety we denote x simply by (w, B) (resp. (B, w )). Notice also that from the paper of Trotter and Weil [22] one can dedce that DReG LDG (resp. DRG LDG, DLG LDG) is the greatest sbpsedovariety of DReG (resp. DRG, DLG) whose intersection with B is NB (resp. LNB, RNB). In order to complete or notation for reglar elements of semigrops ˆF A (V), we will now consider the case where V is a sbpsedovariety of DG containing Sl. It is known (say by Propositions 2.4 and 2.5) that, in this case, a reglar element x of ˆFA (V) is characterized by its content B and by its restriction g to V G. So we denote x by [B, g]. If V is aperiodic (i.e., V J), then every reglar element x is idempotent and it is characterized by its content B. So we denote x simply by (B). Ths, we se the notation ( ) for idempotent elements of aperiodic psedovarieties and [ ] for the reglar elements of the non-aperiodic psedovarieties. The reglar elements of ˆF A (DReG LDG) enjoy the following important properties. Proposition 3.3 Let A be an alphabet, let B, C, D A be sch that B C and D B. Let also b B. Then, in ˆF A (DReG LDG), (1) [w, B, g, w ]b = [w, B, gb, w b], b[w, B, g, w ] = [bw, B, bg, w ], [w, B, g, w ][v, D, f, v ] = [w, B, gf, v ] and [v, D, f, v ][w, B, g, w ] = [v, B, fg, w ]; (2) if one of c(w ) and c(z) is contained in B C, then [w, B, g, w ][z, C, h, z ] = [w, B, g, w ][z, C, h, z ] for every w B N and z C N sch that at least one of c(w ) and c(z ) is contained in B C. In particlar, ˆF A (DRG LDG) satisfies (1 ) [w, B, g]b = [w, B, gb], b[w, B, g] = [bw, B, bg], [w, B, g][v, D, f] = [w, B, gf] and [v, D, f][w, B, g] = [v, B, fg]; (2 ) [w, B, g][z, C, h] = [w, B, g][z, C, h] for every z, z C N. Proof. (1) Is an immediate conseqence of Proposition 2.4 (3) and of Corollary 3.2. (2) Sppose, for instance, that c(w ) B C and let w B N and z C N be sch that c(w ) B C or c(z ) B C. If c(z ) B C, we dedce from (1) that [w, B, g, w ] = [w, B, g, w ][w, B, 1, w ][z, B C, 1, w ]. So [w, B, g, w ][z, C, h, z ] = ([w, B, g, w ][w, B, 1, w ])([z, B C, 1, w ][z, C, h, z ]) = [w, B, g, w ][z, C, h, z ] from (1). Sppose now that c(z ) B C and let a B C. Then c(w ) B C and sing what we proved above, we dedce [w, B, g, w ][z, C, h, z ] = [w, B, g, w ][a +, C, h, z ] = [w, B, g, w ][a +, B C, 1, w ][z, C, h, z ] from (1) = [w, B, g, w ][z, C, h, z ].

9 free profinite semigrops over some classes 9 For the proof of (1 ) and (2 ), it sffices to consider the canonical projection of DReG LDG to DRG LDG and note that the restriction to DRG LDG of a reglar element [w, B, g, w ] of ˆF A (DReG LDG) is the reglar element [w, B, g]. Note that if V is a sbpsedovariety of DReG LDG containing Sl and LI, the restriction to V of a reglar element [w, B, g, w ] of ˆF A (DReG LDG) is also denoted by [w, B, g, w ]. Hence, the previos reslt is also valid in ˆF A (V). In the following sections, we will proceed to the description of the semigrops of implicit operations on varios sbpsedovarieties of DReG LDG, namely the semigrops: ˆF A (DA LDG) and ˆF A (R LDG); ˆF A (V W) with V {DReH, DRH, DH} and W {LECom, LZE, L(Sl G), Com D}; ˆF A (DH W ECom) with W {LZE, L(Sl G), Com D}. Note that the non-aperiodic cases ˆF A (V LDG) with V {DReH, DRH, DH} (and H a non-trivial psedovariety of grops) are not inclded here, becase we were not able to solve them. To give an idea of the inclsion relations between the psedovarieties involved, we note the following inclsions: LSl Com D LCom LZE LDG; LSl L(Sl G) LZE; LZE LECom, LECom LDG bt DReG LECom DReG LDG. 4 Implicit operations on DA LJ We begin or stdy with the description of the semigrops ˆF A (DA LJ) and ˆF A (R LJ). We prove that every element of each of these semigrops, can be written in a niqe form as a prodct of words and idempotents. We note that, since J = DG A, we have immediately DA LDG = DA LJ and R LDG = R LJ. Note also that J is a sbpsedovariety of both DA LJ and R LJ. Let s begin by considering the case DA LJ. Let x ˆF A (DA LJ) and let an order be fixed for the letters of the alphabet A. We say that a factorization of x of the form x = 0 (w 1, A 1, w 1) 1 n 1 (w n, A n, w n) n is normal if i A, 0 1 if x = 0 ; for each 1 i n sch that i (resp. i 1 ) is not the empty word, the first (resp. last) letter of i (resp. i 1 ) does not lie in A i. if i (1 i n 1) is the empty word, then A i and A i+1 are -incomparable;

