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MONOID KERNELS AND PROFINITE TOPOLOGIES ON THE FREE ABELIAN GROUP BENJAMIN STEINBERG

Abstract. To each pseudovariety of abelian groups residually contain-

ing the integers, there is naturally associated a pro nite topology on any nite rank free abelian group. We show in this paper that if the pseudovariety in question has a decidable membership problem, then one can eectively compute membership in the closure of a subgroup and more generally, in the closure of a rational subset of such a free abelian group. Several applications to monoid kernels and nite monoid theory are discussed.

1. Introduction In the early 1990's, the Rhodes' type II conjecture was positively answered by Ash [2] and independently by Ribes and Zalesskii [10]. The type II submonoid, or G-kernel, of a nite monoid is the set of all elements which relate to 1 under any relational morphism with a nite group. Equivalently, an element is of type II if 1 is in the closure of a certain rational language associated to that element in the pro nite topology on a free group. The approach of Ribes and Zalesskii was based on calculating the pro nite closure of a rational subset of a free group. They were also able to use this approach to calculate the Gp -kernel of a nite monoid. In both cases, it turns out the the closure of a rational subset of the free group in the appropriate pro nite topology is again rational. See [7] for a survey of the type II theorem and its motivations. Delgado [4], taking the proof of Ribes and Zalesskii as a model, computed the Gcom -kernel of a monoid by determining the closure of a rational subset of the free abelian group in the pro nite topology. Again, the closure of a rational subset was rational. In this paper, we give for any pseudovariety H of abelian groups, having decidable membership problem and which generates the variety of abelian groups, an algorithm for computing the pro-H closure of a rational subset of a nite rank free abelian group. Again the closure is rational. While this is a problem of independent interest, we show as a consequence that the H-kernel, H-liftable k-tuples, and H-pointlike Date : January 19, 1999. 1991 Mathematics Subject Classi cation. 20M35, 20E18, 20M07, 20K45. Key words and phrases. Abelian groups, pro nite topologies, monoids, rational languages, monoid kernels, pseudovarieties, Mal'cev products. The author was supported in part by Praxis XXI scholarship BPD 16306 98. 1

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pairs are decidable for any pseudovariety of abelian groups H with decidable membership problem. In addition, we obtain several join results for pseudovarieties of monoids. The key step is to rst compute the closure of a subgroup. The algorithm is based on linear algebra and the fundamental theorem of nitely generated abelian groups. The ideas of this proof arose from Delgado's [4] and the author's [14] where a large number of cases is handled and in fact, most of the arguments are similar to those used there. 2. Profinite Topologies A pseudovariety of monoids is a class V of nite monoids closed under the operations of taking nite products, submonoids, and homomorphic images. Examples include the pseudovarieties of nite commutative monoids, nite groups, and nite abelian groups. In this paper, we will mostly be concerned with pseudovarieties of abelian groups. We will use the symbol H exclusively for pseudovarieties of groups. Let G be any group. We now de ne the pro-H topology on G. One takes as a basis of neighborhoods of 1 all normal subgroups N with G=N 2 H and makes G a topological group in the standard way. We say that G is residually in H if f1g is closed, or equivalently, the topology is Hausdor. For example, a group is residually nite if and only if it is residually in G, where G denotes the pseudovariety of all nite groups. This topology has the following alternative description. Let g 2 G. Then we de ne r(g) = minf[G : N ]jG=N 2 H; g 2= N g with the convention that if no such N exists, then r(g) = 1. Then the H-pseudonorm is de ned by jgjH = 2?r(g) and satis es jg1 g2 jH maxfjg1 jH ; jg2 jH g. One then de nes an ultrametric ecart by d(g1 ; g2 ) = jg1 g2?1 jH and can easily check that the pro-H topology is de ned by this ecart. Furthermore, the topology is metric (and the pseudonorm a true norm) if and only if G is residually in H. For example, if G = Z, and H is the pseudovariety of p-groups, this topology is just the usual p-adic topology and the above norm is equivalent to the usual p-adic norm. In the cases of interest, the topology will indeed be metric. We use clH (X ) to denote the closure of a subset X G in this topology. The following elementary results are due to Hall [6]. Proposition 2.1. Let H be a pseudovariety of groups, G a group, and H a subgroup. T 1. H is open if and only if G=HG 2 H, where HG = g2G g?1 Hg, or equivalently, T if H is closed of nite index. 2. clH (H ) = open K H K .

