Profinite semigroups Dominique Perrin 4 d ecembre 2015 Dominique - - PowerPoint PPT Presentation

Profinite semigroups

Dominique Perrin 4 d´ ecembre 2015

Dominique Perrin Profinite semigroups

Outline

Motivating examples Profinite semigroups Profinite codes Uniform recurrence

Dominique Perrin Profinite semigroups

p-adic numbers

Let p be a prime number and let Zp denote the ring of p-adic integers, namely the completion of Z under the p-adic metric. This metric is defined by the norm |x|p =

• p− ordp(x)

if x = 0

• therwise

where ord(x)p(x) is the largest n such that pn divides x. Any element γ ∈ Zp has a unique p-adic expansion γ = c0 + c1p + c2p2 + . . . = (. . . c3c2c1c0)p Note that infinite expansions may represent ordinary integers. For example (. . . 111)2 = −1.

Dominique Perrin Profinite semigroups

We can express the expansion of the elements in Zp as Zp = lim Z/pnZ = {(an)n≥0 ∈ Πn≥0Z/pnZ | for all n, an+1 ≡ an mod pn} This expresses the ring Zp as a projective limit of the rings Z/pnZ.

Dominique Perrin Profinite semigroups

Profinite integers

The direct product of all rings Zp ˆ Z = ΠZp

• ver all prime numbers p is the ring of profinite integers. One may

define it equivalently as the projective limit of all cyclic groups ˆ Z = Πn≥1Z/nZ = {(an)n≥1 ∈ Πn≥1Z/nZ | for all n|m, am ≡ an mod n}

• r as the projective limit of the cyclic groups Z/n!Z

ˆ Z = Πn≥1Z/n!Z = {(bn)n≥1 ∈ Πn≥1Z/n!Z | for all n, an+1 ≡ an mod n!}

Dominique Perrin Profinite semigroups

The factorial number system

The last representation corresponds to an expansion of the form γ = c1 + c22! + c33! + . . . = (. . . c3c2c1)! with digits 0 < ci ≤ i. Note that this time −1 = (. . . 321)! which holds because 1 + 2.2! + . . . + n! = (n + 1)! − 1, as one may verify by induction on n.

Dominique Perrin Profinite semigroups

The Fibonacci morphism

The Fibonacci morphism is the morphism ϕ : A∗ → A∗ with A = {a, b} defined by ϕ(a) = ab and ϕ(b) = a. The sequence of ϕn(a) ϕ(a) = ab ϕ2(a) = aba ϕ3(a) = abaab ϕ4(a) = abaababa · · · is the Fibonacci sequence of words of length equal to the Fibonacci numbers defined by F0 = 0, F1 = 1 and Fn = Fn−1 + Fn−2. One has Fn = |ϕn−2(a)|. The Fibonacci sequence of words converges in the space AN to the Fibonacci infinite word x = abaababa · · · It is a fixed-point of ϕ in the sense that ϕ(x) = x.

Dominique Perrin Profinite semigroups

Form another point of view, this sequence is not convergent. Indeed, the terms of the sequence end alternately with a or b and thus can be distinguished by a morphism from A∗ into a monoid with 3 elements. A sequence converging in this stronger sense is ϕn!(a) ϕ(a) = ab ϕ2(a) = aba ϕ6(a) = abaababaabaababaababa ϕ24(a) = abaababa · · · abaababa · · · The limit in the sense of profinite topology, to be defined below, is a profinite word denoted ϕω(a) which begins by the Fibonacci infinite word

− →

(ϕn (a))n≥0 and ends with the left infinite word

← −

(ϕ2n (b))n≥0. Its length is a profinite number.

Dominique Perrin Profinite semigroups

Topological spaces

We begin with an introduction to the basic notions of topology. A topological space is a set S with a family F of subsets such that (i) it contains ∅ and S, (ii) it is closed under union, (iii) it is closed under finite intersection The elements of F are called open sets. The complement of an

• pen set is called a closed set. A clopen set is both open and

closed. A map ϕ : X → Y between topological spaces X, Y is continuous if for any open set U ⊂ Y , the set ϕ−1(U) is open in X.

