ON POWER GROUPS AND EMBEDDING THEOREMS FOR

ON POWER GROUPS AND EMBEDDING THEOREMS FOR RELATIVELY FREE PROFINITE MONOIDS K. AUINGER AND B. STEINBERG Abstract. We determine those pseudovarieties of groups H for which the power monoids P (G), ranging over all groups G in H, satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of J -trivial monoids by groups in H. That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation PH = J ∗ H. The characterization is in terms of the geometry of the Cayley graphs of the free pro-H groups as well as in terms of the pro-H topology of a finitely generated free group.

1. Introduction One of the most celebrated results in finite monoid theory [17, 12, 11, 21], due to Henckell, Margolis, Pin and Rhodes (modulo Ash’s solution to pointlike conjecture [2]), is the equation m G = BG. PG = J ∗ G = J

Here J is the pseudovariety of J -trivial monoids, G is the pseudovariety of groups, m is the Mal’cev product, PG is the pseudovariety ∗ is the semidirect product, generated by power groups and BG is the pseudovariety of block groups. In [28] the second author intiated an investigation into which pseudovarieties of groups H satisfy the equation (1.1)

m H. J∗H=J

This work was extended in [30]. A characterization of all solutions to (1.1) was obtained by the authors in [6]. In this paper, we turn to the equation (1.2)

PH = J ∗ H.

Besides the groundbreaking work of Margolis and Pin [17] for the case of G, and a slight generalization by the second author [29, 27] to pseudovarieties closed under extension with p-group kernel for some fixed prime p, not much was known about this equation up until now. In particular, the only known non-solutions were pseudovarieties of Abelian groups, where inequality is obvious. The main result of this paper is that the solutions to (1.2) are precisely what are called Hall pseudovarieties in [30, 6]. These are exactly the pseudovarieties of groups H for which the notions of being H-extendible (in the sense of Margolis, The authors gratefully acknowledge support from INTAS projects 99–1224 Combinatorial and geometric theory of groups and semigroups and its applications to computer science. The second author was supported in part by NSERC and by the FCT and POCTI approved projects POCTI/32817/MAT/2000 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER.. 1

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Sapir and Weil [18]) and being H-closed coincide for finitely generated subgroups of a free group. In [6], it is shown that this is necessary for being a non-trivial solution to (1.1). Thus, all non-trival solutions to (1.1) are solutions to (1.2). The set of all solutions to (1.2) shares many of the properties of the set of all nontrivial solutions to (1.1): they form a right ideal in the monoid of pseudovarieties of groups with ∗ [7]; any extension-closed pseudovariety is a solution [29, 27]; any pseudovariety defined by a pseudoidentity of the form x π = 1 with π an infinite supernatural number is a solution [6]; they are join-irreducible; and they satisfy no non-trivial group identities. For example, m Gnil PGnil ( J ∗ Gnil ( J

holds for the pseudovariety Gnil of all nilpotent groups. An important and challenging open problem is whether or not there exists a solution to (1.2) that is not a solution to (1.1). Our main result can be phrased as follows: Theorem 1.1. For a pseudovariety of groups H, the following are equivalent: (1) PH = J ∗ H; (2) all inverse monoids from Sl ∗ H belong to PH; (3) H is Hall. The proof uses a mixture of techniques. To show that being Hall is necessary for H to satisfy (1.2), we make use of ordered monoids and prove an ordered analogue of Theorem 1.1. To show the sufficiency, we exhibit an explicit embedding of a finitely generated free pro-J ∗ H monoid into the pro-PH monoid of closed subsets of some free pro-H group. This is the main technical result and is of interest in its own right. The paper is organized as follows. We begin with a section containing background and notation. Then we introduce the notion of a Hall pseudovariety of groups and establish several fundamental properties of such pseudovarieties. We then prove our main result; the necessity in Section 4 and the sufficiency in Section 5. An additional section provides a language theoretic characterization of Hall pseudovarieties: we show that H is Hall if and only if for each finite alphabet A, the recognizable Hopen subsets of A∗ are the H-polynomials. We also establish an embedding theorem for free pro-J monoids. The final section contains some factorization results used to establish join decompositions of various pseudovarieties. 2. Background and Notation The reader is referred to [8, 1] for standard background in semigroup theory. We use A∗ for the free monoid on a set A; the free profinite monoid on a profinite set A c∗ . An element of a free profinite monoid is sometimes called a profinite is denoted A word. For a pseudovariety V of monoids and A a (pro)finite set, the A-generated free pro-V monoid is denoted Fc V (A); if V is locally finite and A is finite, then c FV (A) is finite and shall be denoted FV (A) — it is a free object in the usual sense. c∗ and an A-generated (pro)finite monoid M , [u]M will denote the For u ∈ A image of u in M under the natural homomorphism; in case M = FbV (A) we shall also use the notation [u]V for [u]M . For an element m of a profinite monoid M , mω−1 , as usual, denotes lim mn!−1 , while mω = lim mn! .

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For a profinite monoid M , we use P (M ) to denote the monoid of closed nonempty† subsets of M endowed with its natural multiplication. If M = lim Mi , then ←− P (M ) = lim P (Mi ) and so P (M ) is a profinite monoid. We frequently consider ←− P (M ) as an ordered monoid with the reverse inclusion ordering. Notice that ⊇ is a closed order. For convenience, we identify M with the set of all singletons in P (M ). Set P1 (M ) = {C ∈ P (M ) | 1 ∈ C}. This is a closed submonoid of P (M ) and is the order ideal generated by {1} in (P (M ), ⊇). Again, P1 (M ) = lim P1 (Mi ) whenever M = lim Mi . ←− ←− Let G be a (pro)finite group. Then the idempotents of P (G) are precisely the (closed) subgroups of G. Hence P (G) satisfies the pseudoidentity x ω ≤ 1. The submonoid P1 (G) satisfies the stronger pseudoidentity x ≤ 1. Clearly G acts continuously on P1 (G) by conjugation, so we can form the semidirect product P1 (G) o G. Margolis and Pin [17] observed that there is a continuous surjective morphism P1 (G) o G  P (G) given by (C, g) 7→ Cg. Moreover, if G = lim Gi , ←− then P1 (G) o G = lim P1 (Gi ) o Gi . Since any ordered monoid satisfying x ≤ 1 is ←− easily seen to be J -trivial [23], P (G) is a pro-J ∗ H monoid for any pro-H group G. Consequently, PH ⊆ J ∗ H [17] for each group pseudovariety H. Let J+ denote the pseudovariety of ordered monoids satisfying x ≤ 1 and P + H denote the pseudovariety of ordered monoids generated by all ordered monoids (P (G), ⊇) with G ∈ H. Since conjugation by G is order-preserving, in fact, we have P+ H ⊆ J+ ∗ H [27]. Recall also that every J -trivial monoid is a quotient of an element of J+ [25, 31]. Hence one can verify that every element of J ∗ H is a quotient of an element of J+ ∗ H. In particular, the inclusion J+ ∗ H ⊆ P+ H implies the inclusion J ∗ H ⊆ PH. Denote by Sl+ the pseudovariety of semilattice monoids, endowed with their natural order. Each naturally ordered inverse monoid in Sl ∗ H belongs Sl+ ∗ H. Recall that for u, v ∈ A∗ , the word u = a1 · · · an is a subword of v if v ∈ ∗ A a1 A∗ · · · A∗ an A∗ . This leads to the definition of a compatible order on A ∗ by setting v ≤ u if u is a subword of v. The order ideals for this order are often called shuffle ideals. They are always finitely generated [14] and so are finite unions of sets of the form A∗ a1 A∗ · · · A∗ an A∗ . Simon’s theorem states that the shuffle ideals generate the Boolean algebra of J-recognizable sets. Moreover, the positive variety of languages corresponding to the pseudovariety J + [23] comprises precisely the set of all shuffle ideals. Let X be any alphabet. For each n ∈ N, define a congruence on X ∗ by u ≡n v if u and v have the same subwords of length at most n. This is clearly a fully invariant congruence and is finite index if X is finite. Hence, ≡ n defines a locally finite ˇ n . Similarly we can define a variety of monoids Jˇn whose finite trace we denote J fully invariant compatible quasi-order ≤n on X ∗ by setting v ≤n u if all subwords of length at most n of u are subwords of v. The associated locally finite pseudovariety ˇ + . Simon’s theorem [25, 8, 19, 23] essentially of ordered monoids will be denoted J n states that [ [ ˇ n and J+ = ˇ+. J= J J n