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If G = FG(A), the free group on A, then clH (f1g) is the verbal subgroup (invariant under all endomorphisms) de ning the variety generated by H. We use FGH (A) to denote the relatively free group FG(A)=clH (f1g) in this variety. For example, if H = Gcom the pseudovariety of abelian groups, then clH (f1g) is the commutator subgroup and FGGcom (A) = A Z, the free abelian group generated by A. More generally, if G is any group, clH (f1g) is a normal subgroup and GH = G=clH (f1g) is the maximal, residually H image of G. One can of course take GbH , the pro nite completion of G in this topology, and in this case, GH is the image of G under the natural map. However, we will have no need to consider pro nite completions in this paper. Proposition 2.2. Let H be a pseudovariety and G a group. Then the natural projection ' : G ! GH is continuous, open, and closed where both groups are given the pro-H topology. Proof. That the map is open and continuous follows easily from the standard isomorphism theorems. To see that the map is closed, let X G be closed. Let y 2 clH (X') and g 2 G be such that '(g) = y. Let N be an open normal subgroup of G. Then N is closed as well, so clH (f1g) N . Also, K = '(N ) is an open normal subgroup of GH . So '(Ng) \ '(X ) = Ky \ '(X ) 6= ;. Since N contains ker ', we have that Ng \ X 6= ;. So g 2 clH (X ) = X and thus, y = '(g) 2 '(X ) as desired. 3. Supernatural Numbers and Pseudovarieties of Abelian Groups

By the fundamental theorem of nitely generated abelian groups, a pseudovariety of abelian groups is completely determined by its cyclic members. Q A supernatural number is a formal product p prime pnp where 0 np 1. There are evident notions for supernatural numbers of divides, lcm (least common multiple), and gcd (greatest common divisor). We use Nb for the set of supernatural numbers, which is actually a complete lattice, ordered by the relation divides. We want to establish a lattice isomorphism between Nb and the lattice of pseudovarieties of abelian groups. In this paper, we will use N to denote the positive integers. Being a subset of Nb , N is also a lattice under the relation divides (although no longer a complete lattice). We will call a subset X N a lter if m 2 X and njm, implies that n 2 X and, n; m 2 X , implies lcm(n; m) 2 X . The set of lters is a complete lattice ordered by inclusion. Proposition 3.1. There is a lattice isomorphism between Nb and the set of lters of N . Proof. To each supernatural number , one can associate the lter N of all natural numbers which divide it. Conversely, to any lter X , one can associate lcm(X ). It is easy to see that these associations are inverse lattice homomorphisms.