Dominique Perrin Profinite semigroups

A basis of the family of open sets is a family B of sets such that any open set is a union of elements of B. Given a fmily of topological spaces Xi indexed by a set I, the product topology on the direct product X = Πi∈IXi is defined as the coarsest topology such that the projections πi : X → Xi are

• continuous. A basis of the family of open sets are the sets of the

form Πi∈IUi where Ui = Xi only for a finite number of indices i.

Dominique Perrin Profinite semigroups

Metric spaces

Metric spaces form a vast family of topological spaces. A metric space is a space S with a function d : S × S → R, called a distance, such that for all x, y, z ∈ S, (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x) (iii) d(x, z) ≤ d(x, y) + d(y, z). Any metric space can be considered as a topological space, considering as open sets the unions of open balls B(x, ε) = {y ∈ S | d(x, y) < ε} for x ∈ S and ε ≥ 0. For example, the set Rn is a metric space for the Euclidean distance. A topological space is separated (or Hausdorff) if any two distinct points belong to disjoint open sets. A topological space is compact if it is separated and if from any family of open sets whose union is S, one may extract a finite subfamily with the same property.

Dominique Perrin Profinite semigroups

Topological semigroups

A topological semigroup is a semigroup S endowed with a topology such that the semigroup operation S × S → S is continuous. A topological monoid is a topological semigroup with identity. A finite semigroup can always be view as a topological semigroup under the discrete topology. As a less trivial example, the set R of nonnegative real numbers is a topological semigroup for the addition and the interval [0, 1] is a topological semigroup for the multiplication. A compact monoid is a topological monoid which is compact (as a topogical space). Note that we assume a compact space to satisfy Hausdorff separation axiom (any two distinct points belong to disjoint open sets). Note also the following elementary property of compact monoids.

Dominique Perrin Profinite semigroups

Proposition The set of factors of an element of a compact monoid is closed. Let M be a compact monoid and let (un)n≥0 be a sequence of factors of x ∈ M converging to some u ∈ M. Let pn, qn be such that x = pnunqn for all n ≥ 1. Since M is compact, the sequences (pn), (qn) have converging subsequences. If p, q are the limits of these subsequences, we have x = puq and thus u is a factor of x.

Dominique Perrin Profinite semigroups

Projective limits

We want to define profinite semigroups as some kind of limit of finite semigroups in such a way that properties true in all finite semigroups will remain true in profinite semigroups. For this we need the notion of projective limit. An A-generated topological semigroup is a mapping ϕ : A → S into a topological semigroup whose image generates a subsemigroup dense in S. A morphism between A-generated topological semigroups ϕ : A → S and ψ : A → T is a continuous morphism θ : S → T such that θ ◦ ϕ = ψ. We denote θ : ϕ → ψ such a morphism. S T A ϕ ψ θ

Figure: A morphism of A-generated semigroups

Dominique Perrin Profinite semigroups

A projective system in this category of objects is given by (i) a directed set I, that is poset in which any two elements have a common upper bound. (ii) for each i ∈ I, an A-genrerated topological semigroup ϕi : A → Si, (iii) for each pair i, j ∈ I with i ≥ j, a connecting morphism ψi,j : ϕi → ϕj such that ψi,i is the identity on Si and for i ≥ j ≥ k, ψi,k = ψi,j ◦ ψj,k. The projective limit of this projective system is a topological semigroup Φ : A → S together with morphisms Φi : Φ → ϕi such that for all i, j ∈ I with i ≥ j, ψi,j ◦ Φi = Φj, and for any A-generated topological semigroup Ψ : A → S and morphisms with morphisms Ψi : Ψ → ϕi such that for all i, j ∈ I with i ≥ j, ψi,j ◦ Ψi = Ψj, the exists a morphism θ : ψ → Φ such that Φi ◦ θ = Ψi for all i ∈ I.

