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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 44 (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...
A Conjecture on the Concatenation Product
"... In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Malcev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure  this oper ..."
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Cited by 8 (3 self)
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In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Malcev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure  this operation corresponds to passing to the upper level in any concatenation hierarchy . Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the StraubingThérien hierarchy and of the dotdepth hierarchy. Consequences for the other levels are also discussed.
Hyperdecidability of Pseudovarieties of Orthogroups
 Glasgow Math. J
"... Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product B fl m V of the pseudovariety of bands with a pseudovariety of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that ..."
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Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product B fl m V of the pseudovariety of bands with a pseudovariety of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in case say only terms (i.e., terms involving only multiplication and the (! \Gamma 1)power) are considered. It is also shown that, if V is a hyperdecidable (respectively reducible) pseudovariety of groups, then so is W. 1 Introduction Motivated by the KrohnRhodes complexity problem [22], the search for uniform algorithms for computing semidirect products of pseudovarieties has led to substantial research in the theory of finite semigroups. Even though there is no universal solution, since the semidirect product of decidable pseudovarieties is not necessarily decidable [1], under suitable assumptions on the factors, the semidirect product might be decidable. The no...
New results on the conjecture of rhodes and on the topological conjecture
 J. of Pure and Applied Algebra
, 1992
"... the present version, the bibliography has been updated and alphabetically ordered. The Conjecture of Rhodes, originally called the “type II conjecture” byRhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most ..."
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the present version, the bibliography has been updated and alphabetically ordered. The Conjecture of Rhodes, originally called the “type II conjecture” byRhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases. 1 The conjecture of Rhodes and the topological conjecture In this paper, all semigroups (respectively monoids, groups) are finite except in the case of free monoids or free groups. If M is a monoid, E(M) (respectively Reg(M)) denotes the set ofidempotents (respectively regularelements) of M. If x ∈ M, x ω denotes the unique idempotent of the subsemigroup of M generated by x. A blockgroup monoid is a monoid in which every Rclass and every Lclass contain at most one idempotent. A number of equivalent conditions are given in [11]. For instance, a monoid M is a blockgroup monoid if and only if, for every regular Jclass D of M, the semigroup D 0 is a Brandt semigroup, or if and only if the submonoid generated by E(M) is Jtrivial. The class of all blockgroup monoids forms a (pseudo)variety of monoids, denoted by BG. We refer to [17] for an introduction to the conjecture of Rhodes and for all undefined notations. Let M be a finite monoid. Recall that the kernel of M is the submonoid
Covers for Monoids
"... . In this contribution to the structure theory of semigroups, we propose a unified generalisation of a string of results on group extensions, originating on the one hand in the seminal structure and covering theorems of McAlister and on the other, in Ash's celebrated solution of the Rhodes conj ..."
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. In this contribution to the structure theory of semigroups, we propose a unified generalisation of a string of results on group extensions, originating on the one hand in the seminal structure and covering theorems of McAlister and on the other, in Ash's celebrated solution of the Rhodes conjecture in finite semigroup theory. McAlister proved that each inverse monoid admits an Eunitary cover, and gave a structure theorem for Eunitary inverse monoids. Subsequent generalisations extended one or both results to orthodox monoids (McAlister, Szendrei, Takizawa), regular monoids (Trotter), Edense semigroups in which the idempotents form a semilattice (Margolis and Pin, Fountain), and Edense semigroups in which the idempotents form a subsemigroup (Almeida, Pin and Weil, Zhonghao Jiang). We show that any Edense monoid admits a Dunitary Edense cover and we provide a structure theorem for Dunitary Edense monoids, in terms of groups acting on a category. Here D(M) is the least weakly s...
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 Malce ..."
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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
and
"... Edense monoids, regular monoids. * The authors gratefully acknowledge support from the FrancoBritish joint research programme ALLIANCE (contract 96069), of the British Council and the Ministère des Affaires Étrangères. 1 Abstract. A monoid M is an extension of a submonoid T by a group G if there ..."
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Edense monoids, regular monoids. * The authors gratefully acknowledge support from the FrancoBritish joint research programme ALLIANCE (contract 96069), of the British Council and the Ministère des Affaires Étrangères. 1 Abstract. A monoid M is an extension of a submonoid T by a group G if there is a morphism from M onto G such that T is the inverse image of the identity of G. Our first main theorem gives descriptions of such extensions in terms of groups acting on categories. The theory developed is also used to obtain a second main theorem which answers the following question. Given a monoid M and a submonoid T, under what conditions can we find a monoid � M and a morphism θ from � M onto M such that � M is an extension of a submonoid � T by a group, and θ maps � T isomorphically onto T. These results can be viewed as generalisations of two seminal theorems of McAlister in inverse semigroup theory. They are also closely related to Ash’s celebrated solution of the Rhodes conjecture in finite semigroup theory. McAlister proved that each inverse monoid admits an Eunitary inverse cover, and gave a structure theorem for Eunitary inverse monoids. Many researchers have extended one or both of these results to wider classes of semigroups. Almost all these generalisations can be recovered from our two main theorems.