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Ash's type II theorem, profinite topology and Malcev products Part I
"... This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by t ..."
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Cited by 49 (10 self)
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This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture  also verified by Ash  it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH 1 H 2 \Delta \Delta \Delta Hn , where each H i is a finitely generated subgroup of G. This significantly extends classical results by M. Hall. Final...
Polynomial closure and unambiguous product
 Theory Comput. Systems
, 1997
"... This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of language ..."
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Cited by 49 (7 self)
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This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of languages
Extending partial automorphisms and the profinite topology on free groups
 Tran. AMS
, 2000
"... Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1,p2,...,pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and ..."
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Cited by 44 (0 self)
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Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1,p2,...,pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1,α2,...,αn of C3 extending the pi. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiĭ stating that a finite product of finitely generated subgroups is closed for this topology. 1.
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 43 (17 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.
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.
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
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 23 (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.
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 22 (7 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...