Dominique Perrin Profinite semigroups

ϕi Φ Ψ ϕj ψi,j Φi Φj θ Ψi Ψj Si S T Sj A ψi,j Φi Φj θ Ψi Ψj Ψ ϕi ϕj

Figure: The projective limit.

Dominique Perrin Profinite semigroups

The uniqueness of the projective limit can be verified (“as a standard diagram chasing exercise”). The existence can be proved by considering the subsemigroup S of the product Πi∈ISi consisting of all (si)i∈I such that, for all i, j ∈ I with i ≥ j, ψi,j(si) = sj endowed with the product topology. The map Φ : A → S is given by Φ(a) = (ϕi(a))i∈I and the maps Φ : S → Si are the projections.

Dominique Perrin Profinite semigroups

Profinite semigroups

A profinite semigroup is a projective limit of a projective system of finite semigroups. A topological space is (i) connected if it is not the union of two disjoint open sets (ii) totally disconnected if its connected components are singletons (iii) zero-dimensional if it admits a basis consisting of clopen sets. The following result, gives a possible direct definition of profinite semigroup without using projective limits. Theorem The following conditions are equivalent for a compact semigroup S. (i) S is profinite, (ii) S is residually finite as a topological semigroup, (iii) S is a closed subsemigroup of a direct product of finite semigroups, (iv) S is totally disconnected, (v) S is zero-dimensional.

Dominique Perrin Profinite semigroups

The explicit construction of the projective limit shows that (i) ⇒ (ii) and (ii) ⇒ (iii) results from the definitions. For (iii) ⇒ (i), see (Almeida, 2005). Since a product of totally disconnected spaces is totally disconnected, we have (iii) ⇒ (iv). The equivalence (iv) ⇔ (v) holds for any compact space. Finally, the implication (v) ⇒ (ii) results from Hunter’s Lemma (see Almeida, 2005). Corollary A closed subsemigroup of a profinite semigroup is also profinite. The product of profinite semigroups is also profinite.

Dominique Perrin Profinite semigroups

A subset K of a semigroup S is recognized by a morphism ϕ : S → M if K = ϕ−1ϕ(K). Proposition Let S be a profinite semigroup. A subset K ⊂ S is clopen if and

• nly if it recognized by a continuous morphism ϕ : S → M into a

finite monoid M. The condition is sufficient since the set K is the inverse image under a continuous function of a clopen set. Conversely assume that K is clopen and that S is a closed subsemigroup of a direct product Πi∈ISi of finite semigroups Si. Then K may be expressed as K = S ∩ (K1 ∪ . . . ∪ Kn) where each Kℓ is is a product of the form Πi∈IXi with Xi ⊂ Si and Xi = Si except on a finite set Jℓ of

• indices. Let J = J1 ∪ . . . ∪ Jn. The projection ϕ : S → Πi∈JSi is a

continuous morphism recognizing K.

Dominique Perrin Profinite semigroups

The free profinite monoid

Consider the projective system formed by representatives of isomorphism classes of all A-generated finite monoids taking the unique connecting morphisms with respect to this set of

• generators. The free profinite monoid on a finite alphabet A,

denoted A∗ is the projective limit of this family. It has the following universal property. Proposition The natural mapping ι : A → A∗ is such that for any map ϕ : A → M into a profinite monoid there exists a unique continuous morphism ˆ ϕ : A∗ → M such that ˆ ϕ ◦ ι = ϕ. S

• A∗

A ϕ ι ˆ ϕ

Figure: The universal property of A∗.