10 10 j. costa the first letter of w i+1 does not lie in A i ; if c(w i ) A i+1, then w i = and w i+1 = v + where and v are the least linear (i.e., sch that each letter occrs exactly once) words in alphabetical order of content, respectively, A i A i+1 and A i+1 sch that the first letter of v does not lie in A i. Proposition 4.1 Every element of ˆF A (DA LJ) admits a normal factorization. Proof. Let x ˆF A (DA LJ). As a conseqence of Proposition 2.4, x admits a factorization of the form x = 0 (w 1, A 1, w 1 ) 1 n 1 (w n, A n, w n) n as a prodct of words i A and idempotents (w i, A i, w i ), sch that for each 1 i n sch that i (resp. i 1 ) is not the empty word, the first (resp. last) letter of i (resp. i 1 ) does not lie in A i and, if i (1 i n 1) is the empty word, then A i and A i+1 are -incomparable. Now sppose that 1 i n 1 is sch that i = 1. Then either one of c(w i ) and c(w i+1 ) is contained in A i A i+1, or c(w i ) and c(w i+1) are both not contained in A i A i+1. In the first case, letting and v be the least linear words in alphabetical order of content, respectively, A i A i+1 and A i+1 sch that the first letter of v does not lie in A i, we have from Proposition 3.3 that the factor (w i, A i, w i )(w i+1, A i+1, w i+1 ) is eqal to (w i, A i, )(v +, A i+1, w i+1 ). In the second case, w i+1 = zz for some words z A i+1 and z A N i+1 sch that c(z) A i (if z 1) and the first letter of z does not lie in A i. Frthermore, (w i, A i, w i )(w i+1, A i+1, w i+1 ) = (w i, A i, w i z)(z, A i+1, w i+1 ) by Proposition 3.3. So, for each 1 i n 1 sch that i = 1, sbstitting in the factorization of x the factor (w i, A i, w i )(w i+1, A i+1, w i+1 ) by (w i, A i, )(v +, A i+1, w i+1 ) in the first case and by (w i, A i, w i z)(z, A i+1, w i+1 ) in the second case, we obtain a normal factorization of x. We now describe some atomata which we will se to constrct test semigrops (the syntactic semigrops of the langages recognized by these atomata) to separate distinct factorizations of elements of ˆF A (DA LJ). Let r, n 0 be two integers and let 0,..., n A and = A 1,..., A n A be sch that, for all 1 i n 1: if i 1 then c( i ) is not contained in either A i or A i+1 ; if i = 1 then A i and A i+1 are -incomparable. Let A = A(r; 0, A 1, 1,..., A n, n ) be the following atomaton q 0 0 X q 1 A 1 q 1 1 X q 2 A 2 q 2... q n 1 A n 1 q n 1 n 1 q n n q n+1 where, for each 1 i n 1, and the atomaton A i is either X i = { i if i 1 A i+1 \ A i if i = 1 A n A i A i A i A i+1 q i or A i \A i+1 A q i i A i+1 A q i,0 i A i+1 q i,1 q i,2... q i,r when, respectively, i 1 or i = 1. Note that the state q i is q i in the first case and is

11 free profinite semigrops over some classes 11 q i,r in the second one. Note also that A i is an atomaton on the alphabet A i. In the figre of atomaton A, the initial, q 0, and final, q n+1, states are pointed ot by arrows. We will follow this convention throghot the paper. Lemma 4.2 Let L be the langage recognized by the atomaton A above. Then S(L) lies in DA LJ. Moreover, if w A +, k > 0 n + 3n 2 + lr (where l is the nmber of indices 1 i n 1 sch that i = 1) and w k is the label of a path T in A, then there exists 1 i n sch that w A + i and T visits state q i (or state q i,r when it exists) and does not visit either state q i 1, if i > 1, or state q i+1, if i < n. In particlar, if r = 0 and the first letter of j (1 j n) does not lie in A j, then S(L) R. In this case, if w, k and