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 Z p denote the ring of p -adic integers, namely the completion of Z under the p -adic metric. This metric is defined by the norm � p − ord p ( x ) if x � = 0 | x | p = 0 otherwise where ord( x ) p ( x ) is the largest n such that p n divides x . Any element γ ∈ Z p has a unique p -adic expansion γ = c 0 + c 1 p + c 2 p 2 + . . . = ( . . . c 3 c 2 c 1 c 0 ) 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 Z p as lim Z / p n Z = Z p { ( a n ) n ≥ 0 ∈ Π n ≥ 0 Z / p n Z | for all n , a n +1 ≡ a n mod p n } = This expresses the ring Z p as a projective limit of the rings Z / p n Z . Dominique Perrin Profinite semigroups
Profinite integers The direct product of all rings Z p ˆ Z = Π Z p over all prime numbers p is the ring of profinite integers. One may define it equivalently as the projective limit of all cyclic groups ˆ = Π n ≥ 1 Z / n Z Z = { ( a n ) n ≥ 1 ∈ Π n ≥ 1 Z / n Z | for all n | m , a m ≡ a n mod n } or as the projective limit of the cyclic groups Z / n ! Z ˆ = Π n ≥ 1 Z / n ! Z Z = { ( b n ) n ≥ 1 ∈ Π n ≥ 1 Z / n ! Z | for all n , a n +1 ≡ a n mod n ! } Dominique Perrin Profinite semigroups
The factorial number system The last representation corresponds to an expansion of the form γ = c 1 + c 2 2! + c 3 3! + . . . = ( . . . c 3 c 2 c 1 ) ! with digits 0 < c i ≤ 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 ϕ n ( a ) ϕ ( a ) = ab ϕ 2 ( a ) = aba ϕ 3 ( a ) = abaab ϕ 4 ( a ) = abaababa · · · is the Fibonacci sequence of words of length equal to the Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n = F n − 1 + F n − 2 . One has F n = | ϕ n − 2 ( a ) | . The Fibonacci sequence of words converges in the space A N 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 ϕ n ! ( a ) ϕ ( a ) = ab ϕ 2 ( a ) = aba ϕ 6 ( a ) = abaababaabaababaababa ϕ 24 ( a ) = abaababa · · · abaababa · · · The limit in the sense of profinite topology, to be defined below, is a profinite word denoted ϕ ω ( a ) which begins by the Fibonacci − → ( ϕ n ( a )) n ≥ 0 and ends with the left infinite word infinite word ← − ( ϕ 2 n ( 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 open 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 X i indexed by a set I , the product topology on the direct product X = 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 i ϕ j 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 i ϕ j 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
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