The elements of A∗ are called pseudowords and the elements of

Dominique Perrin Profinite semigroups

Recognizable sets

A subset X of a monoid M is recognizable if there is a morphism ϕ : M → S into a finite monoid S which recognizes X. Proposition The following conditions are equivalent for a set X ⊂ A∗. (i) X is recognizable. (ii) the closure ¯ X of X in A∗ is open and X = ¯ X ∩ A∗. (ii) X = K ∩ A∗ for some clopen set K ⊂ A∗. Assume that X is recognized by a morphism ϕ : A∗ → S from A∗ into a finite monoid S. By the universale property of A∗, there is a unique continuous morphism ˆ ϕ extending ϕ. Then X = ˆ ϕ−1ϕ(X) is open and satisfies X = ¯ X ∩ A∗. Thus (1) ⇒ (2). The implication (2) ⇒ (3) is trivial. Finally, assume that (3) holds. By Proposition 2 there exists a continuous morphism ψ : A∗ → S into a finite monoid S which recognizes K. Let ϕ be the restriction of ψ to A∗. Then X = A∗ ∩ K = A∗ ∩ ψ−1ψ(K) and so X is recognizable.

Dominique Perrin Profinite semigroups

The natural metric

The natural metric on a profinite monoid M is defined by d(u, v) =

• 2−r(u,v)

if u = v

• therwise

where r(u, v) is the minimal cardinality of a monoid N for which there is a continuous morphism ϕ : M → N such that ϕ(u) = ϕ(v). It is actually a ultrametric since it satisfies the condition d(u, w) ≤ min(d(u, v), d(v, w)) stronger than the triangle inequality.

Dominique Perrin Profinite semigroups

Proposition For a finitely generated profinite semigoup S, the topology is induced by the natural metric. Denote Bε = (u) = {v ∈ S | d(u, v) < ε}. Let K be a clopen set in S. By Proposition 2 there is a continuous morphism ϕ : S → T into a finite semigroup T which recognizes K. For t ∈ T, the ball B2− Card(T)(t) is contained in ϕ−1(t) and thus K is a union of open

• balls. Thus, since the clopen sets form a basis of the topology, any

closed set is a union of open balls. Conversely, consider the open ball B = B2−n(u). Since there is finite number of isomorphism types of semigroups with at most n elements, there are finitely many kernels of continuous morphisms into such semigroups and so their intersection is a clopen congruence on S. It follows that there exists a continuous morphism ϕ : S → T into a finite semigroup such that ϕ(u) = ϕ(v) if and only if r(u, v) > n. Hence B = ϕ−1ϕ(B) so that B is open.

Dominique Perrin Profinite semigroups

This leads to an alternative definition of the free profinite monoid. Theorem For a finite set A, the completion of A∗ for the natural metric is the free profinite monoid A∗. In a profinite semigroup S, the closure of the semigroup generated by an element s ∈ S contains a unique idempotent, denoted sω. If S is profinite, it is the limit of the sequence sn!.

Dominique Perrin Profinite semigroups

Profinite codes

We will expore the notion of a code in the free profinite monoid. Let A, B be finite alphabets. Any morphism β : B∗ → A∗ extends uniquely by continuity to a continuous morphism ˆ β : B∗ → A∗. A finite set X ⊂ A∗ is called a profinite code if the continuous extension ˆ β of any morphism β : B∗ → A∗ inducing a bijection from B onto X is injective. The following statement is from [?]. Theorem Any finite code X ⊂ A+ is a profinite code.

Dominique Perrin Profinite semigroups

Let β : B∗ → A∗ be a coding morphism for X. We have to show that for any pair u, v ∈ B∗ of distinct elements, we have ˆ β(u) = ˆ β(v), that is, there is a continuous morphism ˆ α : A∗ → M into a finite monoid M such that ˆ αˆ β(u) = ˆ α ˆ β(v). For this, let ψ : B∗ → N be a continuous morphism into a finite monoid N such that ψ(u) = ψ(v). Let P be the set of proper prefixes of X and let T be the prefix transducer associated to β (see [?]). Let α be the morphism from A∗ into the monoid of P × P-matrices with elements in N ∪ 0 defined as follows. For x ∈ A∗ and p, q ∈ P, we have α(x)p,q =

• ψ(y)

if there is a path p

x|y

− − → q

• therwise.