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We call a supernatural number recursive if N is a recursive set of natural numbers. Note that if n 2 N , then Nn is nite and hence recursive. Proposition 3.2. There is a lattice isomorphism between the set of lters of N and the lattice of pseudovarieties of abelian groups. Furthermore, a lter N is recursive if and only if the corresponding pseudovariety of abelian groups has decidable membership problem. Proof. Let H be a pseudovariety of abelian groups. Then we let NH = fnjZ=nZ 2 Hg. We show that this is a lter. Indeed, if m 2 NH and njm, then Z=mZ 2 H and mn generates a cyclic subgroup of Z=mZ of order n, so n 2 NH . If n; m 2 NH , then gcd(nn;m) 2 NH and so gcd(nn;m) m = lcm(n; m) is in NH by the Chinese remainder theorem. Conversely, if N is a lter, we can associate to it the pseudovariety HN = hZ=nZjn 2 N i. It is easy to show that these maps are inverse lattice homomorphisms using standard facts about products, subgroups, and quotients of cyclic groups. Suppose N is a recursive lter. Then if G is a nite abelian group, we can eectively, from its multiplication table, decompose it into a product of cyclic groups. Since N is recursive, we can check which of these are in HN . Conversely, if H has decidable membership, one can check if n 2 NH by just checking whether Z=nZ 2 H. We have thus established a lattice isomorphism between Nb and the lattice of pseudovarieties of abelian groups. If 2 Nb , we use H to represent the pseudovariety of all nite abelian groups whose torsion coecients divide . Conversely, to any pseudovariety H of abelian groups we can associate H = lcm(fnjZ=nZ 2 Hg). Corollary 3.3. The associations 7! H and H 7! H are inverse lattice isomorphisms. Furthermore, is recursive if and only if H has decidable membership problem. Normally, one uses supernatural numbers to represent orders of procyclic groups [5]. One can show that if Z is given the pro-H topology, then H is the order of the pro-H completion Zb H. Proposition 3.4. Let 2 Nb and A a set of cardinality n. Then if 2 N , FGH (A) = A (Z=Z). Otherwise, FGH (A) = FGGcom (A) = AZ. Proof. The rst statement is obvious since if 2 N , then any group in the variety generated by H satis es xy = yx and x = 1 and hence, is a Z=Z-module. The second statement follows upon noting that has arbitrarily large divisors. So if (a1 ; : : : ; an ) 2 nZ, then by choosing a divisor m of with m > ai all i, we see that (a1 ; : : : ; an ) 2= m(n Z), but nZ=m(nZ) 2 H . Thus we see that if n 2 N , then Hn is locally nite (that is, has a free object generated by any nite set), that the pro-H topology on that free object is discrete, and that membership in H is trivially decidable. Hence,


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all questions which we shall address in this paper are either uninteresting or trivial for such pseudovarieties. So for the rest of this paper, we will assume is an \in nite" supernatural number. 4. Closures of Subgroups Let A be a nite set of cardinality n, an in nite supernatural number, and H = H . We now describe the closure of a subgroup of F = A Z in the pro-H topology. The proof scheme generalizes techniques of the author's [14]. If G is a nitely generated abelian group, we use Gtor for its torsion subgroup. We will use frequently that given a nite presentation of G, one can eectively nd Gtor . See [9] for a proof of the fundamental theorem of nitely generated abelian groups which shows that everything can be done eectively. In this paper, we will write abelian groups additively. Lemma 4.1. Suppose that H F is closed. Then (F=H )tor 2 H. T Q Proof. SinceQH = open K H , F=H open K H F=K . Let g = (gK ) be an element of ( F=K )tor . Then ord(g) = lcm(ford(gK )jopen K H g) < 1 and divides , since each F=K 2 H. Now, we work towards the converse of this result. Lemma 4.2. Suppose H F is a subgroup with (F=H )tor 2 H and x 2 F n H . Then there is an open subgroup K H such that x 2= K . Proof. By the fundamental theorem of nitely generated abelian groups, there is a basis fe1 ; e2 ; : : : en g for F and positive integers a1 ; : : : ; ak such that fa1 e1 ; : : : ; ak ek g is a basis for H . Since (F=H )tor 2 H, the ai j. If x = b1 e1 + + bnen with bi > 0 some i > k, choose m > bi such that mj. This can be done since is in nite. Then K = ha1 e1 ; : : : ; ak ek ; ek+1 ; : : : ; ei?1 ; mei ; ei+1 ; : : : ; eni is as desired. Otherwise, if there is no such index i, then K = ha1 e1 ; : : : ; ak ek ; ek+1 ; : : : ; en i works. Corollary 4.3. Suppose H F and is an in nite supernatural number. Then H is closed if and only if (F=H )tor 2 H. In particular, if is recursive, it is decidable whether a subgroup is closed (given a nite generating set as input). Proof. We have already seen that the condition is necessary for being closed. But the above lemma shows that it is sucient, since any element of F n H can be separated from H by an open subgroup. The last statement follows since one can eectively compute the torsion coecients of F=H via row and column operations applied to the matrix whose columns are the generators of H .

