Profinite semigroups Dominique Perrin 13 novembre 2015 Dominique Perrin Profinite semigroups
Outline Profinite semigroups Profinite codes Uniform recurrence 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 complement of an open set is called a closed set. A clopen set is both open and closed. A map ϕ : X → Y between topological spaes 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 X i indexed by a set I , the product topology on the direct product Π i ∈ I X i is defined as the coarsest topology such that the projections π i : X → X i are continuous. A basis of the family of open sets are the sets of the form Π i ∈ I U i where U i � = X i 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 R n 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 ( u n ) n ≥ 0 be a sequence of factors of x ∈ M converging to some u ∈ M . Let p n , q n be such that x = p n u n q n for all n ≥ 1. Since M is compact, the sequences ( p n ) , ( q n ) 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. A ϕ ψ θ S T 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 ϕ i : A → S i , (iii) for each pair i , j ∈ I with i ≥ j , a connecting morphism ψ i , j : ϕ i → ϕ j such that ψ i , i is the identity on S i and for i ≥ j ≥ k , ψ i , k = ψ i , j ◦ ψ j , k . The projective limit of this projective system is a topological semigroup Φ : A → S together with morphisms Φ i : Φ → ϕ i such that for all i , j ∈ I with i ≥ j , ψ i , j ◦ Φ i = Φ j , and for any A -generated topological semigroup Ψ : A → S and morphisms with morphisms Ψ i : Ψ → ϕ i such that for all i , j ∈ I with i ≥ j , ψ i , j ◦ Ψ i = Ψ j , the exists a morphism θ : ψ → Φ such that Φ i ◦ θ = Ψ i for all i ∈ I . Dominique Perrin Profinite semigroups
A Ψ Ψ T ϕ j ϕ i θ θ Ψ j Ψ j Ψ i Ψ i Φ S Φ j Φ j Φ i Φ i ψ i , j ψ i , j ϕ j ϕ i S i S j 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 Π i ∈ I S i consisting of all ( s i ) i ∈ I such that, for all i , j ∈ I with i ≥ j , ψ i , j ( s i ) = s j endowed with the product topology. The map Φ : A → S is given by Φ( a ) = ( ϕ i ( a )) i ∈ I and the maps Φ : S → S i 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 − 1 ϕ ( K ). Proposition Let S be a profinite semigroup. A subset K ⊂ S is clopen if and only 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 Π i ∈ I S i of finite semigroups S i . Then K may be expressed as K = S ∩ ( K 1 ∪ . . . ∪ K n ) where each K ℓ is is a product of the form Π i ∈ I X i with X i ⊂ S i and X i = S i except on a finite set J ℓ of indices. Let J = J 1 ∪ . . . ∪ J n . The projection ϕ : S → Π i ∈ J S i 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 ϕ : � A ∗ → M such that ˆ continuous morphism ˆ ϕ ◦ ι = ϕ . A ϕ ι ˆ ϕ � S A ∗ 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. X of X in � (ii) the closure ¯ 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 ϕ − 1 ϕ ( X ) unique continuous morphism ˆ ϕ extending ϕ . Then X = ˆ 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 − 1 ψ ( K ) and so X is recognizable. Dominique Perrin Profinite semigroups
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