Free Profinite Monoids Research Articles

A profinite approach to complete bifix decodings of recurrent languages

Abstract We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of free profinite monoids. We show that the complete bifix decoding of a uniformly recurrent language F by an F-charged rational complete bifix code is uniformly recurrent. An analogous result is obtained for recurrent languages. As an application of the machinery developed within this approach, we show that the maximal pronilpotent quotient of the Schützenberger group of an irreducible symbolic dynamical system is an invariant of eventual conjugacy.

Pronilpotent quotients associated with primitive substitutions

We describe the pronilpotent quotients of a class of projective profinite groups, that we call ω-presented groups, defined using a special type of presentations. The pronilpotent quotients of an ω-presented group are completely determined by a single polynomial, closely related with the characteristic polynomial of a matrix. We deduce that ω-presented groups are either perfect or admit the p-adic integers as quotients for cofinitely many primes. We also find necessary conditions for absolute and relative freeness of ω-presented groups. Our main motivation comes from semigroup theory: the maximal subgroups of free profinite monoids corresponding to primitive substitutions are ω-presented (a theorem due to Almeida and Costa). We are able to show that the composition matrix of a primitive substitution carries partial information on the pronilpotent quotients of the corresponding maximal subgroup. We apply this to deduce that the maximal subgroups corresponding to primitive aperiodic substitutions of constant length are not absolutely free.

On the group of a rational maximal bifix code

Abstract We give necessary and sufficient conditions for the group of a rational maximal bifix code Z to be isomorphic with the F-group of Z ∩ F {Z\cap F} , when F is recurrent and Z ∩ F {Z\cap F} is rational. The case where F is uniformly recurrent, which is known to imply the finiteness of Z ∩ F {Z\cap F} , receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of F.

ON PROFINITE UNIFORM STRUCTURES DEFINED BY VARIETIES OF FINITE MONOIDS

We consider uniformities associated with a variety of finite monoids V, but we work with arbitrary monoids and not only with free or free profinite monoids. The aim of this paper is to address two general questions on these uniform structures and a few more specialized ones. A first question is whether these uniformities can be defined by a metric or a pseudometric. The second question is the description of continous and uniformly continuous functions. We first give a characterization of these functions in term of recognizable sets and use it to extend a result of Reutenauer and Schützenberger on continuous functions for the pro-group topology. Next we introduce the notion of hereditary continuity and discuss the behaviour of our three main properties (continuity, uniform continuity, hereditary continuity) under composition, product or exponential. In the last section, we analyze the properties of V-uniform continuity when V is the intersection or the join of a family of varieties and we discuss in some detail the case where V is commutative.

Maximal subgroups of the minimal ideal of a free profinite monoid are free

We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. More generally, if H is variety of finite groups closed under extension and containing ℤ/pℤ for infinitely may primes p, the corresponding result holds for free pro-\( \bar H \) monoids.

A combinatorial property of ideals in free profinite monoids

A combinatorial property of ideals in free profinite monoids is proved.

Rational codes and free profinite monoids

It is well known that clopen subgroups of finitely generated free profinite groups are again finitely generated free profinite groups. Clopen submonoids of free profinite monoids need not be finitely generated nor free. Margolis, Sapir and Weil proved that the closed submonoid generated by a finite code (which is, in fact, clopen) is a free profinite monoid generated by that code. In this note we show that a clopen submonoid is free profinite if and only if it is the closure of a rational free submonoid. In this case its unique closed basis is clopen, and is, in fact, the closure of the corresponding rational code. More generally, our results apply to free pro- H ¯ monoids for H an extension-closed pseudovariety of groups.

Closed subgroups of free profinite monoids are projective profinite groups

We prove that the class of closed subgroups of free profinite monoids is precisely the class of projective profinite groups. In particular, the profinite groups associated to minimal symbolic dynamical systems by Almeida are projective. Our result answers a question raised by Lubotzky during the lecture of Almeida at the Fields Workshop on Profinite Groups and Applications, Carleton University, August 2005. We also prove that any finite subsemigroup of a free profinite monoid consists of idempotents.

On power groups and embedding theorems for relatively free profinite monoids

We determine those pseudovarieties of groups ${\bf H}$ for which the power monoids $P(G)$ , ranging over all groups $G$ in ${\bf H}$ , satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of ${\cal J}$ -trivial monoids by groups in ${\bf H}$ . That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation ${\bf PH}\,{=}\,{\bf J}\ast {\bf H}$ . The characterization is in terms of the geometry of the Cayley graphs of the free pro- ${\bf H}$ groups as well as in terms of the pro- ${\bf H}$ topology of a finitely generated free group.