Literal Varieties of Languages Induced by

Literal Varieties of Languages Induced by Homomorphisms onto Nilpotent Groups Ondˇrej Kl´ıma and Libor Polák

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Department of Mathematics, Masaryk University Jan´ aˇckovo n´ am 2a, 662 95 Brno, Czech Republic

Abstract. We present here new hierarchies of literal varieties of languages. Each language under consideration is a disjoint union of a certain collection of “basic” languages described here. Our classes of languages correspond to certain literal varieties of homomorphisms from free monoids onto nilpotent groups of class ≤ 2. Keywords: literal varieties of languages, homomorphisms onto monoids, nilpotent groups MSC 2000 Classification: 68Q45 Formal languages and automata

1

Introduction

By the classical Eilenberg’s theorem, the (Boolean) varieties of recognizable languages correspond to pseudovarieties of finite monoids. These appear exactly as finite members of unions of varieties of monoids. Therefore, it is natural to start our investigations with varieties of monoids. Recognizable languages over Xn = {x1 , . . . , xn } corresponding to certain varieties of groups are well-known (the notation is explained in the next section) – see [4], [13], [14], [9], [3] : 1. Boolean combinations of { u ∈ Xn∗ | |u|i ≡ `0 mod ` }, i ∈ {1, . . . , n}, ` ∈ N, `0 ∈ {0, . . . , ` − 1} for the class of all abelian groups. 2. Boolean combinations of { u ∈ Xn∗ | |u|i ≡ `0 mod ` }, i ∈ {1, . . . , n}, `0 ∈ {0, . . . , ` − 1} for the class of all abelian groups satisfying x` = 1. 3. Boolean combinations of u ∗ { u ∈ Xn | ≡ r0 mod r }, v ∈ Xn∗ , r ∈ N, r0 ∈ {0, . . . , r − 1} v for the class of all nilpotent groups. ?

Both authors were supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and by the Grant no. 201/06/0936 of the Grant Agency of the Czech Republic

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4. Boolean combinations of u { u ∈ Xn∗ | ≡ r0 mod r }, v ∈ Xn∗ , |v| ≤ c, r ∈ N, r0 ∈ {0, . . . , r − 1} v for the class of all nilpotent groups of class ≤ c. Such characterizations can be refined as follows : 1’. Disjoint unions of { u ∈ Xn∗ | |u|1 ≡ `1 , . . . , |u|n ≡ `n mod ` }, ` ∈ N, `1 , . . . , `n ∈ {0, . . . , ` − 1} for the class of all abelian groups. It is not difficult to refine the results 2, 3 and 4 in a similar way. Recent investigations in language theory lead to the notion of a literal vari´ ety of languages (Esik and Ito) [6] and Straubing [12]. Such classes of languages generalize the classical varieties, we postulate only the closeness with respect to inverse literal homomorphisms, not with respect to all inverse homomorphisms. Numerous examples are given in paper quoted above and in [7] and [5]. The alge´ braic counterpart are invented by (Esik and Larsen)[7] and [12]. The appropriate equational logic was invented by Kunc [8]. The aim of our contribution is to find rich families of literal varieties of homomorphisms onto nilpotent groups of class ≤ 2 and to present corresponding languages in the finer form. All the varieties of nilpotent groups of class ≤ 2 are well-known and one can find (refined) formulas for all of them. Our new classes are disjoint unions of { u ∈ Xn∗ | |u| ≡ `0 mod `, |u|1 ≡ k1 , . . . , |u|n ≡ kn mod k, |u|j,i ≡ rj,i mod r for all 1 ≤ i < j ≤ n } , where n, `, k, r ∈ N with r | k | ` are fixed and `0 ∈ {0, . . . , ` − 1}, k1 , . . . , kn ∈ {0, . . . , k − 1} satisfying k1 + · · · + kn ≡ `0 mod k, rj,i ∈ {0, . . . , r − 1} for 1 ≤ i < j ≤ n. The next section fixes notation. In Sections 3 and 4 we recall the basics of classical and literal universal algebra. Next section summarizes known results about abelian groups. The main body of our contribution is Section 6 dealing with homomorphisms onto nilpotent groups. Finally, the last section describes how to check membership to our classes of languages using minimal deterministic automata.