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We now compute the closure of a subgroup H of F in the pro-H topology. The procedure is as follows; let e1 ; : : : ; en and a1 ; : : : ; ak be as in the proof of Lemma 4.2. Then the closure of H is the subgroup obtained by replacing each ai with bi = gcd(ai ; ). Of course, we must show that this works. Proposition 4.4. Let F = AZ with basis fe1 ; : : : ; en g and let H = ha1 e1 ; : : : ; ak ek i. For each i, let bi = gcd(ai ; ). Then clH (H ) = hb1e1 ; : : : ; bk ek i. Proof. By the above corollary, K = hb1 e1 ; : : : ; bk ek i is closed and clearly, H K . To show the converse, by continuity of addition, it suces to show that bi ei 2 clH (H ) for each i. Let ' : F ! G 2 H be a homomorphism. We show that '(bi ei ) 2 '(hai ei i). The result will then follow. If '(ei ) = 0, we are done. Otherwise h'(ei )i = Z=mZ where mj. Let ai = mi bi with gcd(mi ; ) = 1. Then mi is relatively prime to m, so there exists di such that di mi 1 mod m. So '(di ai ei) = di mi'(bi ei ) = '(bi ei ) and thus '(bi ei ) 2 '(hai ei i) as desired. Theorem 4.5. Let A be a nite set, F = AZ, a recursive, in nite supernatural number, H = H , and X F a nite subset. Then one can eectively compute a basis for and membership in clH (hX i). Proof. Let H = hX i. By the proof of the fundamental theorem of nitely generated abelian groups, we can eectively nd a basis fe1 ; : : : ; en g for F and positive integers a1 ; : : : ; ak such that fa1 e1 ; : : : ; ak ek g is a basis for H . Furthermore, we can eectively change between this basis and the standard basis. Since is recursive and we can eectively factor the ai , we can eectively compute each of the bi = gcd(ai ; ). So by the above proposition, fb1 e1; : : : ; bnen g is a basis for clH (H ). We can then eectively write this basis in terms of the original basis if we so desire. To check whether an element x 2 F is in clH (H ), we write it as x = c1 e1 + + cn en and determine if bi jci for 1 i k and ci = 0 for i > k. Next we show that if H is a proper subpseudovariety of Gcom corresponding to an in nite supernatural number, then in general for H-closed subgroups H and K of F , H + K is not closed. Indeed, let n be an integer such that Z=nZ 2= H. Let F = Z Z, H = h(1; 0)i, and K = h(1; n)i. Then since F=H = F=K = Z, by Corollary 4.3, H and K are closed. But H + K = h(1; 0); (1; n)i = h(1; 0); (0; n)i: So F=(H + K ) = Z=nZ 2= H. Hence H + K is not closed, again by Corollary 4.3. Thus the approach of [4] cannot immediately be generalized the way [11] generalizes [10]. Corollary 4.6. Let A be a nite set, X a nite collection of reduced words over A, a recursive, in nite supernatural number, and H = H . Then there is an algorithm to compute membership in clH (hX i) FG(A).


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Proof. Let H = hX i and ' : FG(A) ! FGGcom (A) be the canonical projection. We claim w 2 clH (H ) if and only if '(w) 2 clH ('(H )). To see this, rst note that Proposition 2.2 implies that clH ('(H )) = '(clH (H )). So we just need to show that if '(w) 2 clH ('(H )), then w 2 clH (H ). Let N be an open normal subgroup of FG(A). Then '(N ) is a normal open subgroup of FGGcom (A) and so '(Nw) \ '(H ) 6= ;. Since N contains ker ', it follow that Nw \ H 6= ; and so, w 2 clH (H ). See [8, 15] for some consequences of this result, although subsequent results will subsume these applications.