† Some authors include the empty set in P (M ); in the cases of interest to us, things work out the same on the pseudovariety level.

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Let A be a finite alphabet and set Jn = A∗ /≡n = FJˇ n (A). For an element s of Jn , the notion of being a subword of s is well defined for any word w of length ˇ + -monoid by setting s ≤ p if each at most n. Then Jn becomes an ordered free J n subword of length at most n of p is a subword of s. For a (profinite) monoid M generated by a (profinite) set A, we use Γ A (M ) to denote its (profinite) Cayley graph; the reader is referred to [9, 6, 4] for basic notions concerning profinite graphs. Recall that for an A-generated group, the monoid M (G) denotes the Meakin-Margolis [15] expansion of G. It is an A-generated inverse monoid and consists of all pairs (X, g) where g ∈ G and X is a connected finite subgraph of ΓA (G), containing 1 and g. The projection M (G) → G, (X, g) 7→ g is a morphism and the induced congruence on M (G) is usually denoted σ. For a profinite group G, the profinite monoid M (G) is defined similarly, but all graph theoretic terms have to be interpreted correctly in the profinite context. For c a free pro-H group Fc H (A), M (FH (A)) is a free pro-Sl ∗ H inverse monoid, which we denote F\ IMH (A). Throughout this paper, the action of a group G on its Cayley graph will be denoted simply by multiplication on the left; in particular, for g ∈ G and Ξ a subgraph of the Cayley graph, gΞ is the image of Ξ under the action by the element g.

3. Hall Pseudovarieties The original motivation for considering Hall pseudovarieties comes from the paper [18] and an attempt by the second author [30] to extend the Ribes and Zalesski˘ı product theorem. Their importance was further emphasized in [6] where all nontrivial solutions to (1.1) turned out to be Hall. There are several equivalent definitions of a pseudovariety H of groups being Hall. Let F = F G(A) be the free group on A. Recall [26, 18, 13] that one can canonically assign a (finite) inverse automaton Γ H , with distinguished vertex 1, to any (finitely generated) subgroup of F . Let H be a pseudovariety of groups; a finitely generated subgroup H of F is H-extendible if Γ H can be embedded into a finite permutation automaton with transition group in H; see [18]. A pseudovariety H is Hall if each H-extendible subgroup H of F is closed in the pro-H topology of F . Hall’s celebrated theorem [10] stating that each finitely generated subgroup H of F is closed in the profinite topology of F can be formulated as: each G-extendible subgroup of F is closed in the pro-G topology of F , thus motivating the name of this definition. Margolis et. al. [18] have shown that each pro-H closed, finitely generated subgroup H of F is H-extendible. Consequently, a pseudovariety H is Hall if and only if being H-extendible and being H-closed are equivalent for each finitely generated subgroup of the free group F . Equivalently, H is Hall, if and only if the fundamental group of any connected finite inverse automaton over A whose transition inverse monoid has an E-unitary cover over H, is closed in the pro-H topology of F . The following result, proved in [6], is key to understanding Hall pseudovarieties. Theorem 3.1. A pseudovariety H of groups is Hall if and only if, for each finite alphabet A, any one of the following equivalent conditions is satisfied.

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(1) For each connected subgraph ∆ of a quotient graph of the Cayley graph of an A-generated group in H, the fundamental group of ∆ is H-closed in F G(A). (2) For each connected subgraph ∆ of the Cayley graph of an A-generated group in H, the fundamental group of ∆ is H-closed in F G(A). (3) Each free generator of F\ IMH (A) is the (unique) greatest element in its σ-class. (4) Any connected subgraph Λ of ΓA (Fc H (A)) containing both endpoints of an edge e contains e.

We recall various basic properties of Hall pseudovarieties and establish some new ones. Since the trivial subgroup {1} of F is extendible for each pseudovariety H, {1} must be closed for the pro-H topology if H is Hall. In particular, a Hall pseudovariety does not satisfy any non-trivial group identity and therefore residually contains all free groups. Each arboreous pseudovariety in the sense of [6] is Hall; these are precisely the non-trivial solutions to (1.1). In particular, each locally extensible pseudovariety H [7] is also Hall; a pseudovariety is called locally extensible if, for each G ∈ H, there exists a cyclic group C for which the wreath product C o G also belongs to H. This latter class contains each pseudovariety H which is defined by a pseudoidentity of the form xν = 1 where ν is any infinite supernatural number [7]. It turns out that the class of all Hall pseudovarieties shares many properties with the class of all arboreous pseudovarieties. The first result requires Theorem 1.1 and the authors are not aware of a direct proof of it; the next two results are easily established by slightly modifying the respective proofs for the arboreous case in [6].

Theorem 3.2. The set of all Hall pseudovarieties forms a right ideal in the monoid of pseudovarieties of groups. Proof. It is shown in [7, Corollary 2.4] that the set of all solutions to (1.2) forms a right ideal.  Theorem 3.3. If a Hall pseudovariety H is contained in the join V ∨ W of two pseudovarieties of groups V and W, then H is contained in V or W. In particular, H is (strictly) join irreducible. Proof. Let H be Hall and V and W be pseudovarieties of groups such that H ⊆ V ∨ W but H 6⊆ W. We need to show that H ⊆ V. There exists a finite alphabet Y and σ ∈ Fc G (Y ) such that W |= σ = 1 but H 6|= σ = 1.

Take any finite alphabet X disjoint from Y and any π ∈ Fc G (X) for which V |= π = 1 holds. We need to show that H |= π = 1 holds as well. By choice of σ and π, V ∨ W |= πσ = σπ

hence also

V ∨ W |= π a σ = σπ a

where π a = a−1 πa and a is a letter not in X ∪ Y . From H ⊆ V ∨ W we have H |= πaσa−1 π −1 aσ −1 = a.

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c Let Π and Σ be, respectively, the Cayley graphs of Fc H (X) and FH (Y ), both c considered as subgraphs of the Cayley graph of FH (A) where A = X ∪ {a} ∪ Y . Then ∆ = Π ∪ {(π, a)} ∪ πaΣ ∪ {(πaσa−1 , a)}

∪ πaσa−1 Π ∪ {(πaσa−1 π −1 , a)} ∪ πaσa−1 π −1 aΣ

is a subgraph of the Cayley graph of Fc H (A) containing 1 and a. Since H is Hall, ∆ contains the edge (1, a). But the only edges in ∆ labeled by a are (π, a), (πaσa−1 , a), (πaσa−1 π −1 , a).