2

Our languages

Let N = {1, 2, . . . }, and N0 = N∪{0} be the sets of all positive integers, and nonnegative integers. The relation of divisibility is denoted by |, i.e., for k, ` ∈ N0 , k | l if and only if there exists m ∈ N0 such that mk = ` and the meaning of k ≡ ` mod m is that m | (k − `). The greatest common divisor of k, ` ∈ N0 is

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denoted by gcd(k, `). For a mapping f : B → A, we write imf = {f (b) | b ∈ B} and ker f = { (b, c) ∈ B × B | f (b) = f (c) }. Our alphabets/sets of variables will be X = {x1 , x2 , . . . } and, for n ∈ N, Xn = {x1 , . . . , xn }. The free semigroup (monoid) over the set Y is denoted by Y + and Y ∗ , respectively. We have Y ∗ = Y + ∪ {1} where 1 is the empty word. When using only several first variables, we write x, y, z, . . . instead of x1 , x2 , x3 , . . . . Let u, v ∈ X ∗ , i, j ∈ N, i < j. We denote : |u| – the length of the word u, u v – the number of occurrences of v in u as a subword, in particular, we write |u|i = xui – the number of occurrences of xi in u |u|j,i = xjuxi – the number of occurrences of xj xi in u as a subword, i.e. the number of different factorizations u = pxj qxi r, p, q, r ∈ X ∗ . Our basic ingredients are the following languages : let n, `, k, r ∈ N with r | k | `, let `0 ∈ {0, . . . , ` − 1}, k1 , . . . , kn ∈ {0, . . . , k − 1} satisfying k1 + · · · + kn ≡ `0 mod k, rj,i ∈ {0, . . . , r − 1} for 1 ≤ i < j ≤ n. We put A(n; `, `0 ; k, k1 , . . . , kn ) = = { u ∈ Xn∗ | |u| ≡ `0 mod `, |u|1 ≡ k1 , . . . , |u|n ≡ kn mod k } , N (n; `, `0 ; k, k1 , . . . , kn ; r, r2,1 , . . . , rn,1 , . . . , rn,n−1 ) = = { u ∈ Xn∗ | |u| ≡ `0 mod `, |u|1 ≡ k1 , . . . , |u|n ≡ kn mod k, |u|j,i ≡ rj,i mod r for all 1 ≤ i < j ≤ n } . Already at this place we can mention that : (i) Preimages of a given A(n; `, `0 ; k, k1 , . . . , kn ) by literal homomorphisms ∗ into Xn∗ are disjoint unions of (i.e. letter goes to letter) from Xm A(m; `, · · · ; k, · · · , . . . , · · · )’s. (ii) Preimages of a given A(n; `, `0 ; k, k1 , . . . , kn ) by arbitrary homomorphisms ∗ from Xm into Xn∗ are not of the above form since, for instance, it is not the case for A(2; 6, 3; 2, 0, 1) and f : X2∗ → X2∗ , x1 7→ 1, x2 7→ x2 .

3

Classical Universal Algebra

We recall here the basis of universal algebra of monoids – for more information see Almeida’s book [1]. Let M denote the class of all monoids. For V ⊆ M, let Fin V denote the class of all finite members from V. A class of monoids is a variety if it is closed with respect to the forming of homomorphic images, submonoids and products. Similarly, a class of finite monoids is a pseudovariety if it is closed with respect to the forming of homomorphic images, submonoids and products of finite families.