5. Closures of Rational Subsets Let M be a monoid. A subset L M is said to be recognizable if there exists a homomorphism ' : M ! N with N nite and L = '?1 (B ) with B N . The collection of recognizable subsets is denoted Rec(M ). If A is a set, we use A for the free monoid on A. If M is any monoid, and X M , we use X for the submonoid generated by X . A subset of M is called rational if it is in the smallest collection of subsets of M containing the nite subsets and closed under nite unions, nite products, and the operation X 7! X . The collection of rational subsets of M is denoted Rat(M ). A rational expression for a rational set is an expression like ((ab [ c)d ) which shows how to build the set up from the \rational operations". Kleene's theorem says that Rat(A ) = Rec(A ) for A nite and more generally, implies that Rec(M ) Rat(M ) if M is nitely generated. It is easy to show that if ' : M ! N is a homomorphism, then the image of a rational set is rational while the inverse image of a recognizable set is recognizable [3]. Let be a recursive, in nite supernatural number and H the associated pseudovariety of abelian groups. We now give an algorithm to compute the closure of a rational subset of F = A Z for A a nite set. First we note that if such an algorithm exists, then must be recursive. Indeed, the membership problem for H is the same as asking for A nite and N a nite index subgroup of F = AZ, with a given nite generating set, whether N is closed. But it is easy to see that if g + N = g0 + N , then g 2 clH (N ) if and only if g0 2 clH (N ). So to verify whether N is closed, it suces to check whether any element of a nite set of coset representatives of N in F is in clH (N ), but not in N . But if Y is a generating set of N , then N = (Y [ ?Y ) and hence is rational, so we can check this. A subset of a monoid M is said to be semilinear if it is a nite union of sets of the form ab1 b2 bn (n 0; b1 ; : : : ; bn 2 M ): The following proposition is straightforward and can be found in [4]. Proposition 5.1. Let M be a nitely generated commutative monoid. Then L 2 Rat(M ) if and only if it is semilinear.

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Given a nite presentation of a commutative monoid M , one can eectively place a rational subset in the above form by induction on the star height. Also note that since M is commutative, b1 + + bn = fb1 ; : : : ; bn g . So we have the following corollary. Corollary 5.2. Let M be a nitely presented commutative monoid. Then S any rational subset L of M can be eectively placed in the form i (ai + Bi ) with ai 2 M , Bi M a nite subset, and the union nite. Proposition 5.3. Let L = Si(ai + Bi) be a rational subset of F . Then S clH (L) = i (ai + clH (hBi i)). Proof. Since the pro-H topology is metric, taking the closure commutes with taking nite unions. So it suces to show that clH (a + B ) = a + clH (hB i). Since translation by a is a continuous automorphism, it suces to show that clH (B ) = clH (hB i). The inclusion from left to right is clear. By continuity of multiplication, it is easy to see that clH (B ) is a submonoid. So it suces to show ?B clH (B ). But if b 2 B , then (n! ? 1)b ! ?b so ?b 2 clH (B ). Corollary 5.4. Let be a recursive, in nite supernatural number, H the associated pseudovariety of abelian groups, and F a nitely generated free abelian group. The one can compute membership in the closure of a rational subset of F given a rational expression as input. Note that since any subgroup of a nite rank free abelian group is nitely generated, and hence a rational subset, we see that the closure of a rational subset is rational. Observe that in the case of Gcom , our results show that every subgroup is closed and hence, [ [ clGcom ( (ai + Bi)) = (ai + hBi i): i

i

This result was originally obtained by Delgado [4]. 6. Monoid Kernels and Applications to Monoid Theory We now give several applications to the theory of monoids. Let M and N be monoids. A relational morphism : M ?! N is a relation M N which is a submonoid projecting onto M . The most important example is the case where M and N are generated by a set A and = f(a; a)ja 2 Ag . If : M ?! G is a relational morphism with G a group, then ?1(1) is a submonoid of M . Fix a pseudovariety H of groups. Then for M nite, we let \ ?1(1): KH (M ) = :M ?! G2H This set is called the H-kernel of M and is a submonoid containing the idempotents of M . If V is a pseudovariety of monoids, then the Mal'cev


