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Closed subgroups of free profinite monoids are projective profinite groups
 Bull. London Math. Soc
"... Abstract. 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 duri ..."
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Abstract. 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. 1.
Profinite groups associated with weakly primitive substitutions, Fundam
 Prikl. Mat
, 2007
"... Abstract. A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword which is a factor of all its infinite factors, that is one which lies ..."
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Cited by 4 (1 self)
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Abstract. A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword which is a factor of all its infinite factors, that is one which lies
Characterization of group radicals with an application to Mal’cev products, in preparation
"... Abstract. Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups. 1. ..."
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Abstract. Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups. 1.
Profinite semigroups and applications
 IN STRUCTURAL THEORY OF AUTOMATA, SEMIGROUPS, AND UNIVERSAL ALGEBRA
, 2003
"... Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which ..."
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Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial problems on rational languages translate into algebraictopological problems on profinite semigroups. The aim of these lecture notes is to introduce these topics and to show how they intervene in the most recent developments in the area.
ON FREE PROFINITE SUBGROUPS OF FREE PROFINITE MONOIDS
, 712
"... Abstract. 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 p ..."
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Abstract. 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 proH monoids. 1.
PSEUDOVARIETIES DEFINING CLASSES OF SOFIC SUBSHIFTS CLOSED FOR TAKING SHIFT EQUIVALENT
, 2005
"... For a pseudovariety V of ordered semigroups, let S (V) be the class of sofic subshifts whose syntactic semigroup lies in V. It is proved that if V contains Sl − then S (V∗D) is closed for taking shift equivalent subshifts, and conversely, if S (V) is closed for taking conjugate subshifts then V con ..."
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For a pseudovariety V of ordered semigroups, let S (V) be the class of sofic subshifts whose syntactic semigroup lies in V. It is proved that if V contains Sl − then S (V∗D) is closed for taking shift equivalent subshifts, and conversely, if S (V) is closed for taking conjugate subshifts then V contains LSl − and S (V) = S (V∗D). Almost finite type subshifts are characterized as the irreducible elements of S (LInv), which gives a new proof that the class of almost finite type subshifts is closed for taking shift equivalent subshifts.
RATIONAL CODES AND FREE PROFINITE MONOIDS
"... Abstract. 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 ..."
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Abstract. 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 proH monoids for H an extensionclosed pseudovariety of groups. 1.
A COMBINATORIAL PROPERTY OF IDEALS IN FREE PROFINITE MONOIDS
, 2008
"... The reader is referred to [8] for all undefined notation concerning finite and profinite semigroups. We assume throughout this note that V is a pseudovariety of monoids [1, 5, 8] closed under Mal’cev product with the pseudovariety A of aperiodic monoids, i.e., A ○m V = V. Denote by ̂ FV(A) the free ..."
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The reader is referred to [8] for all undefined notation concerning finite and profinite semigroups. We assume throughout this note that V is a pseudovariety of monoids [1, 5, 8] closed under Mal’cev product with the pseudovariety A of aperiodic monoids, i.e., A ○m V = V. Denote by ̂ FV(A) the free proV monoid on a profinite space A [1,3,8]. In this note we prove the following theorem. Theorem 1. Suppose that α1,...,αm ∈ ̂ FV(A) and I1,...,In are closed ideals in ̂ FV(A) where m ≤ n. If α1 · · · αm ∈ I1 · · · In, then αi ∈ Ij for some i and j. Before proving the theorem, we state a number of consequences. Recall that an ideal I in a semigroup is prime if ab ∈ I implies a ∈ I or b ∈ I. Corollary 2. Let I = I 2 be a closed idempotent ideal of ̂ FV(A). Then I is prime. Proof. Suppose that ab ∈ I = I 2. Then a ∈ I or b ∈ I by Theorem 1. An element a of a semigroup S is said to be regular if there exists b ∈ S so that aba = a. Any regular element of a profinite semigroup generates a closed idempotent ideal. Hence we have: Corollary 3. Every regular element of ̂ FV(A) generates a prime ideal. In particular, the minimal ideal of ̂ FV(A) is prime. The second statement of Corollary 3 was first proved by Almeida and Volkov using techniques coming from symbolic dynamics [2]. Our next result generalizes a result of Rhodes and the author showing that all elements of finite order in ̂ FV(A) are group elements, which played a key role in proving that such elements are in fact idempotent [7]. Let ̂ N denote the profinite completion of the monoid of natural numbers; it is in fact a profinite semring. We use ω for the nonzero idempotent of ̂ N. Corollary 4. Let α ∈ ̂ FV(A) satisfy a n = a n+λ for some positive integer n and some 0 ̸ = λ ∈ ̂ N. Then a is a group element, i.e., a = a ω a.
PRESENTATIONS OF SCHÜTZENBERGER GROUPS OF MINIMAL SUBSHIFTS
"... Abstract. In previous work, the first author established a natural bijection between minimal subshifts and maximal regularJclasses of free profinite semigroups. In this paper, the Schützenberger groups of such Jclasses are investigated in particular in respect to a conjecture proposed by the first ..."
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Abstract. In previous work, the first author established a natural bijection between minimal subshifts and maximal regularJclasses of free profinite semigroups. In this paper, the Schützenberger groups of such Jclasses are investigated in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for several types of minimal subshifts associated with substitutions. The Schützenberger subgroup of the Jclass corresponding to the ProuhetThueMorse subshift is shown to admit a somewhat simpler presentation, from which it follows that it satisfies the conjecture, that it has rank three, and that it is nonfree relatively to any pseudovariety of groups. 1.