Consequently, one of the following pseudoidentities must be valid in H: π = 1, πaσa−1 = 1, πaσa−1 π −1 = 1. If it were true that H |= πaσa−1 π −1 = 1, we could erase the letters from X and a to obtain H |= σ = 1, contradicting to the choice of σ. Hence H satisfies π = 1 or πaσa−1 = 1; in the first case we are done while in the second case we may erase all letters from Y and a to obtain π = 1 as well. Altogether, H |= π = 1. Since π = 1 was chosen arbitrarily among the pseudoidentities valid in V, we arrive at H ⊆ V.  The property of Hall pseudovarieties established in the above theorem goes by such names as: complete join irreducibility, being co-prime and join irreducibility. Like the set of all arboreous pseudovarieties, the set of all Hall pseudovarieties is not a left ideal. The following proposition shows that, for example, the pseudovariety Ab2 ∗ G3 is not Hall, where Abp is the pseudovariety of elementary Abelian p-groups and Gp is the pseudovariety of p-groups. The argument extends the arboreous case [6]. Proposition 3.4. Suppose that V is a pseudovariety satisfying a non-trivial group identity w = 1 and W is a pseudovariety of groups such that V ∗ W ) W. Then V ∗ W is not Hall.

Proof. Let H = V ∗ W; since W ( H there exists a finite alphabet A and π ∈ F\ V∗W (A) such that π 6= 1, but W |= π = 1. Write π = π(A) to indicate the “variables” occurring in π. Moreover there exists a reduced word w = w(z1 , . . . , zn ) ∈ F G(z1 , . . . , zn ) such that V satisfies the identity w = 1. Choose w to be as short as possible. Without loss of generality, we may assume that z n is the last letter of w. Now choose 2n pairwise disjoint finite alphabets A 1 , B1 , . . . , An , Bn of size |A|, set X = A1 ∪ B1 ∪ · · · ∪ An ∪ Bn and form π(A1 ), π(B1 ), . . . , π(An ), π(Bn ) ∈ F\ V∗W (X). Let ν = w(π(A1 )π(B1 ), . . . , π(An )π(Bn )) ∈ F\ V∗W (X)

be obtained from w by substituting each zi with π(Ai )π(Bi ). Let a be a letter that is not in X and ν 0 be obtained by replacing all π(Bn ) with aπ(Bn )a−1 . Then H |= ν 0 = 1. The element ν 0 can be represented as a “reduced product” in the terms π(Ai )±1 , π(Bi )±1 , a±1 and as such ν 0 is of the form µa−1 and µ is again such a product. Multiplying both sides by a, we obtain H |= µ = a. Using the minimality of w and an argument similar to that used in the proof of Theorem 3.3, one can construct a subgraph of the Cayley graph of F\ V∗W (X ∪ {a})

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which contains both endpoints 1, a of the the edge e = (1, a) but not e itself. In particular, V ∗ W cannot be Hall.  The next result will be of essential use in sections 5 and 6, but seems to be remarkable in its own right. Lemma 3.5. Let H be a Hall pseudovariety, A, B disjoint alphabets and D = A∪B. Take any ξ0 , . . . , ξt ∈ Fc H (A) and b1 , . . . , bt ∈ B, and set δ = ξ0 b1 ξ1 b2 · · · ξt−1 bt ξt .

Then each sequence (wn ) of words wn ∈ D ∗ which converges in Fc H (D) to δ has a subsequence (wnk ) for which each member admits a factorization wnk = x0,k b1 x1,k · · · xt−1,k bt xt,k

with limk→∞ xj,k = ξj for each j = 0, . . . , t.

Proof. Let H be a Hall pseudovariety. Write [ H= Hn

where the pseudovarieties Hn form an ascending chain of locally finite subpseuc dovarieties of H. Set G = Fc H (A), H = FH (D), Hn = FHn (D), Γ = ΓD (H) and Γn = ΓD (Hn ). The proof is by induction on t, the case t = 0 being trivial. First observe that the edges (ξ0 , b1 ), (ξ0 b1 ξ1 , b2 ), . . . , (ξ0 b1 ξ1 · · · ξt−1 , bt ) of Γ are pairwise distinct. To prove this, it suffices to show that their respective initial vertices are distinct. Indeed, suppose that (3.1)

ξ 0 b 1 ξ1 · · · b i ξi = ξ 0 b 1 ξ1 · · · b i ξj

for some i < j. Erasing in (3.1) all letters from A, we find that the identity b1 · · · b i = b 1 · · · b j

holds in H, which contradicts the fact that H does not satisfy any non-trivial identity. Let (wn ) be a sequence of words wn ∈ D ∗ with lim[wn ]H = δ. By going to c∗ to some ρ for which a subsequence we may assume that (wn ) converges in D [ρ]H = δ; set (∆, δ) = [ρ]M (H) . Then, in particular, (wn ) converges in M (H) to (∆, δ). Let Ξ = ΓA (G) considered as a subgraph of Γ and for any j ∈ {0, . . . , t − 1} set Let

η j = ξ 0 b 1 ξ1 · · · b j ξj .

Ψ = η0 b1 Ξ ∪ {(η1 , b2 )} ∪ η1 b2 Ξ · · · ∪ {(ηt−1 , bt )} ∪ ηt−1 bt Ξ. Then (ξ0 , b1 ) ∈ / Ξ ∪ Ψ. Also, Ξ and Ψ are connected subgraphs of Γ. On the other hand, the graph ∆ ∪ Ξ ∪ Ψ is connected and contains both endpoints of the edge (ξ0 , b1 ); hence ∆ contains (ξ0 , b1 ) by Theorem 3.1. Since (wn ) converges to (∆, δ) in M (H) and since the edge (ξ 0 , b1 ) belongs to ∆, there exists an ascending sequence N0 ≤ N1 ≤ · · · of positive integers such that, for each l ≥ 0 and each i ≥ Nl , the path in Γl starting at 1, labeled wi contains the edge ([ξ0 ]Hl , b1 ).

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It follows that, for each l ≥ 0 and each i ∈ {Nl , Nl + 1, . . . , Nl+1 − 1} the word wi admits a factorization wi = z i b 1 y i such that [zi ]Hl = [ξ0 ]Hl . In particular, in G and as well as in H, lim zi = ξ0 .

i→∞

For the sequence of suffixes (yi ), there exists a subsequence (yik ) which converges in H to some element α. By going to an appropriately chosen subsequence of (w i ), we may as well assume that (yi ) converges to α. Then ξ0 b1 ξ1 b2 · · · ξt−1 bt ξt = δ = lim [zi b1 yi ]H = ξ0 b1 α. i→∞

In particular lim[yi ]H = α = ξ1 b2 · · · ξt−1 bt ξt .

By the inductive hypothesis there exists a subsequence (y ik ) of (yi ) for which each member admits a factorization yik = xk,1 b2 · · · bt xk,t such that lim[xk,j ]H = ξj for each j = 1, . . . , t. Setting xk,0 = zik then yields the desired factorization of the words wik .  The following Corollary is immediate from Lemma 3.5 and, in a sense, asserts the ability of each Hall pseudovariety to witness certain finite subwords of profinite words. Corollary 3.6. Let H be a Hall pseudovariety, A, B disjoint alphabets and D = c∗ be such that, in H, ρ admits a factorization A ∪ B. Let ρ ∈ D [ρ]H = ξ0 b1 ξ1 b2 · · · ξt−1 bt ξt

with ξi ∈ Fc H (A) and bi ∈ B for all i. Then b1 · · · bt is a subword of ρ, that is, there c∗ (i = 0, . . . , t) such that exist Xi ∈ D (3.2)

ρ = X0 b1 X1 b2 · · · Xt−1 bt Xt .