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Result 1 (Baldwin and Berman [2]) The pseudovarieties of finite monoids are exactly the classes of the form Fin U where U is a union of a chain of varieties of monoids. Recall that an n-ary identity is a pair u = v where u, v ∈ Xn∗ , n ∈ N. A monoid M ∈ M satisfies u = v if ( ∀ α : Xn∗ → M ) α(u) = α(v). In fact, the choice of n is not significant and we write M |= u = v in this case. For a class V ⊆ M, we put Id V = { (u, v) ∈ X ∗ × X ∗ | ( ∀ M ∈ V ) M |= u = v } . Let Π ⊆ X ∗ × X ∗ be a set of identities. We put Mod Π = { M ∈ M | ( ∀ π ∈ Π ) M |= π } . Further, for π ∈ X ∗ × X ∗ , the meaning of Π |= π is ( ∀ M ∈ M ) ( M |= Π implies M |= π ) , and Π is closed if ( ∀ π ∈ X ∗ × X ∗ ) ( Π |= π implies π ∈ Π ). A congruence % on Y ∗ is fully invariant if for each u, v ∈ Y ∗ and each endomorphism g : Y ∗ → Y ∗ , u % v implies g(u) % g(v). The following classical theorems are credited to Birkhoff : Result 2 The varieties of monoids are exactly the classes of the form ModΠ where Π ⊆ X ∗ × X ∗ Result 3 Let Π ⊆ X ∗ × X ∗ . Then Π is an closed set of identities if and only if it is an fully invariant congruence. Result 4 The mappings V 7→ Id V and Π 7→ Mod Π are mutually inverse bijections between the class of all varieties of monoids and the class of all fully invariant congruences on X ∗ .

4

Literal Universal Algebra

Let L be the category having all Y ∗ ’s, Y is a set, as objects and f ∈ L(Z ∗ , Y ∗ ) if and only if f (Z) ⊆ Y . We speak about literal homomorphisms. Note that we can take an arbitrary category of free monoids instead of L and that classes of homomorphisms from finitely generated free monoids onto finite monoids were studied before the results from [10] which we present here. Let M denote the class of all homomorphisms from free monoids onto monoids. Such homomorphism φ : A∗ M is finite if both A and M are

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finite. For V ⊆ M, let Fin V denote the class of all finite members from V and we define : HV = { σφ : Y ∗ N | (φ : Y ∗ M ) ∈ V, N ∈ M, σ : M N surj. homom.}, SL V = { φf : Z ∗ im(φf ) | Z a set, f ∈ L(Z ∗ , Y ∗ ), (φ : Y ∗ M ) ∈ V } , PV = { (φγ )γ∈Γ : Y ∗ im((φγ )γ∈Γ ) | Γ a set, (φγ : Y ∗ Mγ ) ∈ V for γ ∈ Γ } Q (here (φγ )γ∈Γ : Y ∗ → γ∈Γ Mγ , u 7→ (φγ (u))γ∈Γ ). A more transparent definition of the product is the following : consider kernels ker φγ of all φγ ’s and take the canonical homomorphism Y ∗ Y ∗ / ∩γ∈Γ ker φγ . A class V ⊆ M is a L-variety (or literal variety) if it is closed with respect to the operators H, SL and P. Similarly, a class X ⊆ Fin M is an L-pseudovariety (or literal pseudovariety) of finite homomorphisms onto monoids if it is closed with respect to H, SL and Pf (products of finite families). Result 5 ([10], Theorem 3) The literal pseudovarieties of finite homomorphisms onto monoids are exactly the classes of the form Fin U where U is a union of a chain of literal varieties of homomorphisms onto monoids. Let u, v ∈ Xn∗ . A homomorphism ( φ : Y ∗ M ) ∈ M L-satisfies (or literally satisfies) the identity u = v if ( ∀ f ∈ L(Xn∗ , Y ∗ ) ) (φf )(u) = (φf )(v). We write φ |=L u = v. Let (φ : Y ∗ M ) |= u = v mean M |= u = v. For a class V ⊆ M, we put IdL V = { (u, v) ∈ X ∗ × X ∗ | ( ∀ φ ∈ V ) φ |=L u = v } . Let Π ⊆ X ∗ × X ∗ be a set of identities. We set φ |=L Π if ( ∀ π ∈ Π ) φ |=L π , and ModL Π = { φ ∈ M | φ |=L Π } . Further, for π ∈ X ∗ × X ∗ , the meaning of Π |=L π is ( ∀ φ ∈ M ) ( φ |=L Π implies φ |=L π ) , and Π is L-closed (or literally closed) if ( ∀ π ∈ X ∗ × X ∗ ) ( Π |=L π implies π ∈ Π ) . A congruence % on Y ∗ is L-invariant if for each u, v ∈ Y ∗ and each g ∈ L(Y ∗ , Y ∗ ), u % v implies g(u) % g(v). The following statements are modifications of Results 2–4 : Result 6 ([10], Theorem 1) The varieties of homomorphisms from free monoids onto monoids are exactly the classes of homomorphisms of the form ModL Π where Π ⊆ X ∗ × X ∗ . Result 7 ([11], Lemma 2.1) Let Π ⊆ X ∗ × X ∗ . Then Π is an L-closed set of identities if and only if it is an L-invariant congruence. Result 8 ([11], Theorem 2.1.) The mappings V 7→ IdL V and Π 7→ ModL Π are mutually inverse bijections between the class of all literal varieties of homomorphisms onto monoids and the class of all literally invariant congruences on X ∗.

