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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 the Ma ..."
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Cited by 43 (9 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...
Free Profinite Semigroups Over Semidirect Products
, 1995
"... We give a general description of the free profinite semigroups over a semidirect product of pseudovarieties. More precisely,\Omega A (V W) is described as a closed subsemigroup of a profinite semidirect product of the form\Omega \Omega AW\ThetaA V \Omega AW. As a particular case, the free pro ..."
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Cited by 23 (11 self)
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We give a general description of the free profinite semigroups over a semidirect product of pseudovarieties. More precisely,\Omega A (V W) is described as a closed subsemigroup of a profinite semidirect product of the form\Omega \Omega AW\ThetaA V \Omega AW. As a particular case, the free profinite semigroup over J 1 V is described in terms of the geometry of the Cayley graph of the free profinite semigroup over V (here J 1 is the pseudovariety of semilattice monoids). Applications are given to the calculations of the free profinite semigroup over J 1 Nil and of the free profinite monoid over J 1 G (where Nil is the pseudovariety of finite nilpotent semigroups and G is the pseudovariety of finite groups). The latter free profinite monoid is compared with the free profinite inverse monoid, which is also calculated here.
Free profinite Rtrivial monoids
"... This article is concerned with the structure of semigroups of implicit operations on R, the pseudovariety of all Rtrivial semigroups. We give two complementary descriptions for these semigroups: first by means of labeled ordinals, and second by means of labeled infinite trees of finite depth. In th ..."
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Cited by 12 (11 self)
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This article is concerned with the structure of semigroups of implicit operations on R, the pseudovariety of all Rtrivial semigroups. We give two complementary descriptions for these semigroups: first by means of labeled ordinals, and second by means of labeled infinite trees of finite depth. In the first representation, the product of implicit operations has a simple characterization. In the second one, the topology on implicit operations is best visible. A similar study is conducted for the implicit operations on DRG, the nonaperiodic analogue of R. This leads to a description of the corresponding variety of recognizable languages and to the computation of certain joins of pseudovarieties.
Profinite Methods in Finite Semigroup Theory
, 2001
"... This paper is a survey of the authors' recent results in the theory of finite semigroups using profinite techniques. This involves the study of free profinite semigroups, whose structure encodes algebraic and combinatorial properties of finite semigroups. ..."
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Cited by 4 (4 self)
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This paper is a survey of the authors' recent results in the theory of finite semigroups using profinite techniques. This involves the study of free profinite semigroups, whose structure encodes algebraic and combinatorial properties of finite semigroups.
Equations for pseudovarieties
 Formal Properties of Finite Automata and Applications
, 1989
"... Tameness of some locally trivial ..."
Tameness of Some Locally Trivial Pseudovarieties
, 1999
"... kappatameness is a strong property of semigroup pseudovarieties related to the membership problem. We prove the kappatameness of the following pseudovarieties: N, D, K and LI, which are associated through Eilenberg's correspondence with the varieties of finite and cofinite languages, suffix testab ..."
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Cited by 2 (1 self)
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kappatameness is a strong property of semigroup pseudovarieties related to the membership problem. We prove the kappatameness of the following pseudovarieties: N, D, K and LI, which are associated through Eilenberg's correspondence with the varieties of finite and cofinite languages, suffix testable languages, prefix testable languages and factor testable languages respectively.
Semidirect Products With the Pseudovariety of All Finite Groups
, 2000
"... This is a survey of recent results related to semidirect products of an arbitrary pseudovariety with the pseudovariety of all nite groups. The main avour is the establishment of links between various operators on pseudovarieties, some obviously computable, others known not to be so. This not only le ..."
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This is a survey of recent results related to semidirect products of an arbitrary pseudovariety with the pseudovariety of all nite groups. The main avour is the establishment of links between various operators on pseudovarieties, some obviously computable, others known not to be so. This not only leads to decidability results but does so in a sort of uniform way which has a structural tint even though the arguments are mostly syntactical.