Moreover, the elements Xi can be chosen such that [Xi ]H = ξi for all i. c∗ to ρ. Then lim[wn ]H = [ρ]H and Proof. Let (wn ) be a sequence converging in D so, by Lemma 3.5, there exists a subsequence (wnk ) such that wnk = xk,0 b1 xk,1 b2 · · · xk,t−1 bt xk,t

and lim[xk,j ]H = ξj for each j. By going to a further subsequence we may assume c∗ to some element Xj . Then [Xj ]H = ξj that each sequence (xk,j )k∈N converges in D for each j and ρ = lim wnk = X0 b1 X1 b2 · · · Xt−1 bt Xt , as required.



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It should be mentioned that Lemma 3.5 and Corollary 3.6 hold under the more general assumptions of Lemma 8.1 and Corollary 8.2 formulated in Section 8. We end this section by pointing out that certain join decompositions established by Almeida and Weil [3] for the case of so-called arborescent pseudovarieties (and that hold more generally for arboreous pseudovarieties [6]) remain valid in the setting of Hall pseudovarieties. Recall that ECom is the pseudovariety of monoids with commuting idempotents, DH is the pseudovariety of monoids whose regular D-classes belong to the pseudovariety of groups H, ACom is the pseudovariety of aperiodic commutative monoids, and ZE(H) is the pseudovariety of monoids with central idempotents and subgroups in H. The proof of these results depends on another observation concerning Hall pseudovarieties and is deferred to Section 8. Theorem 3.7. For each Hall pseudovariety H, the join decompositions (J ∩ ECom) ∨ H = DH ∩ ECom and ACom ∨ H = ZE(H) are valid. 4. The Necessity of Being Hall In this section, we show that being Hall is necessary for a pseudovariety H to be a solution to (1.2). The first lemma is a generalization to the context of ordered monoids of the well-known result that any permutative identity satisfied by a semigroup S is satisfied by P (S) [1]. Lemma 4.1. Let M be a monoid and suppose that M satisfies a pseudoidentity π = w where w ∈ A∗ has no repeated letters. Then (P (M ), ⊇) satisfies the pseudoidentity π ≤ w. Proof. Suppose that w = a1 · · · an . Let f : A → P (M ) be any map; we also denote c∗ → P (M ). Take any by f the induced morphism A x ∈ wf = (a1 f ) · · · (an f ).

By the assumption on w, we may choose for each i a well-defined element m ai ∈ ai f such that x = ma1 · · · man ; for a ∈ A \ {a1 , . . . , an } choose any element ma ∈ af . We thus have a mapping g : A → M given by a 7→ ma . Denote also by g the c∗ → M . induced morphism A c∗ . By construction, vg ∈ vf for each v ∈ A∗ whence σg ∈ σf for each σ ∈ A Since M |= π = w, we also have x = (a1 · · · an )g = πg. Therefore x = πg ∈ πf . Since x was arbitrarily chosen from (a1 · · · an )f , we obtain πf ⊇ (a1 · · · an )f = wf , as desired.  Corollary 4.2. Let V be a pseudovariety. If V |= π = w where w ∈ A∗ has no repeated letters, then P+ V |= π ≤ w. Next let A be a finite alphabet, A0 = {a0 | a ∈ A} be a disjoint copy of A with a 7→ a0 a bijection from A to A0 . Set A = A ∪ A0 and for a0 ∈ A0 let a00 = a; then a 7→ a0 (a ∈ A) is a permutation of A. In the usual way, this extends to an involution of A∗ by setting (a1 · · · an )0 = a0n · · · a01 .

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c∗ if and only if the sequence Moreover, a sequence (wi ) of words in A∗ converges in A 0 (wi ) converges, and if two sequences (wi ) and (vi ) have the same limit then so do (wi0 ) and (vi0 ). This follows immediately from the two facts that (i) the map a 7→ a0 induces an automorphism on each relatively free A-generated monoid M and hence (ii) for two words u, v ∈ A∗ and an A-generated relatively free monoid M , [u]M = [v]M if and only if [u0 ]M ρ = [v 0 ]M ρ where M ρ is the reverse (or dual) monoid, obtained by transposing the multiplication table. Altogether, the involution 0 : c∗ → A c∗ . Using the A∗ → A∗ extends uniquely to a (continuous) involution 0 : A 0 + + involution , we are able to establish that the equality J ∗ H = P H is valid only in case H is Hall. Corollary 4.3. If H is not Hall then Sl+ ∗ H 6⊆ P+ H. In particular, J+ ∗ H 6= P+ H. Proof. Suppose that H is not Hall. According to Theorem 3.1, there exists an alphabet B and an element (∆, b) ∈ F\ IMH (B) with b a letter of B such that the edge (1, b) does not belong to ∆. Let H = Fc H (B) be the free pro-H group on B; c∗ be such that, subject to the then F\ IMH (B) = M (H). Let A = B and σ ∈ A substitution a 7→ [a]M (H) , a0 7→ [a]−1 (for all a ∈ B), [σ]M (H) = (∆, a). Then, M (H) −1 0 subject to a 7→ [a]H , a 7→ [a]H , [σ]H = [a]H . c∗ be obtained from σ by substituting (a0 a)ω−1 a0 for each a0 ∈ B 0 ; that Let π ∈ A c∗ determined by a 7→ a, a0 7→ is, π is the composition of the endomorphism of A 0 ω−1 0 (a a) a (a ∈ B) with σ. In other words, if B = {a1 , . . . , an } then π(a1 , . . . , an , a01 , . . . , a0n ) = σ(a1 , . . . , an , (a01 a1 )ω−1 a01 , . . . , (a0n an )ω−1 a0n ).

Then again subject to the original substitution a 7→ [a] M (H) , a0 7→ [a]−1 M (H) , we have that ω−1 0 [(a0 a)ω−1 a0 ]M (H) = ([a]−1 [a]−1 M (H) [a]M (H) ) M (H) = [a ]M (H) for each a ∈ B,

whence [π]M (H) = [σ]M (H) = (∆, b) 6≤ [b]M (H)

and the latter just states that the naturally ordered pro-Sl + ∗ H inverse monoid M (H) does not satisfy the pseudoidentity π ≤ b. On the other hand, it follows from [σ]H = [b]H and the construction of π that H satisfies the pseudoidentity π = b. Trivially, the single letter word b has no repeated letters, whence PH+ |= π ≤ b by Corollary 4.2. Consequently, Sl+ ∗ H 6⊆ P+ H, and thus J+ ∗ H 6= P+ H, as required.  We extend this result to the unordered setting. To do this, recall that each power group satisfies xω ≤ 1 subject to the order ⊇ and so the following lemma applies.

Lemma 4.4. Suppose that (M, ≤) is an ordered monoid in [[xω ≤ 1]]. Then ≤ extends the natural order on idempotents; that is, for e, f ∈ E(M ), e ≤ f ⇐⇒ ef = f e = e.

Proof. Suppose that e ≤ f . Then multiplying both sides on the left by e, we obtain e ≤ ef . But since f ≤ 1, ef ≤ e. So e = ef . The equality f e = e is dual. Conversely, if ef = f e = e, then e = ef ≤ f since e ≤ 1.  Corollary 4.5. If H is not Hall, then Sl ∗ H 6⊆ PH. In particular, J ∗ H 6= PH.