T are as above, then T ends in state q i (or state q i,0 ), with i as above. Proof. Becase of the choice of k, it is clear that the path T visits a state p having a loop and stays in p for at least w steps. If p = q i,r for some 1 i n 1, then w (A i A i+1 ) + and so, in particlar, w A + i. Otherwise, p = q i for some 1 i n and so w A + i. In both cases T does not visit either state q i 1 (when i > 1) or state q i+1 (when i < n) becase in that case T wold contain a transition labeled with a word not in A + i. Let s now show that S(L) lies in DA LJ. For that, it sffices to show that, for all x, y, z A + and m large enogh, (xy) m (yx) m (xy) m L (xy) m, x m+1 L x m and (x m yx m zx m ) m L (x m zx m yx m ) m. Withot loss of generality, we may sppose that m is an exponent of S(L) (so that, for all w A +, w m L w m w m, that is, the syntactic image of w m is an idempotent of S(L)). Let x, y, z A +. To prove that (xy) m (yx) m (xy) m L (xy) m it sffices to show the following condition: (xy) m (yx) m (xy) m is the label of a path P in A (say from a state p to a state q) if and only if there is a path Q in A labeled (xy) m and co-terminal with P (that is, from p to q). So let s first sppose that P exists. Then from the above, xy A + i for some 1 i n and P visits state q i or state q i,r (when it exists) and does not visit either state q i 1 (if i > 1) or state q i+1 (if i < n). Sppose that q i,r exists (i.e., that 1 i < n and i = 1) and that P visits the states of the form q i,j (0 j r) for at least xy steps. Then xy (A i A i+1 ) + and so P is entirely between the states q i,0 and q i,r. Therefore, the sbpath of P labeled (yx) m (xy) m is entirely in q i,r and so the existence of Q is clear. Now sppose that P visits the states of the form q i,j (0 j r) for at most xy 1 steps so that P visits state q i. Therefore, since P does not visit state q i 1 (when i > 1) and, as above, it can not visit the states of the form q i 1,j (0 j r), if they exist, for more than xy 1 steps, we dedce that at most max{ xy, i 1 } 1 of the steps of P take place strictly between the states q i 1 and q i. Hence, the sbpath of P labeled (yx) m is entirely in q i. So the existence of Q is also clear in this case. (In fact, what is clear is the existence of a path labeled (xy) m (xy) m co-terminal with P. Bt, since we are considering m sch that (xy) m L (xy) m (xy) m, the existence of Q is garanteed.) The case when q i,r does not exist can be treated analogosly. Similarly, one can show that the existence of Q implies the existence of P, proving that (xy) m (yx) m (xy) m L (xy) m. That x m+1 L x m can be proved analogosly.

12 12 j. costa Now sppose that P is a path in A labeled (x m yx m zx m ) m so that c(x) c(y) c(z) A i for some 1 i n. Then either i < n, i = 1 and P takes place entirely between the states q i,0 and q i,r, and the existence of a path Q in A co-terminal with P and labeled (x m zx m yx m ) m is immediate, or at least x m yx m zx m steps of P take place in state q i and at most yx m zx m (= x m yx m z ) steps of P take place strictly between the states q i and q i+1 (resp. between the states q i 1 and q i ). In this case, let P 1, P 2 and P 3 be the sbpaths of P labeled, respectively, x m yx m z, x m (x m yx m zx m ) m 2 x m and yx m zx m. Hence, P 2 is entirely in q i so that P 1 ends in q i and P 3 begins in q i. Moreover, the sbpath of P 1 labeled yx m z begins in q i or in q i 1,r (when it exists). In both cases it is clear that there is a path P 1 co-terminal with P 1 and labeled x m zx m y. Analogosly, there is a path P 3 co-terminal with P 3 and labeled zx m yx m. Since, trivially, there is a path labeled x m (x m zx m yx m ) m 2 x m, entirely in q i, we dedce the existence of a path Q (co-terminal with P and labeled (x m zx m yx m ) m ). By symmetry, we dedce that (x m yx m zx m ) m L (x m zx m yx m ) m. Finally, sppose that r = 0 and that the first