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5

Abelian Groups

The parts (i) and (ii) of the following result are well-known and the item (iii) appears in the sources quoted in the introduction. Result 9 (i) The varieties of monoids consisting of abelian groups are exactly A(`) = Mod ( xy = yx, x` = 1 ), where ` ∈ N . Moreover, A(`) ⊆ A(`0 ) if and only if ` | `0 . (ii) The corresponding fully invariant congruences are α(`) = { (u, v) ∈ X ∗ × X ∗ | ( ∀ i ∈ N ) |u|i ≡ |v|i mod ` } . (iii) For the corresponding varieties of languages A(`) we have : (A(`))(Xn∗ ) consists of disjoint unions of { u ∈ Xn∗ | |u|1 ≡ `1 , . . . , |u|n ≡ `n mod ` }, `1 , . . . , `n ∈ {0, . . . , ` − 1} The case of literal varieties of homomorphisms is solved by the following : Result 10 ([11]) (i) The literal varieties of homomorphisms onto abelian groups are exactly A(`, k) = ModL ( xy = yx, x` = 1, xk = y k ), where k, ` ∈ N, k | ` . Moreover, A(`, k) ⊆ A(`0 , k 0 ) if and only if k | k 0 and ` | `0 . (ii) The corresponding literally invariant congruences are α(`, k) = = { (u, v) ∈ X ∗ × X ∗ | |u| ≡ |v| mod ` and ( ∀ i ∈ N ) |u|i ≡ |v|i mod k } . (iii) For the corresponding literal varieties of languages A(`, k), we have : (A(`, k))(Xn∗ ) consists of disjoint unions of A(n; `, `0 ; k, k1 , . . . , kn ) where `0 ∈ {0, . . . , ` − 1}, k1 , . . . , kn ∈ {0, . . . , k − 1}, k1 + · · · + kn ≡ `0 mod k .

6

Nilpotent groups

Let (G, ·) be a group. For g, h ∈ G, we define the commutator [g, h] of g and h by [g, h] = g −1 h−1 gh. Further, we put [g1 , . . . , gs ] = [g1 , [g2 , . . . , gs ]] for s ∈ N, s ≥ 3, g1 , . . . , gs ∈ G. A group (G, ·) is said to be nilpotent of the class ≤ s (s ∈ N) if it satisfies the identity S [x1 , . . . , xs+1 ] = 1. We denote by Ns the class of all such groups. Let N = s∈N Ns be the class of all nilpotent groups and note that A = N1 be the class of all abelian groups. Notice that each group satisfies gh[h, g] = hg and that both [gg 0 , h] = [g, h][g 0 , h] and [g, hh0 ] = [g, h][g, h0 ] hold in the class N2 . We denote by N s the class of all homomorphisms from M which are onto groups from Ns . We consider the case s = 2. By the following lemma we can restrict our attention to literal varieties of homomorphisms onto members of N2 . Moreover, we can suppose that our homomorphisms are “literally periodic”.