ON POWER GROUPS

11

Proof. Suppose that H is not Hall and let B, A, π, b, H, M (H) be as in the proof of Corollary 4.3; then subject to the substitution a 7→ [a] M (H) , a0 → 7 [a]−1 M (H) , one −1 0 has [π ]M (H) = [π]M (H) . By choice of b and π, −1 [π]M (H) [π]−1 M (H) 6≤ [b]M (H) [b]M (H) ,

whence

−1 −1 [π]M (H) [π]−1 M (H) [b]M (H) [b]M (H) 6= [π]M (H) [π]M (H) .

This implies, in particular, that and so

F\ IMH (B) = M (H) 6|= (ππ 0 )ω (bb0 )ω = (ππ 0 )ω

Sl ∗ H 6|= (ππ 0 )ω (bb0 )ω = (ππ 0 )ω . On the other hand, H |= π = b implies H |= π 0 = b0 (as each group is isomorphic to its reverse via the map g 7→ g −1 ). Let G ∈ H; by Corollary 4.2, whence and therefore also So, by Lemma 4.4,

P (G) |= π ≤ b, π 0 ≤ b0 , P (G) |= ππ 0 ≤ bb0

P (G) |= (ππ 0 )ω ≤ (bb0 )ω .

P (G) |= (ππ 0 )ω (bb0 )ω = (ππ 0 )ω . Since G ∈ H was arbitrarily chosen, and the result follows.

PH |= (ππ 0 )ω (bb0 )ω = (ππ 0 )ω



As a consequence of this result, [6, Theorem 8.3] and the fact that Hall pseudovarieties are join-irreducible (Theorem 3.3) we obtain: Theorem 4.6. For each non-Hall pseudovariety H, the inequalities (4.1)

m H PH ( J ∗ H ( J

hold. In particular, (4.1) is valid for H = Gnil the pseudovariety of all nilpotent groups and more generally for each join reducible pseudovariety H. 5. Profinite Embeddings and the Sufficiency of Being Hall In this section, we prove our main result: namely, for each Hall pseudovariety H and each finite alphabet A, the free profinite monoid F[ J∗H (A) (order) embeds This establishes the sufficiency of the Hall condition for H being into P (Fc (A)). H a solution to (1.2). The reader should compare with the argument proving PG = J ∗ G [17, 21], which implies that any finite A-generated member of J ∗ G divides the power group of an A-generated group. We begin with a lemma on profinite power monoids. Lemma 5.1. Let {ϕm,n : Mm → Mn | m, n ∈ N, m ≥ n} be an inverse system of finite monoids and M = lim Mn . Let X, Xk ∈ P (M ), k ∈ N; suppose that (Xk ) ←− converges to X. Then, for x ∈ M , x ∈ X if and only if, for each k, there exists xk ∈ Xk such that (xk ) converges to x.

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K. AUINGER AND B. STEINBERG

Proof. Let ϕn : M → Mn be the canonical projection and x ∈ X. By the assumption that Xk converges to X and since the monoids Mi are finite, there exists an ascending sequence K1 < K2 < . . . of positive integers such that for each i and each k ≥ Ki , Xk ϕi = Xϕi . For a given i and k ∈ {Ki , . . . , Ki+1 − 1}, choose an element xk ∈ Xk such that xk ϕi = xϕi . For k < K1 let xk ∈ Xk be chosen arbitrarily. We claim that lim xk = x. Let i be a positive integer. In order to verify the claim, it suffices to show that xk ϕi = xϕi for all k ≥ Ki . Indeed, let k ≥ Ki ; then there exists a (unique) j ≥ i such that Kj ≤ k < Kj+1 . By definition, xk ϕj = xϕj and thus xk ϕi = xk ϕj ϕj,i = xϕj ϕj,i = xϕi .

Conversely, let, for each k, xk ∈ Xk and suppose that the sequence (xk ) converges to x in M . As above, we have integers K1 < K2 < . . . such that, for each i, and, for each k ≥ Ki , Xk ϕi = Xϕi . −1 is closed, In particular, xk ∈ Xϕi ϕi for all k ≥ Ki . Since the set Xϕi ϕ−1 i T −1 x ∈ Xϕi ϕ−1 . This holds for each i, whence x ∈ Xϕ ϕ = X, the latter equality i i i being valid because X is closed.  Let us establish some notation. For a Hall pseudovariety H let (H n )n∈N be an ascending chain of locally finite pseudovarieties such that [ Hn = H.

Fix a finite alphabet A. Set G = Fc H (A) and Gn = FHn (A). Then G = lim Gn for ←− the natural inverse system G1 ← G2 ← · · · ← Gn ← Gn+1 ← · · ·

whose morphisms we do not explicitly name. Let Γ = Γ A (G) be the profinite Cayley graph of G and Γn = ΓA (Gn ) be the Cayley graphs of the groups Gn (both with respect to A). The morphisms Gn+1 → Gn induce graph morphims Γn+1 → Γn and again, Γ = lim Γn . ←− We fix the following notation: C the free pro-J monoid on the profinite set G × A ˇ n -monoid on Gn × A Cn = (Gn × A)∗ /≡n , the free J F the free pro-J ∗ H monoid on A = the closed submonoid of C o G generated by {((1, a), a) | a ∈ A} ˇ n ∗ Hn monoid on A Fn the free J = the submonoid of Cn o Gn generated by {((1, a), a) | a ∈ A}, and note that C = lim Cn and F = lim Fn . Moreover, Cn with its natural order ←− ←− ˇ + . Hence C with the inverse limit order is a (see Section 2) is free on Gn × A in J n free pro-J+ monoid on the profinite set G × A. One can verify by a compactness argument† that the order on C is given by π ≤ ρ if and only if every finite subword of ρ is a subword of π, but we shall not use this fact. If we give Cn o Gn the product order and Fn the induced order, then Fn is free ˇ + ∗ Hn on A. Hence if we give F the inverse limit order, we see that F is a in J n free pro-J+ ∗ H monoid on A. Notice that the inverse limit order on F is just the restriction of the product order on C o G. †

The authors are grateful to J. Almeida for pointing this out to them.

ON POWER GROUPS

13

Theorem 5.2. Let H be a Hall pseudovariety and A be a finite alphabet; then the map a 7→ {a, a0 } τ : A → P (Fc H (A)), induces a continuous order embedding

c (F[ J∗H (A), ≤) ,→ (P (FH (A)), ⊇).

Proof. We retain the notation introduced above and set H = Fc H (A). Within the profinite monoid P (H), let P denote the closed submonoid generated by Aτ = {{a, a0 } | a ∈ A}; note that P is then an A-generated profinite monoid via τ . We consider P as a profinite ordered monoid with the induced order ⊇. For convenience, c∗ and G; likewise, A∗ shall be viewed as a subset we view A∗ as a subset of both A c∗ and H. Our goal is to show that τ induces a continuous order isomorphism of A from F onto P ; that is, P is a free pro-J ∗ H, in fact, a free pro-J + ∗ H monoid. First note that τ extends to a unique, surjective, continuous, order-preserving morphism τˆ : F → P since P is a pro-J+ ∗ H monoid and F is free. It suffices therefore to show that τˆ is an order embedding. That is, we must show that if c∗ are such that [σ]P ≤ [π]P , then [σ]F ≤ [π]F . π, σ ∈ A We know that [σ]F = (b σ , [σ]G ) where σ b is a uniquely determined element of C; we use the analogous notation for π. We must then show that [σ] G = [π]G and that π ]Cn for all n. σ b≤π b; the latter inequality is equivalent to asking that [b σ ] Cn ≤ [b Denote by ψ : H → G ⊆ H the unique morphism induced by a 7→ a, a 0 7→ a for each a ∈ A. Then ψ induces an endomorphism of P (H), which, abusing notation, we also denote by ψ, given by X 7→ {xψ | x ∈ X}. Since, for each a ∈ A, [a]P ψ = aτ ψ = {a} it follows that (5.1)

[µ]P ψ = µˆ τ ψ = {[µ]G } for each µ ∈ F.