letter of j (1 j n) does not lie in A j. As above, these conditions clearly imply that, for some 1 i n, w A + i and T ends in state q i or state q i,0 (in this case i < n). To prove that S(L) R, let s show that (xy) m x L (xy) m. For that, let P be a path in A labeled (xy) m x. This path ends in some state q i or state q i,0. In the first case, the assertion that there is some path Q in A labeled (xy) m and co-terminal with P is immediate. In the second case, either xy (A i A i+1 ) +, and so P is entirely in q i,0 (and the existence of sch a path Q is trivial), or there is at least a letter of xy in A i \ A i+1 and P stays in q i,0 for at most yx 1 steps. In this case, the existence of the desired path Q is also ensred (this path can pass from state q i to state q i,0 sing, for instance, the last occrrence not in A i+1 of a letter of the word (xy) m ). The proof of the converse is similar and so we conclde that (xy) m x L (xy) m, proving that S(L) R. Now we are able to prove the following characterization of the semigrops of implicit operations on DA LJ. Theorem 4.3 Let x, y ˆF A (DA LJ) and let x = 0 (w 1, A 1, w 1 ) 1 (w n, A n, w n) n and y = v 0 (z 1, B 1, z 1 )v 1 (z m, B m, z m)v m be factorizations in normal form. Then x = y if and only if n = m, i = v i, w i = z i, A i = B i and w i = z i for all i. Proof. Let r 1 be an integer sch that r > v i for every 1 i n and c(s r (w i )) A i+1 for every 1 i n 1 sch that i = 1 and c(w i ) A i+1. Consider the atomaton A = A(r; 0 p r (w 1 ), A 1, 1,..., n 1, A n, s r (w n) n ) where, for each 1 i n 1, i is eqal to: s r (w i ) ip r (w i+1 ) if i 1, or i = 1 and c(w i ) A i+1; 1 if i = 1 and c(w i ) A i+1. Note that by definition of normal factorization of x, for each 1 i n 1, if i = 1 then A i and A i+1 are -incomparable and if i 1 then c( i ) A i, A i+1. Let L be the langage recognized by A and let µ : A + S be its syntactic homomorphism. By Lemma 4.2, S DA LJ. So let ˆµ : ˆF A (DA LJ) S be the niqe continos homomorphic extension of µ, and let k > 0 n + 3n 2 + lr (where l is the nmber of indices 1 i n 1 sch that i = 1) be an exponent of S (so that for all w A + the syntactic image of w k is an idempotent of S).

13 free profinite semigrops over some classes 13 For each 1 i n, let w i A N i and w i A N i be sch that w i = p r (w i ) w i and w i = w i s r(w i ) so that (w i, A i, w i ) = p r(w i )( w i, A i, w i )s r(w i ). Since ( w i, A i, w i ) is idempotent, its image in S, ˆµ( w i, A i, w i ) is also idempotent. By density of A+ in ˆF A (DA LJ), there is a word x i sch that c(x i ) = A i and ˆµ( w i, A i, w i ) = µ(xk i ). Now it is not very difficlt to verify that w = 0 p r (w 1 )x k 1s r (w 1) 1 p r (w 2 )x k 2 s r (w n 1) n 1 p r (w n )x k ns r (w n) n is a word recognized by A, whence w L. On the other hand, we have ˆµ(x) = µ(w). Consider now words z i Bi N (1 i m) and z i B N i sch that z i = p r (z i ) z i and z i = z i s r(z i ) so that (z i, B i, z i ) = p r(z i )( z i, B i, z i )s r(z i ). Consider also words y i sch that c(y i ) = B i and ˆµ( z i, B i, z i ) = µ(yk i ). Let w = v 0 p r (z 1 )y k 1s r (z 1)v 1 p r (z 2 )y k 2 y k ms r (z m)v m. We have ˆµ(y) = µ(w ) and, as x = y, ˆµ(x) = ˆµ(y). Therefore, µ(w) = µ(w ) whence w L and so w is recognized by A. Let P be a sccessfl path in A (i.e., which goes from q 0 to q n+1 ) labeled w and, for each 1 i m, let P i be the sbpath of P labeled v 0 p r (z 1 )y k 1 s r(z 1 )v 1p r (z 2 )y k 2 yk i. From Lemma 4.2, we dedce that the path P i visits state q ji for some 1 j i n sch that B i A ji and does not visit state q ji +1 (if j i < n). Frthermore, the sbpath P i of P i labeled p r (z i )y k i does not visit state q ji 1 (if j i > 1). In particlar, the path P 1 visits state q 1 and so the word 0 p r (w 1 ) is a prefix of v 0 p r (z 1 )y k 1. Now since r > v 0, also the path P 1 visits state q 1. Hence, j 1 = 1 and B 1 A 1. By symmetry