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49
Closed subgroups in proV topologies and the extension problem for inverse automata
 INT. J. ALGEBRA COMPUT
, 1999
"... We relate the problem of computing the closure of a finitely generated subgroup of the free group in the proV topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial onetoone maps on a finite set, can they ..."
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Cited by 42 (7 self)
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We relate the problem of computing the closure of a finitely generated subgroup of the free group in the proV topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial onetoone maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extensionclosed. Turning to practical computations, we modify Ribes and Zalesski i's algorithm to compute the prop closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pronilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups.
Profinite semigroups, Mal'cev products and identities
 J. ALGEBRA
, 1996
"... We compute a set of identities defining the Mal'cev product of pseudovarieties of finite semigroups or finite ordered semigroups. We also characterize the pointlike subsets of a finite semigroup by means of a relational morphism into a profinite semigroup. Finally, we apply our results to the p ..."
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Cited by 41 (16 self)
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We compute a set of identities defining the Mal'cev product of pseudovarieties of finite semigroups or finite ordered semigroups. We also characterize the pointlike subsets of a finite semigroup by means of a relational morphism into a profinite semigroup. Finally, we apply our results to the proof of the decidability of the Mal'cev products of a decidable pseudovariety with the pseudovarieties of nilpotent and of J trivial semigroups.
Möbius functions and semigroup representation theory. II. Character formulas and multiplicities
 Adv. Math
"... Abstract. We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota’s theory of Möbius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idemp ..."
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Cited by 38 (9 self)
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Abstract. We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota’s theory of Möbius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigaire, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing
Finite state automata: A geometric approach
 Trans. Amer. Math. Soc
"... Abstract. Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various probl ..."
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Cited by 22 (13 self)
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Abstract. Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was nontrivial), is whether it is true, for a pseudovariety of groups H, thataJtrivial coextension of a group in H must divide a semidirect product of a Jtrivial monoid and a group in H. We show the answer is affirmative if H is closed under extension, and may be negative otherwise. 1.
Hyperdecidable Pseudovarieties and the Calculation of Semidirect Products
 Internat. J. Algebra Comput
"... This note introduces the notion of a hyperdecidable pseudovariety. This notion appears naturally in trying to prove decidability of the membership problem of semidirect products of pseudovarieties of semigroups. It turns out to be a generalization of a notion introduced by C. J. Ash in connection ..."
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Cited by 21 (6 self)
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This note introduces the notion of a hyperdecidable pseudovariety. This notion appears naturally in trying to prove decidability of the membership problem of semidirect products of pseudovarieties of semigroups. It turns out to be a generalization of a notion introduced by C. J. Ash in connection with his proof of the "type II" theorem. The main results in this paper include a formulation of the definition of a hyperdecidable pseudovariety in terms of free profinite semigroups, the equivalence with Ash's property in the group case, the behaviour under the operator g of taking the associated global pseudovariety of semigroupoids, and the decidability of V W in case gV is decidable and has a given finite vertexrank and W is hyperdecidable. A further application of this notion which is given establishes that the join of a hyperdecidable pseudovariety with a locally finite pseudovariety with computable free objects is again hyperdecidable. 1. Introduction A typical problem in...
Profinite Methods in Semigroup Theory
 Int. J. Algebra Comput
, 2000
"... this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun. ..."
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Cited by 21 (3 self)
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this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun.
On the Decidability of Iterated Semidirect Products With Applications to Complexity
, 1997
"... The notion of hyperdecidability has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidab ..."
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Cited by 20 (9 self)
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The notion of hyperdecidability has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly verified property of the first factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free profinite objects). As an application of the main results, the decidability of the KrohnRhodes (group) complexity is shown to follow from nonalgorithmic abstract properties likely to be satisfied by the pseudovariety of all finite aperiodic semigroups, thereby suggesting a new approach to answer (affirmativ...
On the extension problem for partial permutations
 PROC. AMER. MATH. SOC
, 2003
"... A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no nontrivial group identity and a metabelian pseudovariety Q. For each of these pseudovarietie ..."
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Cited by 14 (2 self)
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A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no nontrivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl∗V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators ∗, ∗∗, m○, ♦, ♦n, andP do not preserve decidability. In addition, several joins, including A ∨ U, are shown to be undecidable.
Syntactic and Global Semigroup Theory, a Synthesis Approach
 in: Algorithmic Problems in Groups and Semigroups
, 2000
"... This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of semigroups. We introduce various abstract and algorithmic properties that a pseudovariety of semigroups mig ..."
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Cited by 13 (8 self)
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This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of semigroups. We introduce various abstract and algorithmic properties that a pseudovariety of semigroups might possibly satisfy. The main theorem states that given a finite collection of pseudovarieties, each satisfying certain properties of the sort alluded to above, any iterated semidirect product of these pseudovarieties is decidable. In particular, the pseudovariety G of finite groups satisfies these properties. J. Rhodes has announced a proof, in collaboration with J. McCammond, that the pseudovariety A of finite aperiodic semigroups satisfies these properties as well. Thus, our main theorem would imply the decidability of the complexity of a finite semigroup. 1. Introduction In virtually any discipline, there will arise various schools or approaches to the development of that discip...