Consequently, [σ]P ≤ [π]P implies {[σ]G } = [σ]P ψ ⊇ [π]P ψ = {[π]G }, establishing that [σ]G = [π]G ; call this common element γ. We are left with showing that σ b≤π b in C, i.e. showing that [b σ ] Cn ≤ [b π ]Cn for all positive integers n. So, let us fix n ∈ N and set s = [b σ ] Cn , p = [b π ]Cn and g = [σ]Gn = [π]Gn = [γ]Gn . Moreover, [σ]Fn = (s, g), [π]Fn = (p, g). With this notation, we are left with showing that s ≤ p. c∗ . For any word Let (uk ) be a sequence of words in A∗ converging to π in A ∗ u ∈ A , let u denote the unique path in Γn starting at 1 and labeled u, viewed as an element of (Gn × A)∗ . Then [uk ]Fn = ([uk ]Cn , [uk ]Gn ) and since limk→∞ [uk ]Fn = (p, g) we have that (5.2)

[uk ]Cn = p and [uk ]Gn = g

for all sufficiently large k. By going to a subsequence, we may assume without loss of generality that (5.2) holds for all k. By definition of the order ≤n on Cn , we must show that every t-tuple ((g1 , a1 ), (g2 , a2 ), . . . , (gt , at )),

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with (gi , ai ) ∈ Gn × A and t ≤ n, that is a subword of p, is also a subword of s. The key idea of the proof is that the elements of A0 serve as “markers” to encode edges appearing in such a subword; so, for instance, if ξ0 a01 ξ1 a02 · · · ξt−1 a0t ξt ∈ (b π , [π]G )ˆ τ

with ξj ∈ G for all j, then, heuristically speaking,

((ξ0 , a1 ), (ξ0 a1 ξ1 , a2 ), . . . , (ξ0 a1 ξ1 a2 · · · ξt−1 , at ))

should be a subword of π b. So let us suppose that t ≤ n and let (5.3)

((g1 , a1 , ), (g2 , a2 ), . . . , (gt , at ))

be a subword of p (for some g1 , . . . , gt ∈ Gn and a1 , . . . , at ∈ A). It follows that, for each k, uk has a factorization uk = Zk,0 (g1 , a1 )Zk,1 (g2 , a2 ) · · · Zk,t−1 (gt , at )Zk,t

with Zk,i ∈ (Gn × A)∗ . Consequently, uk has a corresponding factorization uk = xk,0 a1 xk,1 a2 · · · xk,t−1 at xk,t

such that (5.4)

[xk,0 ]Gn = g1 , [xk,0 a1 xk,1 ]Gn = g2 , . . . , [xk,0 a1 · · · xk,t−1 ]Gn = gt .

By taking a subsequence, we may assume without loss of generality that, for each i = 0, . . . , t, the sequence (xk,i )k∈N converges in G to some element ξi . In particular, γ = lim uk = lim xk,0 a1 xk,1 a2 · · · xk,t−1 at xk,t = ξ0 a1 ξ1 a2 · · · ξt−1 at ξt . k→∞

k→∞

For i = 0, . . . , t, let hi = [ξi ]Gn . From (5.4), it follows that h0 = g1 , h0 a1 h1 = g2 , . . . , h0 a1 · · · at−1 ht−1 = gt .

Moreover, the word u fk ∈ A∗ defined by

u fk = xk,0 a01 xk,1 a02 · · · xk,t−1 a0t xk,t

belongs to [uk ]P and in H,

lim u fk = lim xk,0 a01 · · · xk,t−1 a0t xk,t = ξ0 a01 ξ1 a02 · · · ξt−1 a0t ξt .

k→∞

k→∞

It follows from Lemma 5.1 that

fk ∈ lim [uk ]P = [π]P δ := ξ0 a01 ξ1 a02 · · · ξt−1 a0t ξt = lim u k→∞

k→∞

so that, by the assumption [σ]P ⊇ [π]P , δ ∈ [σ]P . c∗ ; in particular, Now let (vk )k∈N be a sequence of words in A∗ converging to σ in A [vk ]Fn → (s, g) and [vk ]P → [σ]P . Once more by Lemma 5.1, for each k, we can choose a word wk ∈ [vk ]P such that wk → δ in H. Lemma 3.5 now applies to δ with B = A0 , and so there is a subsequence (wki ) for which each wki admits a factorization (5.5)

wki = zki ,0 a01 zki ,1 · · · zki ,t−1 a0t zki ,t

such that limi→∞ zki ,j = ξj for each j = 0, . . . , t. It follows from (5.1) that, for each i, vki = wki ψ = yki ,0 a1 yki ,1 a2 · · · yki ,t−1 at yki ,t

ON POWER GROUPS

15

with yki ,j = zki ,j ψ. Then, for each j = 0, . . . , t, lim yki ,j = ( lim xki ,j )ψ = ξj ψ = ξj .

i→∞

i→∞

Consequently, for all sufficiently large i and each j = 1, . . . , t, [yki ,0 a1 yki ,1 · · · aj−1 yki ,j−1 ]Gn = h0 a1 h1 · · · aj−1 hj−1 = gj .

This implies that, for i large enough, the t-tuple (5.3) is a subword of v ki . But the sequence ([vki ]Cn )i∈N is eventually constant with limit s. Therefore, the t-tuple (5.3) is a subword of s and so s ≤ p, as desired. Altogether [σ] F ≤ [π]F , completing the proof that τˆ is an order embedding.  As a consequence we obtain the more difficult part of Theorem 1.1. Corollary 5.3. For each Hall pseudovariety H, J+ ∗ H = P+ H and J ∗ H = PH. Theorem 5.2 shows that for a Hall pseudovariety H, the A-generated free pro-PH monoid can be represented as a submonoid of the power monoid of the A-generated free pro-H group. It seems to be of interest to determine which pseudovarieties of monoids V have such a representation. For example, an analogous result holds for H = Ab the pseudovariety of all Abelian groups. Indeed, it is not hard to show that PAb = Com, the pseudovariety of all commutative monoids, and inside 0 P (Fd Ab (A)) the set {{a, a } | a ∈ A} generates a free profinite commutative monoid. 6. Language Theory Results and H-open Recognizable Sets