it follows that A 1 = B 1. Now since the last letter of 0 (if it exists) does not lie in A 1 = B 1, we dedce that 0 is a prefix of v 0. Again by symmetry it follows that 0 = v 0 and conseqently that p r (w 1 ) = p r (z 1 ). Since this holds for r arbitrarily large, we conclde that w 1 = z 1. Now as the first letter of the word v 1 p r (z 2 ) does not lie in B 1 = A 1 (note that, as the factorization of y is normal, if v 1 = 1 then the first letter of z 2 does not belong to B 1 ) we have j 2 > 1. Let s consider the two possible cases for 1. First case Sppose, first, that 1 1, i.e., that 1 1, or 1 = 1 and c(w 1 ) A 2. Then atomaton A begins like this A 1 0 s r (w 1) 1 p r (w 2 ) q 2 q 0 q 1 A 2 Therefore, the word s r (w 1 ) 1p r (w 2 ) is a factor of y1 ks r(z 1 )v 1p r (z 2 )y2 k. Since the first letters of 1 p r (w 2 ) and v 1 p r (z 2 ), respectively, do not lie in A 1 = B 1, we dedce that s r (w 1 ) = s r(z 1 ) and that 1p r (w 2 ) is a prefix of v 1 p r (z 2 )y2 k. Now as above, this implies that w 1 = z 1, j 2 = 2 and B 2 A 2. Moreover, since the last letter of 1 (if it exists) does not lie in A 2, and so does not lie also in B 2, we dedce that 1 is a prefix of v 1. If v 1 1 we can apply symmetry to dedce that 1 = v 1. If v 1 = 1, we have trivially 1 = v 1. Now this eqality implies that p r (w 2 ) is a prefix of p r (z 2 )y2 k so that p r(w 2 ) = p r (z 2 ). Therefore, as above w 2 = z 2. Note that, in this case, it remains to prove the inclsion A 2 B 2.

14 14 j. costa Second case Sppose now that 1 = 1, i.e., that 1 = 1 and c(w 1 ) A 2. In particlar, w 1 = and w 2 = v + where and v are the least linear words in alphabetical order of content, respectively, A 1 A 2 and A 2 sch that the first letter of v does not lie in A 1. We may also sppose that v 1 = 1 since otherwise, we cold apply an argment as above to dedce that v 1 wold be a prefix of 1 and so 1 wold not be eqal to the empty word. In this case, the beginning of the atomaton A is the following. A 1 A 1 A 2 A 2 0 q 0 q 1 A 1 \A 2 A q 1 A 1,0 2 A q 1 A 1,1 2 A 2 \A 1 q 1,2... q 1,r q 2 Therefore, in path P 2, the first letter of p r (z 2 ) is read in the transition from state q 1,r to state q 2, and s r (z 1 ) is read in the transitions between state q 1,0 and state q 1,r. This means, in particlar, that j 2 = 2 so that B 2 A 2. So in both cases ( 1 = 1 and 1 1) we have B 2 A 2. Hence, B 2 A 2 and applying symmetry we dedce that A 2 = B 2. In the case 1 = 1 we are considering, we also dedce that c(s r (z 1 )) A 1 A 2. Since r is arbitrarily large, this implies that c(z 1 ) A 2. So since we are dealing with normal factorizations and v 1 = 1, we have z 1 = = w 1 and z 2 = v + = w 2. Therefore, we have proved that w 1 = z 1, 1 = v 1, w 2 = z 2 and A 2 = B 2. Iterating the above argment, we dedce that n = m, i = v i, w i = z i, A i = B i and w i = z i for all i. This last proof shows, in particlar, that the syntactic semigrops of the langages recognized by the atomata A(r; 0, A 1,..., A n, n ), as above, sffice to separate distinct implicit operations on DA LJ. Corollary 4.4 The psedovariety DA LJ is generated by the syntactic semigrops of the langages recognized by the atomata A(r; 0, A 1,..., A n, n ) where r, n 0 and, for some alphabet A, 0,..., n A and = A 1,..., A n A are sch that, for each 1 i n 1: if i 1 then c( i ) is not contained in either A i or A i+1 ; if i = 1 then A i and A i+1 are -incomparable. Almeida and Azevedo [4] showed that R L = [[(xy) ω x(zx) ω = (xy) ω (zx) ω ]]. If a, b and c are distinct letters of an alphabet A, in ˆF A (DA LJ) we have and (ab) ω a(ca) ω = ((ab) +, {a, b}, (ab) )a((ca) +, {a, c}, (ca) ) = ((ab) +, {a, b}, (ba) )((ca) +, {a, c}, (ca) ) (ab) ω (ca) ω = ((ab) +, {a, b}, (ab) )((ca) +, {a, c}, (ca) ). Hence, by Theorem 4.3, (ab) ω a(ca) ω (ab) ω (ca) ω and so DA LJ does not satisfy the psedoidentity (xy) ω x (zx) ω = (xy) ω (zx) ω. This proves that (R L) LJ DA LJ. Let s now consider the case R LJ. Let x ˆF A (R LJ). We say that a factorization of x of the form x = 0 (w 1, A 1 ) 1 n 1 (w n, A n ) n is normal if