Then M = α(A∗) is a finite monoid and α extends to a continuous morphism ˆ α : A∗ → M. Since, by [?, Proposition 4.3.2], the transducer T realizes the decoding function of X, we have αβ(y)1,1 = ψ(y) for any y ∈ B∗. By continuity, we have ˆ α ˆ β(y)1,1 = ψ(y) for any y ∈ B∗. Then ˆ α is such that ˆ α ˆ β(u) = ˆ αˆ β(v). Indeed ˆ αˆ β(u)1,1 = ψ(u) = ψ(v) = ˆ αˆ β(v)1,1.

Dominique Perrin Profinite semigroups

Example Let A = {a, b} and X = {a, ab, bb}. The set X is a suffix code. It has infinite deciphering delay since abb · · · = a(bb)(bb) · · · = (ab)(bb)(bb) · · · . Nonetheless, X is a profinite code in agreement with Theorem 4. Note that, in A∗, the pseudowords a(bb)ω and ab(bb)ω are distinct (the first one is a limit of words of odd length and the second one of words of even length). Theorem 4 shows that the closure of the submonoid generated by a finite code is a free profinite monoid. This has been extended to rational codes in [?]. Actually, for any rational code X, the profinite submonoid generated by X is free with basis the closure ¯ X of X [?, Corollary 5.7]. Note that we have here a free profinite monoid with infinite basis (see [?]). Example Let A = {a, b} and X = a∗b. Then the profinite monoid X ∗ is free with basis ¯ X = a∗b ∪ aωb.

Dominique Perrin Profinite semigroups

Factorial sets of pseudowords

An infinite pseudoword x is recurrent if for every factor u of x there is some v such that uvu is a factor of x. It is uniformly recurrent if for every factor u of x, there is an n ≥ 1 such that u is a factor of any factor of x of length n. The proof of the following result is straightforward (using Proposition 1). Proposition An infinite pseudoword is uniformly recurrent if and only if all its infinite factors have the same finite factors.

Dominique Perrin Profinite semigroups

Green relations

Recall that the J -order in a monoid M is defined by x ≤J y if x is a factor of y. Two elements x, y are J -equivalent if each one is a factor of the other (this is one of the Green’s relations). Replacing the notion of factor by prefix (resp. suffix), one obtains the R-order (resp. L-order). Thus, two elements x, y of a monoid M are R-equivalent (resp. L-equivalent) if xM = yM (resp. Mx = My). The H-equivalence is the intersection of R and LL. In any monoid, one has RL = LR and one denotes D the equivalence RL = LR which is the supremum of R and L. In a compact monoid, one has D = J (the proof is the same as in a finite monoid. It uses the fact that a compact monoid satisfies the stability condition : if x ≤R y and xJ y, then xRy and dually for L.

Dominique Perrin Profinite semigroups

The following result gives an algebraic characterization of uniform recurrence in the free profinite monoid. Theorem An infinite pseudoword is uniformly recurrent if and only if it is J -maximal. The proof uses three lemmas. An element s of a semigroup S is regular if there is some x ∈ S such that sxs = s. In a compact semigroup, a J -class contains a regular element if and only if all its elements are regular, if and only if it contains an idempotent.

Dominique Perrin Profinite semigroups

For a pseudoword w, we denote X(w) the the set of all infinite pseudowords which are limits of sequences of finite factors of w. Lemma Let x be uniformly recurrent pseudoword over a finite alphabet A.

1 Every element of X(w) is a factor of w. 2 All elements of X(w) lie in the same J -class of

A∗.