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m H is the pseudovariety of all nite monoids M with a relational product V morphism : M ?! G 2 H with ?1(1) 2 V. A simple exercise [7] shows m H if and only if KH (M ) 2 V. Hence, the computability that M 2 V of KH implies the decidability of membership for any pseudovariety of the m H with V having decidable membership. In the case where V is form V m H = V H where local in the sense of Tilson [16], one can show that V the right hand side is the semidirect product of pseudovarieties [7]. We now show that if H is a decidable pseudovariety of abelian groups, then KH is computable. If M is an A-generated monoid, we use [w]M to denote the image of a word w 2 A in M . Proposition 6.1. Let M be a nite monoid generated by a nite set A, an in nite supernatural number, H the associated pseudovariety of abelian groups, and F = A Z. For m 2 M , let Lm = f[w]F jw 2 A and [w]M = mg: Then m 2 KH (M ) if and only if 0 2 clH (Lm ). Proof. Suppose m 2 KH (M ). Let N be an open normal subgroup of F . Consider the relational morphism : M ?! F=N de ned by = f([a]M ; a + N )ja 2 Ag : Then since m 2 KH (M ), there exists w 2 A such that [w]M = m and [w]F 2 N . So N \ Lm 6= ;. It follows that 0 2 clH (Lm ). For the converse, let : M ?! G 2 H be a relational morphism. Choose a~ 2 ([a]M ) for each a 2 A. Let ' : F ! G be the morphism associated to the map a 7! a~. Let N = ker '. Then N is an open normal subgroup and so Lm \ N 6= ;. Thus, there exists w 2 A such that [w]M = m and [w]F 2 N . But by de nition of ', '([w]F ) 2 (m). So m 2 ?1 (0). It follows that m 2 KH (M ). Theorem 6.2. Let H be a pseudovariety of abelian groups with decidable membership. Then KH is computable. Proof. If H is locally nite, the result is trivial. So suppose H corresponds to a recursive, in nite supernatural number. Let M be a nite A-generated monoid. For m 2 M , the set of words w 2 A with [w]M = m is a rational subset and one can eectively compute a rational expression for it by Kleene's algorithm. Hence, Lm is a rational subset of F and one can eectively obtain a rational expression for it. So by Corollary 5.4, one can decide whether 0 2 clH (Lm ). Hence by the above proposition, KH is computable. Corollary 6.3. Let V be a pseudovariety of monoids and H a pseudovariety mH of abelian groups, each with decidable membership problem. Then V has decidable membership problem. A related algorithmic problem is the following. Let M be a nite monoid and H a pseudovariety of groups. Then (m1 ; : : : ; mk ) 2 M k is called