15 free profinite semigrops over some classes 15 i A, 0 1 if x = 0 ; for each 1 i n sch that i (resp. i 1 ) is not the empty word, the first (resp. last) letter of i (resp. i 1 ) does not lie in c(x i ). if i (1 i n 1) is the empty word, then A i and A i+1 are -incomparable; if A i A i+1, then w i+1 = v + where v is the least linear word in alphabetical order of content A i+1 sch that the first letter of v does not lie in A i. Using the (R LJ)-recognizable langages described on Lemma 4.2 and applying similar argments as those of the proof of Theorem 4.3, one can show that the implicit operations on R LJ are characterized by the following reslt. Theorem 4.5 Every element of ˆF A (R LJ) admits a normal factorization. Let x, y ˆF A (R LJ) and let x = 0 (w 1, A 1 ) 1 (w n, A n ) n and y = v 0 (z 1, B 1 )v 1 (z m, B m )v m be factorizations in normal form. Then x = y if and only if n = m, i = v i, w i = z i and A i = B i for all i. Natrally, a left-right dal of this last theorem cold be stated for the psedovariety L LJ. 5 Implicit operations on DReG LECom In this section, we concentrate or attention on sbpsedovarieties of DReG LECom, namely the psedovarieties of the form W LECom where H is a psedovariety of grops and W is one of DReH, DRH and DH. Note that DReG LECom is a sbpsedovariety of DReG LDG. Indeed, we have DReG LECom L(DReG ECom) = L(DG ECom) LDG, since DReG ECom = DG ECom. Also note that J is not a sbpsedovariety of LECom becase it does not satisfy the psedoidentity (exe) ω (eye) ω = (eye) ω (exe) ω which defines LECom. Besides the properties given by Proposition 3.3, the reglar elements of the semigrop ˆF A (DReG LECom) enjoy also the following important one. Proposition 5.1 Let A be an alphabet and let B and C be sbalphabets of A sch that B C. In ˆF A (DReG LECom), if one of c(w ) and c(z) is contained in B C, then [w, B, g, w ][z, C, h, z ] = [w, B C, gh, z ]. In particlar, ˆFA (DRG LECom) satisfies [w, B, g][z, C, h] = [w, B C, gh] for every z C N. Proof. Pt p = [w, B, g, w ] and q = [z, C, h, z ]. Sppose first that both c(w ) and c(z) are contained in B C. Also let r be the idempotent [z, B C, 1, w ]. Then (rpr) ω (rqr) ω is idempotent since V LECom and so (rpr) ω (rqr) ω = [z, B C, 1, w ] by Corollary 3.2. Moreover, p = p(rpr) ω and q = (rqr) ω q by Proposition 3.3. Ths, pq = p(rpr) ω (rqr) ω q = p[z, B C, 1, w ]q = [w, B C, gh, z ] again by Proposition 3.3.

16 16 j. costa Sppose now that, for instance, c(z) B C (and not necessarily c(w ) B C) and let a B C. Then by Proposition 3.3, q = [z, B C, 1, a ][a +, C, h, z ]. So pq = p[z, B C, 1, a ][a +, C, h, z ] = [w, B, g, a ][a +, C, h, z ] = [w, B C, gh, z ] since c(a ) = c(a + ) = {a} B C. The second part of the reslt is a natral conseqence of the first one. The second part of this reslt says that the prodct of any two reglar elements of ˆFA (DRG LECom) with non-disjoint contents, is a reglar element. In the case of the prodct xy of two reglar elements x and y of ˆF A (DReG LECom) with nondisjoint contents, we only are sre to obtain a reglar element if one of c(x ) and c(y ) is contained in c(x) c(y), where x and y are, respectively, the restrictions of x and y to D and K. As we shall see, only nder these conditions will the prodct xy be a reglar element. We begin by considering the cases DReH LECom where H is a psedovariety of grops. We say that a factorization of an element x ˆF A (DReH LECom) of the form x = 0 [w 1, A 1, g 1, w 1] 1 n 1 [w n, A n, g n, w n] n is normal if: i A, 0 1 if x = 0 ; if i (1 i n 1) is the empty word, then the first letter of w i+1 does not lie in A i and c(w i ) A i+1; for each 1 i n sch that i (resp. i 1 ) is not the empty word, the first (resp. last) letter of i (resp. i 