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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.
Closed subgroups of free profinite monoids are projective profinite groups
 Bull. London Math. Soc
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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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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
Profinite groups associated to sofic shifts are free
 Proc. London Math. Soc
"... Abstract. We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup ..."
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Abstract. We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the kernel of the free profinite semigroup and the Jclass associated to an irreducible sofic shift. 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.
Characterization of group radicals with an application to Mal’cev products, in preparation
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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.