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an H-liftable k-tuple if for every relational morphism : M ?! G 2 H, there exists g1 ; : : : ; gk 2 G such that g1 : : : gk = 1 and gi 2 (mi ) for all i. For instance, an H-liftable 1-tuple is just a member of KH (M ). One can show in a similar manner to above that if H is a pseudovariety of abelian groups corresponding to an in nite supernatural number, then (m1 ; : : : ; mk ) is an H-liftable k-tuple if and only if 0 2 clH (Lm1 + + Lmk ). Since Lm1 + + Lmk is a rational subset, we obtain the following. Proposition 6.4. Let H be a decidable pseudovariety of abelian groups. Then one can compute H-liftable k-tuples. If M is a nite monoid, X M , and V a pseudovariety of monoids, then X is said to be V-pointlike if for every relational morphism : M ?! V2 V, one has that X ?1(v) for some v 2 V . For example, KH (M ) is an Hpointlike set. One can show that M 2 V if and only if its only V-pointlike subsets are singletons. Hence, if V has decidable pointlike pairs, then V has decidable membership. The following is proved in the same manner as Proposition 6.1, see for instance [4]. Proposition 6.5. Let M be a nite monoid generated by a nite set A and H be a non-locally nite pseudovariety of abelian groups. Then fm; ng M is H-pointlike if and only if 0 2 clH (Lm ? Ln ). Corollary 6.6. Let H be a pseudovariety of abelian groups with decidable membership problem. Then H-pointlike pairs are decidable. A locally nite pseudovariety V is said to be order computable if there is a computable bound on the size of the free object on any nite set (in this case the pseudovariety is necessarily decidable). Recall if V and W are pseudovarieties, then their join V _ W is the smallest pseudovariety containing them. The following two results are consequences of the author's [12, 13]. Theorem 6.7. If V is an order computable, locally nite pseudovariety and H is a pseudovariety of abelian groups with decidable membership problem, then V _ H has decidable pointlike pairs and hence, is decidable. For unde ned terms in the following theorem, see [1, 13]. Theorem 6.8. Let V be a pseudovariety of J-trivial monoids with a decidable word problem for its monoid of implicit operations and H a pseudovariety of abelian groups with decidable membership problem. Then V _ H has decidable pointlike pairs and hence is decidable. In particular, J _ H is decidable, where J is the pseudovariety of all nite J-trivial monoids. Finally, it is shown in [14] that if H is a pseudovariety of groups with decidable pointlike pairs, then J H is decidable. Theorem 6.9. Let H be a pseudovariety of abelian groups with decidable membership problem. Then J H has decidable membership problem. m H 6= J H for pseudovarieties of abelian It is shown in [15], that J groups, so this result is meaningful.


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References [1] J. Almeida, Finite Semigroups and Universal Algebra, World Scienti c, 1994. [2] C. J. Ash, Inevitable Graphs: A proof of the Type II conjecture and some related decision procedures, Int. J. Algebra and Comput. 1 (1991), 127{146. [3] J. Berstel, Transductions and context-free languages, Teubner, Stuttgart 1979. [4] M. Delgado, Abelian Pointlikes of a Monoid, Semigroup Forum 56 (1998), 339{361. [5] M. Freid and M. Jarden, Field Arithmetic, Springer-Verlag, 1986. [6] M. Hall Jr. A topology for free groups and related groups, Annals of Mathematics 52 (1950) 127{139. [7] K. Henckell, S. Margolis, J. -E. Pin, and J. Rhodes, Ash's type II theorem, pro nite topology and Malcev products. Part I, Int. J. Algebra and Computation 1 (1991) 411{436. [8] S. Margolis, M. Sapir, P. Weil, Closed subgroups in pro-V topologies and the extension problem for inverse automata, preprint 1998. [9] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Menlo Park 1984. [10] L. Ribes and P. A. Zalesski, On the pro nite topology on a free group, Bull. London Math. Soc. 25 (1993) 37{43. [11] , The pro-p topology of a free group and algorithmic problems in semigroups, Int. J. Algebra and Computation 4 (1994) 359{374. [12] B. Steinberg, On pointlike sets and joins of pseudovarieties, Int. J. Algebra and Computation 8 (1998) 203{231. [13] , On algorithmic problems for joins of pseudovarieties, Semigroup Forum (1999). To appear. , Inevitable Graphs and Pro nite Topologies: Some solutions to algorithmic [14] problems in monoid and automata theory stemming from group theory, Int. J. Algebra and Computation (1999). To appear. [15] , Finite state automata: A geometric approach, Tech. Rep. Univ. of Porto, 1999. [16] B. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure and Applied Algebra 48 (1987) 83{198.

Faculdade de Ci^encias, da Universidade do Porto, 4099-002 Porto, Portugal

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