Next we aim to provide a language theoretic characterization of Hall pseudovarieties. Recall that, for a pseudovariety V of monoids, Pol(V) denotes the positive variety of V-polynomials [23, 27, 24]. A V-polynomial over the alphabet A is a finite union of sets of the form L0 a0 L1 · · · an Ln where the Li are V-recognizable subsets of A∗ and the ai are letters. Pin and Weil showed [24], that the pseudovam V; without going into riety of ordered monoids corresponding to Pol(V) is LJ + + + m H=J m H for each pseudovariety the precise definition, we remark that LJ H of groups [27]. Let V be a pseudovariety of monoids. We consider each finitely generated free monoid A∗ with the pro-V topology. Let OV denote the positive variety of V-open recognizable sets; we use OV to denote the corresponding pseudovariety of ordered monoids. The following basis theorem for OV was established by the second author in [27, Theorem 6.2]. Theorem 6.1. Let V be a pseudovariety of monoids. Then OV is defined by all c∗ are such ordered pseudoidentities of the form π ≤ w, where w ∈ A∗ and π ∈ A that V |= π = w.

m G. The second author generalized this Pin proved in [22] that OG = J+ m H holds for each pseudovariety of result in [27] and showed that OH = J+ groups H which is closed under co-extensions by p-groups, while the equation fails for each pseudovariety H satisfying a non-trivial monoid identity. We now show m H holds if and only if H is Hall; that is, we show that the equality OH = J+ that H is Hall if and only if Pol(H) = OH . Recall that, for a monoid M and a pseudovariety of groups H, the H-kernel KH (M ) is the submonoid of M consisting of all elements relating to 1 under every

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relational morphism to a group in H [11, 28]. It is well known that for any Agenerated monoid M and m ∈ M , m ∈ KH (M ) if and only if m = [π]M for some c∗ for which H |= π = 1. Also, for any pseudovariety of (ordered) monoids V, π∈A m H if and only if KH (M ) ∈ V [11]. M ∈V

Theorem 6.2. A pseudovariety of groups H is Hall if and only if, for each finite alphabet B, a recognizable language L ⊆ B ∗ is H-open if and only if L is an Hpolynomial. In other words, H is Hall if and only if Pol(H) = OH if and only if m H. OH = J +

m H is valid for Proof. Suppose that H is Hall. It is shown in [27], that O H ⊆ J+ every pseudovariety of groups H, so we need only establish the reverse inclusion. Using Theorem 6.1 for OH , it suffices to show that whenever H |= π = w with c∗ , w ∈ B ∗ and M ∈ J+ m H is a B-generated monoid, one has [π] M ≤ [w]M . π∈B By assumption on M , KH (M ) ∈ J+ , and hence satisfies the identity x ≤ 1. But, by Corollary 3.6 with A = ∅ and ξi = 1 for all i, if w = b1 · · · bt , then π = X0 b1 X1 · · · bt Xt with [Xi ]H = 1. Therefore [Xi ]M ∈ KH (M ) and so [Xi ]M ≤ 1. It is then immediate that [π]M ≤ w. For the converse, suppose that H is not Hall. From the proof of Corollary 4.3 c∗ such that H |= π = a while there exists a finite alphabet A, a ∈ A and π ∈ A + + m m H is not Sl ∗ H 6|= π ≤ a. A fortiori, J H 6|= π ≤ a, proving that J+ contained in OH by Therorem 6.1. 

Altogether we can formulate the main result of the paper as follows. Theorem 6.3. For a pseudovariety H of groups the following are equivalent. (1) H is Hall (2) J ∗ H = PH (3) Sl ∗ H ⊆ PH (4) J+ ∗ H = P+ H (5) Sl+ ∗ H ⊆ P+ H (6) Pol(H) = OH m H (7) OH = J+ + (8) Sl ∗ H ⊆ OH We end this section with an observation concerning literal morphisms, generalizing a result of Pin [20] for the pseudovariety G. A morphism f : A ∗ → B ∗ is literal if Af ⊆ B. It is well known that if f : A∗ → B ∗ is a surjective literal morphism and L is a V-recognizable subset of A∗ , then the image Lf is a PV-recognizable subset of B ∗ ; in fact [27], Lf is P+ V-recognizable. Proposition 6.4. Let H be a pseudovariety of groups; then each surjective literal morphism f : A∗  B ∗ is an open map for the pro-H topology if and only if P+ H ⊆ OH . In particular, f is open if H is Hall.

Proof. Suppose first that P+ H ⊆ OH and that f is such a literal morphism. Since the H-recognizable sets form a basis for the pro-H topology on A ∗ , it suffices to show that Lf is open whenever L ⊆ A∗ is H-recognizable. But Lf is P+ H-recognizable and hence H-open by hypothesis. We conclude that f is an open map. For the converse, a straightforward modification to positive varieties of the argument given in [1] for the unordered setting shows that the positive variety of P + Hrecognizable sets is generated by sets of the form Lf where L is H-recognizable and

ON POWER GROUPS

17

f is a surjective literal morphism. It follows that if such f are pro-H open, then P+ H ⊆ O H .  7. A related embedding theorem for J Recall that for a monoid M , P1 (M ) denotes the monoid of all sets of M containing 1, which is well-known to be J -trivial. In Section 5 we showed that, for each Hall pseudovariety H the elements {a, a0 } generate a free pro-PH monoid inside the power monoid of the free pro-H group on A. In analogy to this, we now ask, for which pseudovarieties V of monoids does the set {{1, a} | a ∈ A} generate a free pro-J monoid inside the power monoid P (Fc V (A)) of the free pro-V monoid on A? Note that the set of pseudovarieties V for which this is true forms a filter in the lattice of all pseudovarieties. We shall give a sufficient condition, which is in terms of the pro-V topology on the free monoid A∗ . Theorem 7.1. Let V be a pseudovariety of monoids such that all shuffle ideals of A∗ are open for the pro-V topology; then the mapping τ : A → P1 (Fc V (A))

induces an order embedding

a 7−→ {1, a}

cJ (A), ≤) ,→ (P1 (Fc (F V (A)), ⊇).

Proof. Let A be a finite alphabet. Since all shuffle ideals in A ∗ are open, V does not satisfy any non-trivial identity. Indeed, suppose that V |= u = v for some u, v ∈ A∗ ; then u and v cannot be separated by neighborhoods. It follows that u is in the shuffle ideal generated by v, and conversely; that is, v is a subword of u and conversely, implying that u and v are identical words. Consequently, A ∗ can be viewed as a submonoid of Fc V (A). c c Let τˆ : (FJ (A), ≤) → (P1 (FV (A), ⊇) be the unique continuous, order-preserving map extending τ . For each word w ∈ A∗ , wˆ τ consists precisely of all the subwords of w. We claim that in general, πˆ τ ∩ A∗ is precisely the set of finite subwords of π. c∗ . Suppose first that w is a Fix a sequence (un ) of words converging to π in A subword of π. Then for n sufficiently large, w is a subword of u n , hence w ∈ un τˆ for such n. Therefore, by Lemma 5.1, w ∈ lim un τˆ = πˆ τ . Conversely, suppose w ∈ πˆ τ = lim un τˆ. Again appealing to Lemma 5.1, we see that there is a sequence (wn ) in A∗ , with wn ∈ un τˆ for each n, such that wn → w in Fc V (A) and hence in the pro-V topology on A∗ . Let L be the principal shuffle ideal generated by w. Since L is V-open by hypothesis, we have that wn ∈ L for n sufficiently large since L is a neighborhood of w. Hence w is a subword of wn for all n sufficiently large. Since wn is a subword of un , w is a subword of un for n sufficiently large, and hence w is a subword of π [1]. Suppose now that σˆ τ ≤ πˆ τ . Let w be a finite subword of π. Then w ∈ πˆ τ ⊆ σˆ τ by the claim. Hence w is a subword of σ, again by the claim. Since w was arbitrary, σ ≤ π and so τˆ is an order embedding, as desired.  Since a shuffle ideal is an H-polynomial for any pseudovariety of groups H, Theorem 6.2 tells us that the above theorem applies whenever V contains a Hall pseudovariety of groups. Recall that ZE(1) is the pseudovariety of aperiodic monoids with central idempotents. The pro-ZE(1) topology on A ∗ is discrete. Hence the above results apply to any pseudovariety V containing ZE(1).