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Lemma 1. Let G be a variety of groups and let V be a literal variety of homomorphisms from free monoids onto monoids whose finite members are onto groups from G. Then there exists a literal variety W of homomorphisms from free monoids onto groups from G and ` ∈ N such that both V and W literally satisfy x` = 1 and Fin W = Fin V. Proof. Let φ : {a}∗ M be the free object in V over A = {a}, that is, φ is the product of all (we consider representatives of classes giving the same kernels) (φi : {a}∗ Mi ) ∈ V where i ∈ I. If M is infinite, then M = {1, b, b2 , . . . } where b = φ(a), and bk 6= b` for k 6= `, k, ` ∈ N0 . Putting b, b2 , . . . into a single class we get a congruence τ on M such that M/τ is finite and it is not a group, a contradiction. Consequently, M consists of pairwise different 1, b, b2 , . . . , bk+`−1 with bk+` = bk , for some k ∈ N0 , ` ∈ N. Since M is a group, we have k = 0. Now we show that V |=L x` = 1. Indeed, for an arbitrary (ψ : B ∗ N ) ∈ V consider the compositions of ψ with {a}∗ → B ∗ , a 7→ b, b ∈ B. Let W = < FinV > be a literal variety generated by Fin V. All members of W satisfy the identity x` = 1 literally. Notice that if a monoid M with a generating set A satisfies g ` = 1 for all g ∈ A, then M is a group since `−1 . . . g1`−1 is the inverse of g1 . . . gm . Moreover, each homomorphism of W is gm onto a group from G. Since Fin V ⊆ < Fin V > = W, we have Fin V ⊆ Fin W. Conversely, from W ⊆ V it follows that Fin W ⊆ Fin V. t u For any t ∈ N and x, y ∈ X, we define [x, y]t = xt−1 y t−1 xy ∈ X ∗ . If we consider φ : Xn∗ G satisfying x` = 1 literally then φ([x, y]` ) = [φ(x), φ(y)] for x, y ∈ Xn . From that reason [x, y] will always mean [x, y]` for an appropriate `. A crucial role in our considerations is played by certain numerical parameters. The relationships between them are explored in the following result. Lemma 2. Let (φ : Y ∗ G) ∈ N 2 and let there exist t ∈ N such that φ |=L xt = 1. Then there exist m, `, k, r ∈ N, each smallest (with respect to the divisibility), satisfying φ |= xm = 1, φ |=L x` = 1, φ |=L xk = y k , φ |=L [x, y]r = 1 . Moreover, those parameters satisfy (a) k | `, (b) r | k, (c) If ` is odd or 2r | ` then m = `; m = 2` otherwise. Proof. Using the Bezout’s Lemma we see that, for each `, `0 ∈ N0 , the facts φ |=L 0 0 x` = 1, x` = 1 imply φ |=L xgcd(`,` ) = 1. Therefore there exists the smallest ` ∈ N0 (with respect to the divisibility) such that φ |=L x` = 1. Moreover, 0 φ |=L x` = 1 if and only if ` divides `0 . The same is true for the literal satisfaction of xk = y k and [x, y]r = 1 and for the (usual) satisfaction of xm = 1. Let A = φ(Y ). ` | m : ( g m = 1 for all g ∈ G) implies (am = 1 for all a ∈ A).

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k | ` : (for all a ∈ A, a` = 1) implies (for all a, b ∈ A, a` = b` ). r | k : Let a, b ∈ A. Then abk = a · ak = ak · a = bk · a = abk [b, a]k and thus [b, a]k = 1. By the assumptions, ` ∈ N. p p p (p2) The proof of (c) : For each a, b ∈ A, p ∈ N, `we have (ab) = a b `· [b, a] . ` If ` is odd or 2r | ` then r | 2 and thus (ab) = 1. For (a1 . . . , aq ) , q ∈ N, ` use induction. Clearly r | 2` 2 . Conversely, if G |= x = 1 and ` is even, then, in ` particular, for all a, b ∈ A, we have [b, a](2) = 1. Thus r | ` = ` (` − 1). Since r and ` − 1 are relatively prime we have r | 2` .