1 ) does not lie in c(x i ). Propositions 2.4 and 5.1 garantee that every element of ˆFA (DReH LECom) admits a normal factorization. In order to separate distinct factorizations, we will need some adeqate atomata which we now describe. For n 0, let 0,..., n A and = A 1,..., A n A be sch that i 1 (1 i n 1). Let l {1,..., n}, let A l be a permtation atomaton on the alphabet A l with set of states Q l and let q l, q l Q l. Finally, let C = C( 0, A 1, 1,..., A l ; q l ; q l, l,..., A n, n ) be the following atomaton. A 1 l 1... q l 1 q l A l q l l n q l+1... q n q n+1 q 0 0 q 1 A l 1 In order to simplify notations, we denote Q i = {q i } for all 1 i n with i l. Before the proof of a lemma, note that LECom = [[e(exe) ω (eye) ω e = e(eye) ω (exe) ω e]]. Lemma 5.2 Let L be the langage recognized by the atomaton C above, and sppose that it satisfies the following extra condition: for each 1 i n 1, c( i ) is not contained in either A i or A i+1. Then S(L) lies in DReG LECom and its sbgrops lie in the psedovariety generated by the transition grop S(A l ). Moreover, if w A +, k is an exponent of S(L) sch that k > 0 n + n and w k is the label of a path in C, then there exists i {1,..., n} sch that w A + i and the path visits Q i bt does not visit either Q i 1 (if i > 1) or Q i+1 (if i < n). Proof. The second part of the lemma and the fact that S(L) verifies the psedoidentity (xy) ω (yx) ω (xy) ω = (xy) ω defining DReG can be proved as in Lemma 4.2. Now from the remark immediately before the lemma, to show that S(L) lies in LECom it sffices to show that x k (x k yx k ) k (x k zx k ) k x k L x k (x k zx k ) k (x k yx k ) k x k for all x, y, z A +. For this, it sffices to prove that A l+1 A n

17 free profinite semigrops over some classes 17 x k (x k yx k ) k (x k zx k ) k x k is the label of a path P in C if and only if there is a path Q in C labeled x k (x k zx k ) k (x k yx k ) k x k co-terminal with P. Let x, y, z A +, sppose that P exists and consider the two sbpaths P 1 and P 2 of P labeled, respectively, x k (x k yx k ) k and (x k zx k ) k x k. By the second part of the lemma, since P is a path in C, there are 1 i j n sch that P 1 (resp. P 2 ) visits Q i (resp. Q j ) and does not visit Q i 1 nor Q i+1 (resp. Q j 1 nor Q j+1 ). Since P 1 and P 2 are consective paths, it follows that either i = j or i + 1 = j. We claim that i = j. Indeed, let s sppose that i + 1 = j. Then c(x) c(y) A i, c(x) c(z) A i+1 and, becase of the choice of k, the sbpath of P labeled v = yx k x k z is a path from Q i to Q i+1. Hence, i is a factor of v whence it is a factor of one of yx k x k and x k x k z. Bt this contradicts the hypothesis on the content of i since in that case, c( i ) A i or c( i ) A i+1. Hence i = j and so the existence of path Q is clear. Indeed, the sbpath P of P labeled (x k yx k ) k (x k zx k ) k is entirely in Q i. Therefore, if Q i = {q i } this is immediate. If Q i is not singlar (so that i = l and Q l is the state set of atomaton A l ), we dedce, since k is an exponent of S(L), that P is a path in A l from a state q Q l to the same state q. We also dedce that there is a path labeled (x k zx k ) k (x k yx k ) k from q to q. By symmetry, it follows that x k (x k yx k ) k (x k zx k ) k x k L x k (x k zx k ) k (x k yx k ) k x k proving that S(L) LECom. To conclde the proof, consider the syntactic morphism µ : A + S(L). Let w A + and sppose that µ(w) is a reglar element of S(L) so that µ(w) = µ(w k+1 ). Now let w A + be sch that µ(ww w) = µ(w) and µ(w ww ) = µ(w ). Then as above, one can show that, for every 1 i n, w A + i if and only if w A + i. Hence, the sbsemigrop of S(L) consisting of its reglar elements divides the direct prodct of the sbsemilattice of 2 A generated by the A i with the semigrops of the form i 1 G i i, where G i is the trivial grop if i l and is the grop S(A l ) otherwise, and i 1 (resp. i ) is a sfix of i 1 (resp. prefix of i ) with content contained in A i. The sbgrops of these semigrops are sbgrops of S(A l ) and so the proof is conclded. Before we present the characterization of the implicit operations on DReH LECom, we recall the notion of the Cayley graph of a grop. Let G be an A-generated grop. The Cayley graph of G is the labeled graph whose set of vertices is G, and, for every g G and a A, there exists an edge, labeled a, from vertex g to vertex ga. Theorem 5.3 Let H be a psedovariety of grops, let x, y ˆF A (DReH LECom) and let x = 0 [w 1, A 1, g 1, w 1] 1 n 1 [w n, A n, g n, w n] n and y = v 0 [z 1, B 1, h 1, z 1]v 1 v m 1 [z m, B m, h m, z m]v m be factorizations in normal form. Then x = y if and only if n = m, i = v i, w i = z i, A i = B i, g i = h i and w i = z i for all i. Proof. Consider the following atomaton C A 1 A 2 A n 0 p r (w 1 ) q 0 q 1 s r (w 1) 1 p r (w 2 ) q 2... s r (w n) q n n q n+1 where r 1 is an integer sch that r > v j for all 1 j m and sch that, for all 1 i n 1 with i = 1, the content of the word s r (w i ) is not contained in A i+1. This garantees that the content of the word s r (w i ) ip r (w i+1 ) is not contained in either A i or A i+1 and that the atomaton C is as in the conditions of Lemma 5.2. Hence, the

18 18 j. costa syntactic semigrop S of the langage L recognized by C is in DA LECom and so S DReH LECom. Now sing similar (and somewhat simpler) argments to those in the proof of Theorem 4.3, one can show that n = m, i = v i, w i = z i, A i = B i and w i = z i for all i. Now note that H = (DReH LECom) G. For every 1 i n, let w i A N i, w i A N i and ḡ i, h i ˆF Ai (H) be sch that [w i, A i, g i, w i ] = p r(w i )[ w i, A i, ḡ i, w i ]s r(w i ) and [w i, A i, h i, w i ] = p r(w i )[ w i, A i, h i, w i ]s r(w i ). Set x i = [ w i, A i, ḡ i, w i ] and ȳ i = [ w i, A i, h i, w i ]. Let s now fix an i {1,..., n} and consider an A i -generated grop G of H. Let A i be the Cayley graph of G over A i. Note that the transition semigrop of A i is G. Let A i = {a i,1,..., a i,ni } and let C be the following atomaton 0 p r (w q 0 1 ) q 1 A 1 A i 1 A i+1 s r (w i 1) i 1 p r (w... q i 1 i ) s r (w i) i p r (w i+1 ) s q i A i q i r (w n) n q i+1... q n A n q n+1 where the states q i and q i are, respectively, the elements 1 and ( x i ) G (a i,1,..., a i,ni ) of G. Denote by µ : A + S the syntactic homomorphism of the langage recognized by C and by ˆµ its continos homomorphic extension to ˆF A (DReH LECom) (which exists by Lemma 5.2). Moreover, consider an exponent k > 0 n + n of S and, for all j {1,..., n}, words x j and y j sch that c(x j ) = c(y j ) = A j, ˆµ( x j ) = µ(x k j ) and ˆµ(ȳ j) = µ(yj k ). We then have ˆµ( 0 p r (w 1 )ȳ 1 s r (w 1 ) 1p r (w 2 ) ȳ i 1 s r (w i 1 ) i 1p r (w i )) = µ( 0 p r (w 1 )y k 1 s r(w 1 ) 1p r (w 2 ) y k i 1 s r(w i 1 ) i 1p r (w i )), ˆµ(s r (w i ) ip r (w i+1 )ȳ i+1 ȳ n s r (w n) n ) = µ(s r (w i ) ip r (w i+1 )y k i+1 yk ns r (w n) n ), ˆµ(x) = µ( 0 p r (w 1 )x k 1 xk ns r (w n) n ). Using the eqalities proved so far, one can verify that 0 p r (w 1 )y k 1 yk i 1 s r(w i 1 ) i 1p r (w i ), s r (w i ) ip r (w i+1 )y k i+1 yk ns r (w n) n, 0 p r (w 1 )x k 1 xk ns r (w n) n, are the labels of paths in C from, respectively, q 0 to q i, q i to q n+1 and q 0 to q n+1. Since ˆµ(x) = ˆµ(y) = ˆµ( 0 p r (w 1 )ȳ 1 ȳ n s r (w n) n ) and A i is a permtation atomaton, it follows that (ȳ i ) G (a i,1,..., a i,ni ) = ( x i ) G (a i,1,..., a i,ni ), which shows that ḡ i = h i. Hence, g i = h i and the proof is conclded. One can verify, similarly to the case LJ above, that the eqality (R L) LECom = DA LECom does not hold. Let now V be one of the psedovarieties DRH LECom and DH LECom. We say that a factorization of an element x ˆF A (V) of the form x = 0 x 1 1 n 1 x n n is normal if: i A, 0 1 if x = 0 ; x i ˆF A (V) (1 i n) is reglar; if i (1 i n 1) is the empty word, then c(x i ) c(x i+1 ) = ; for each 1 i n sch