3 Every element of X(w) is regular. Dominique Perrin Profinite semigroups

Assertion 1 results directly from Proposition 1. Suppose that u, v ∈ X(w). By Proposition 6, they have the same set of finite factors. Thus, by Assertion 1, they are J -equivalent. Assume that u is the limit of a sequence (un)n≥0 of finite factors

• f w. Since w is recurrent, there are finite words vn such that

unvnun is a factor of w. If v is an accumulation point of the sequence vn, then uvu is a factor of w which belongs to X(w). By Assertion 2, it is J -equivalent to u. In a compact monoid, by the stability condition, this implies that u and uvu are H-equivalent and thus that u is regular (indeed, uHuvu implies uHu(vu)ω and thus (vu)ω is an idempotent in J(u)).

Dominique Perrin Profinite semigroups

Lemma Let w be a uniformly recurrent pseudoword. Each H-class contained in the J -class of w contains some element of X(w). Let u ∈ J(w). Denote by xn and yn the prefix and the suffix of u of length n. Since u is uniformly recurrent by Proposition 6, there is a factor tn of x of length at least 2n having xn as a prefix and yn as a suffix. Taking a subsequence, we may assume that the sequences (xn), (yn) and (tn) converge to x, y, t. Then x, y, t ∈ J(w) by Lemma 8(b). Since x ≥R t and t ≤L y, by stability, we obtain uHt.

Dominique Perrin Profinite semigroups

Lemma Let u be a uniformly recurrent pseudoword and suppose v is a pseudoword such that uv is still uniformly recurrent. Then u and uv are R-equivalent. Suppose first that v is finite. Let un be the suffix of u of length n. Since u is an infinite factor of uv which is uniformly recurrent, they have the same finite factors by Lemma 6. Hence for every n there is some m(n) such that um(n) = xnunvyn. By compactness, we may assume by taking subsequences that the sequences xn, yn, un converge to x, y, u′ respectively. Then by continuity of the multiplication, the sequence um(n) converges to xu′vy. Since the limits of two convergent sequences of suffixes of the same pseudoword are L-equivalent, we obtain that xu′uyLu and thus uRu′v by stability. Since u′ is the limit of a sequence of suffixes of u, there is some factorization of the form u = zu′. Since the R-equivalence is a left congruence, we finally obtain uv = zu′vRzu′ = u.

Dominique Perrin Profinite semigroups

Assume next that v is infinite. We assume by contradiction that u >R uv. Let vn be a sequence of finite words converging to v. Taking a subsequence, we may assume that uvn >R u for all n. Thus for each n, we have a factorization vn = xnanyn with an ∈ A such that uRuxn >R uxnan ≥R uv. Since the alphabet is finite and A∗ is compact we may, up to taking a subsequence, assume that the letter sequence an is constant and the sequences xn, yn converge to x and y respectively. Thus we have p = xay with a ∈ A and uRux >R uxa ≥R uv. On the other hand, since ux and uxa are infinite factors of uv, they are both uniformly recurrent by Proposition 6. By the first part, we have uxRuxa, a contradiction. A dual result holds for the L-order.

Dominique Perrin Profinite semigroups

proof of the theorem

Suppose first that w is J -maximal as an infinite pseudoword. If v is an infinite factor of w, it is J -equivalent to w. Hence v, w have the same factors and, in particular, the same finite factors. By Proposition 6, w is uniformly recurrent. Suppose conversely that u, w ∈ A∗ \ A∗ are such that u ≥J w with w uniformly recurrent. Set w = puq with p, q ∈ A∗. By the dual of Lemma 10, we have puLu. And by Lemma 10, we have puRpuq. Thus u and w are J -equivalent.

Dominique Perrin Profinite semigroups

example

The J -class of aω in A∗ is a singleton. The J -class of (ab)ω has four elements. It is represented in Figure 4. (ab)ω (ab)ωa b(ab)ω (ba)ω

Figure: The J -class of (ab)ω.

Dominique Perrin Profinite semigroups