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2

t u

Remark. In fact, we will not need the parameter m in our considerations. We added it to show that the relationship between identities being satisfied literally and globally is far from being trivial. The next result helps us to understand our languages. Lemma 3. Let n, l, k, r ∈ N satisfy r | k | `. Then the classes N (n; l, l0 ; k, k1 , . . . , kn ; r, r2,1 , . . . , rn,1 , . . . , rn,n−1 ) where `0 ∈ {0, . . . , ` − 1}, k1 , . . . , kn ∈ {0, . . . , k − 1} satisfying k1 + · · · + kn ≡ `0 mod k, rj,i ∈ {0, . . . , r − 1} for 1 ≤ i < j ≤ n form a partition of the set Xn∗ . Proof. Indeed, using sequences of transpositions pxi xj q → pxj xi q on the word xki xkj , one gets exactly the words u with the following parameters |u|i = |u|j = k, |u|j,i ∈ {0, . . . , k 2 }. To get a word u ∈ N (n; l, l0 ; k, k1 , . . . , kn ; r, r2,1 , . . . , rn,1 , . . . , rn,n−1 ) apply an appropriate sequence of transpositions on xk11 . . . xknn · xk1 xk2 · . . . · xk1 xkn · . . . · xkn−1 xkn . Clearly, the union of the classes is Xn∗ and they are pairwise disjoint.

t u

The following result will help us to distinguish varieties with different parameters (even their finite members). Lemma 4. Let `, k, r ∈ N satisfy r | k | `. Then the formula ` ξ : Zr → (Z` × Zk )Z` ×Zk , u 7→ (a, p) 7→ (a + up , p) r

()

correctly defines an action of the group (Zr , +) on (Z` × Zk , +) by automorphisms. Moreover, the formula ` (a, p, u) ◦ (b, q, v) = (a + b + uq , p + q, u + v) r

(∗)

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defines a group operation on the set Z` × Zk × Zr ; in particular (0, 0, 0) is the neutral element and (a, p, u)−1 = (−a + up r` , −p, −u). This group is nilpotent of class 2. Let α = (1, 0, 0), β = (1, 1, 0), γ = (1, 0, 1) . Then the set {α, β, γ} generates (Z` × Zk × Zr , ◦), the number k is the smallest (with respect to the divisibility) with αk = β k = γ k and the order of each of α, β, γ is `. Finally, r is the smallest number such that [β, γ]r = 1, and for each (a, p, u), (b, q, v) ∈ Z` × Zk × Zr , we have [(a, p, u), (b, q, v)]r = 1. Proof. Taking representatives u0 , u00 ∈ Z of u ∈ Zr , a0 , a00 ∈ Z of a ∈ Z` and p0 , p00 ∈ Z of p ∈ Zk , we see that a0 + u0 p0

` ` ≡ a00 + u00 p00 mod ` , r r

which means that the formula () correctly defines a mapping. Let u ∈ Zr . It is easy to see that ξ(u) is a bijective homomorphisms of (Z` × Zk , +) onto itself. Further, for each u, v ∈ Zr , (a, p) ∈ Z` × Zk , we have (ξ(u + v))(a, p) = ξ(u)(ξ(v)(a, p)) and thus ξ is a homomorphism of the group (Zr ) into the group Aut(Z` × Zk , +) of all automorphisms of the group (Z` × Zk , +). The formula (∗) describes the well-known semidirect product of groups. The commutator [(a, p, u), (b, q, v)] equals to ((uq − vp) r` , 0, 0) and therefore it commutes with all elements and r is as mentioned in the lemma. The rest follows from the formula (a, p, u)m = (ma +