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+ If the hypotheses of Theorem 7.1 apply, then P1 V = J and P+ 1 V = J . However, this also holds under much weaker assumptions on V; in particular it is true for various locally finite pseudovarieties V for which there is no embedding as in Theorem 7.1 [16]. Here, for a pseudovariety of monoids V, P 1 V respectively P+ 1V denotes the pseudovariety of (ordered) monoids generated by all P 1 (M ) respectively (P1 (M ), ⊇) with M ∈ V.

8. More on Hall pseudovarieties Here we present the proof of Theorem 3.7; the auxiliary results, however, seem to be interesting in their own right. Throughout this section, H denotes a Hall pseudovariety. The arguments in the following proofs are very geometric in flavor, involving (profinite) Cayley graphs; the reader is recommended to draw pictures. Lemma 8.1. Let A be a finite alphabet, n ∈ N, A0 , . . . , An ⊆ A, x1 , . . . , xn ∈ A be such that, for each i, xi ∈ / Ai−1 ∪ Ai . For i = 0, . . . , n take any πi ∈ Fc H (Ai ); then π0 x1 π1 · · · xn πn 6= 1.

Proof. The proof is by induction on n. For the case n = 1, suppose by contrast that π0 x1 π1 = 1. Since, by assumption, π0 and π1 do not depend on x1 , we may erase all x 6= x1 and obtain the identity x1 = 1, which is absurd. Let n > 1 and suppose that the claim is true for all k < n. Assume that π0 x1 π1 · · · πn−1 xn πn = 1. For each i, denote by Γi the Cayley graph of Fc H (Ai ), considered as a subgraph of ΓA (Fc (A)), and set H η i = π 0 x 1 π1 · · · π i .

Since H is Hall, for the edge (π0 , x1 ), we must have

(π0 , x1 ) ∈ Γ0 ∪ π0 x1 Γ1 ∪ {(η1 , x2 )} ∪ η1 x2 Γ2 ∪ · · · ∪ {(ηn−1 , xn )} ∪ ηn−1 xn Γn . Since x1 ∈ / A0 ∪ A1 , either

(π0 , x1 ) = (ηk , xk+1 )

for some k ∈ {1, . . . , n − 1}, or (8.1)

(π0 , x1 ) ∈ ηk−1 xk Γk

for some k ∈ {2, . . . , n}. In the first case, π0 = π0 x1 π1 · · · xk πk and therefore, for ψ0 = 1, 1 = ψ 0 x 1 π2 · · · x k πk for some k < n, contradicting the induction hypothesis. In the second case, we have π0 = π0 x1 π1 · · · xk γk for some k ≤ n and γk ∈ Fc H (Ak ); note that in this case we also have that x 1 ∈ Ak (because of (8.1)). Cancelling π0 on the left and conjugating by x1 , we obtain 1 = π 1 x2 · · · x k δk

for δk = γk x1 ∈ Fc H (Ak ); this again contradicts to the induction hypothesis.



ON POWER GROUPS

19

Corollary 8.2. Let A, A0 , . . . , An ⊆ A and x1 . . . , xn ∈ A be as in Lemma 8.1. For i = 0, . . . , n, take any πi , σi ∈ Fc H (Ai ); then

(8.2)

π0 x1 π1 · · · xn−1 πn = σ0 x1 σ2 · · · xn−1 σn

implies πi = σi for each i = 0, . . . , n.

Proof. We retain the notation from the above proof. Since H is Hall, (8.2) implies that (σ0 , x1 ) ∈ Γ0 ∪ {(π0 , x1 )} ∪ {(π0 x1 π1 , x2 ), (σ0 x1 σ1 , x2 ), . . .

. . . , (π0 x1 π1 · · · πn−1 , xn ), (σ0 x1 σ1 · · · σn−1 , xn )}

∪ π0 x1 Γ1 ∪ σ 0 x1 Γ1 ∪ · · · ∪ π 0 x1 · · · x n Γn ∪ σ 0 x1 · · · x n Γn .

Since x1 ∈ / A0 , we have (σ0 , x1 ) ∈ / Γ0 . If it were true that (σ0 , x1 ) 6= (π0 , x1 ), then one of the following cases would have to occur: • (σ0 , x1 ) = (π0 x1 · · · πk , xk+1 ) or (σ0 , x1 ) = (σ0 x1 · · · σk , xk+1 ) for some k ∈ {1, . . . , n − 1}, • σ0 = π0 x1 π1 · · · xk−1 γk or σ0 = σ0 x1 σ1 · · · xk−1 γk for some k ∈ {2, . . . , n} with γk ∈ Fc H (Ak ) and x1 ∈ Ak . However each of these cases leads to an identity 1 = α 0 x 1 α1 · · · x k αk

with αi ∈ Fc H (Ai ) each i, that does not hold in H by Lemma 8.1. Hence σ 0 = π0 and so π0 x1 can be cancelled on the left of (8.2). Now we may now argue by induction, the base n = 0 of the induction being trivial.  Corollary 8.2 already proves Theorem 3.7. Indeed, the factorization condition of Corollary 8.2 is necessary and sufficient for the first join decomposition and sufficient for the second [3, 5]. References 1. J. Almeida, “Finite Semigroups and Universal Algebra”, World Scientific, Singapore, 1994. 2. C. J. Ash, Inevitable Graphs: A proof of the Type II conjecture and some related decision procedures, Internat. J. Algebra Comput. 1 (1991), 127–146. 3. J. Almeida and P. Weil, Reduced factorizations in free profinite groups and join decompositions of pseudovarieties, Internat. J. Algebra Comput. 4 (1994), 375–403. 4. J. Almeida and P. Weil, Relatively free profinite monoids: An introduction and examples, pp. 73–117 in: Semigroups, Formal Languages and Groups, J. B. Fountain, ed., Kluwer, Dordrecht, 1995. 5. K. Auinger, Join decomposition of pseudovarieties involving semigroups with commuting idempotents, J. Pure Appl. Algebra 170 (2002), 115–129. 6. K. Auinger and B. Steinberg, The geometry of profinite graphs with applications to free groups and finite monoids, Trans. Amer. Math. Soc., to appear. 7. K. Auinger and B. Steinberg, Constructing divisions into power groups, preprint. 8. S. Eilenberg, “Automata, Languages and Machines”, Academic Press, New York, Vol A, 1974: Vol B, 1976. 9. D. Gildenhuys and L. Ribes, Profinite groups and Boolean graphs, J. Pure Appl. Algebra 12 (1978), 21–47. 10. M. Hall Jr., A topology for free groups and related groups, Ann. of Math. 52 (1950), 127–139. 11. K. Henckell, S. W. Margolis, J.-E. Pin, and J. Rhodes, Ash’s type II theorem, profinite topology and Mal’cev products, Part I, Internat. J. Algebra Comput. 1 (1991), 411–436. 12. K. Henckell and J. Rhodes, The theorem of Knast, the PG = BG and type-II conjectures, pp 453–463 in: “Monoids and semigroups with applications (Berkeley, CA, 1989)”, ed. J. Rhodes, World Scientific, River Edge, NJ, 1991.

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