m(m − 1) ` up , mp, mu), m ∈ N . 2 r t u

Lemma 5. Let G be a finite group with a generating set A, such that a[b, c] = [b, c]a for all a, b, c ∈ A. Then G is nilpotent of the class ≤ 2. Proof. Since G is a finite group, each element of G can be written as a product of generators, i.e. it can be viewed as a word from A∗ . So, it is enough to prove that [u, v] commutes with w for all u, v, w ∈ A∗ . We prove this by the induction with respect to |u|+|v|. It is clear for u = 1 or v = 1. If u, v ∈ A then [u, v] commutes with each generator by the assumption, hence [u, v] commutes with each w ∈ A∗ . So, we have proved the statement for u, v ∈ A∗ such that |u| + |v| ≤ 2. Now suppose that n ∈ N, n > 2 and that the statement is true for all u, v, w ∈ A∗ , |u| + |v| < n. Assume first, that u = u1 u2 , where u1 , u2 ∈ A+ . −1 Then [u1 u2 , v] = u−1 u2 v. Now from the induction assumption we have 2 [u1 , v]v that [u1 , v] commutes with u−1 2 . Hence [u1 u2 , v] = [u1 , v][u2 , v]. Again by the induction assumption both [u1 , v] and [u2 , v] commute with each w ∈ A∗ and we can conclude that also [u1 u2 , v] commutes with each w ∈ A∗ . The case when u ∈ A can be treated in a similar way : we write v as a product of two shorter words. t u

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Theorem 1. (i) The following literal varieties of homomorphisms from free monoids onto nilpotent groups of class ≤ 2 are pairwise different : N (`, k, r) = ModL ( [x, [y, z]] = 1, x` = 1, xk = y k , [x, y]r = 1 ) where `, k, r ∈ N, r | k | `. Moreover, N (`, k, r) ⊆ N (`0 , k 0 , r0 ) if and only if ` | `0 , k | k 0 , r | r0 . (ii) The corresponding literally invariant congruences on X ∗ are of the form ν(`, k, r) = { (u, v) ∈ X ∗ × X ∗ | |u| ≡ |v| mod `, |u|i ≡ |v|i mod k for i ∈ N, |u|j,i ≡ |v|j,i for 1 ≤ i < j } . (iii) For the corresponding literal varieties of languages N(`, k, r), we have : N(`, k, r)(Xn∗ ) consists of disjoint unions of N (n; `, `0 ; k, k1 , . . . , kn ; r, r2,1 , . . . , rn,1 , . . . , rn,n−1 ) where `0 ∈ {0, . . . , ` − 1}, k1 , . . . , kn ∈ {0, . . . , k − 1}, k1 + · · · + kn ≡ `0 mod k , rj,i ∈ {0, . . . , r − 1} for 1 ≤ i < j ≤ n . Proof. (i) It follows from Lemmas 2, 4 and 5. (ii) Claim 1. ν(`, k, r) is a literally invariant congruence on X ∗ . Clearly, the mentioned relation is an equivalence on the set X ∗ . Now notice that for arbitrary u ∈ X ∗ , i, j ∈ N, i 6= j, ξ : X → X, we have : |xi u| = |u| + 1, |xi u|i = |u|i + 1, |xi u|j = |u|j , |xi u|i,j = |u|i,j + |u|j , |ξ(u)| = |u|, P |ξ(u)|i = s=1,...,p |u|is where ξ −1 (xi ) = {xi1 , . . . , xip }, i1 , . . . , ip pairwise different, P (f) |ξ(u)|i,j = s=1,...,p, t=1,...,q |u|is ,jt with i1 , . . . , ip as above and ξ −1 (xj ) = {xj1 , . . . , xjq }, j1 , . . . , jq pairwise different.

(a) (b) (c) (d) (e)

Let (u, v) ∈ ν(`, k, r). Then (xi u, xi v) ∈ ν(`, k, r) by (a) – (c) and (uxi , vxi ) ∈ ν(`, k, r) by their duals. Finally, (ξ(u), ξ(v)) ∈ ν(`, k, r) due to (d) – (f). Claim 2. ν(`, k, r) is generated as literally invariant congruence by the set { ([x, [y, z]], 1), (x` , 1), (xk , y k ), ([x, y]r , 1) | x, y, z ∈ X } . Indeed, let (u, v) ∈ ν(`, k, r) and let all the variables of u and v be among x1 , . . . , xn . The identity u = u0 where |u|1

u0 = x1

|u|2,1 n . . . x|u| . . . [xn , x1 ]|u|n,1 . . . [xn , xn,n−1 ]|u|n,n−1 n [x2 , x1 ]

is valid in each nilpotent group of class ≤ 2. Similarly for v = v 0 where |v|1

v 0 = x1

|v|2,1 n . . . x|v| . . . [xn , x1 ]|v|n,1 . . . [xn , xn,n−1 ]|v|n,n−1 . n [x2 , x1 ]

11

Since [xj , xi ] = x`−1 x`−1 xj xi , we have j i |u0 | ≡ |u| mod `, |u0 |i ≡ |u|i mod ` for each i ∈ N , |u0 |i,j ≡ |u|i,j mod r for each i, j ∈ N, i 6= j . and similarly for v. Now we will rewrite u0 to v 0 . Since [x, y]r = 1 literally we are done with commutators. Then we literally use xk = y k |u|1

to rewrite x1

|v|

|u|1 −|v|1

to x1 1 x2

|u|1 −|v|1 +|u|2

then to rewrite x2 |v|

|v|

|u|1 −|v|1 +|u|2 |u|3 x3

to get x1 1 x2 |v|

n . . . x|u| , n

|u|1 −|v|1 +|u|2 −|v|2

to x2 2 x3

|v|

|u|1 −|v|1 +|u|2 −|v|2 +|u|3 |u|4 n x4 . . . x|u| and so on . n |u| −|v|1 +···+|u|n−1 −|v|n−1 +|u|n literally to rewrite xn 1 to

to get x1 1 x2 2 x3

|v|

Finally, we use x` = 1 xn n . We succeed since |u| ≡ |v| mod ` literally. (iii) : The restriction of the relation ν(`, k, r) onto Xn∗ × Xn∗ has exactly the classes N (n; `, `0 ; k, k1 , . . . , kn ; r, r2,1 , . . . , rn,1 , . . . , rn,n−1 ). The result follows from [11] Theorem 5.1.2. t u Remarks. 1. Putting r = 1 we get all the results concerning homomorphisms onto abelian groups. 2. The case k = ` = m corresponds to the classical varieties of nilpotent groups of the class ≤ 2.

7

Automata

Let A = (Q, A, ·, i, T ) be a complete deterministic automaton, i.e. Q is a nonempty finite set of states, A is an alphabet, · : Q × A → Q is the action by letters, i ∈ Q is the initial state, and T ⊆ Q is the set of all terminal states. For any a ∈ A, we have the action αa : Q → Q, q 7→ q · a. We speak about an action of the first type. If each action of the first type is a permutation of the set Q then we can define actions of the second type in the following way. For given a, b ∈ A we define the action βa,b as a mapping βa,b : Q → Q by the rule p · βa,b = q

if and only if

( ∃r ∈ Q )( r · ba = p and r · ab = q ) .

Because actions by a and b are permutations, a state r in the defining property is uniquely determined. Hence the previous definition is correct and moreover the action βa,b is also a permutation of the set Q. We say that an automaton A is 2-nilpotent if and only if it is a complete deterministic automaton in which the actions of the first type are permutations and commute with the actions of the second type. For a 2-nilpotent automaton A, we define À as the least common multiple of the lengths of all cycles in all actions of the first type; kA as the smallest number such that the kA -th iteration of each action of the first type gives the same mapping from Q to Q; rA as the least common multiple of the lengths of all cycles in all actions of the second type.

12

Proposition 1. Let L be a language over Xn with minimal automaton A(L). Then L ∈ N(`, k, r)(Xn∗ ) if and only if A(L) is 2-nilpotent and ÀL |`, kAL |k and rAL |r. Proof. It is a consequence of Theorem 1 and the well-known fact that transformation monoid of a minimal automaton A(L) is isomorphic to the syntactic monoid of L. t u

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