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15
Coherence, local quasiconvexity and the perimeter of 2complexes
, 2002
"... A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2complexes which is introduced here. ..."
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Cited by 24 (5 self)
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A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that onerelator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3manifold groups. The main application is to establish the coherence
Normal Forms for Free Aperiodic Semigroups
 Int. J. Algebra Comput
, 1999
"... The implicit operation # is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates. Using # there is a welldefined algebra which is known as the free aperiodic semigroup. In this article we show that for each n, the ..."
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Cited by 16 (1 self)
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The implicit operation # is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates. Using # there is a welldefined algebra which is known as the free aperiodic semigroup. In this article we show that for each n, the n generated free aperiodic semigroup is defined by a finite list of pseudoidentities and has a decidable word problem. In the language of implicit operations, this shows that the pseudovariety of finite aperiodic semigroups is #recursive. This completes a crucial step towards showing that the KrohnRhodes complexity of every finite semigroup is decidable.
Groups and semigroups: connections and contrasts
 Proceedings, Groups St Andrews 2005, London Math. Soc. Lecture Note Series 340, Vol
"... Group theory and semigroup theory have developed in somewhat different directions in the past several decades. While Cayley’s theorem enables us to view groups as groups of permutations of some set, the analogous result in semigroup theory represents semigroups as semigroups of functions from a set ..."
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Cited by 2 (1 self)
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Group theory and semigroup theory have developed in somewhat different directions in the past several decades. While Cayley’s theorem enables us to view groups as groups of permutations of some set, the analogous result in semigroup theory represents semigroups as semigroups of functions from a set to itself. Of course both group theory and semigroup theory have developed
Maximal Groups in Free Burnside Semigroups
, 1998
"... . We prove that any maximal group in the free Burnside semigroup defined by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a gr ..."
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Cited by 1 (1 self)
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. We prove that any maximal group in the free Burnside semigroup defined by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the J class containing the group. For n = 2 and for every m 2 we present examples with 2m \Gamma 1 generators. Hence, in these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n = 2, which was almost completely unknown until now. Surprisingly, the case n = 2 presents simultaneously the complexities of the cases n = 1 and n 3: the maximal groups are cyclic of order m for n 3 but they can have more generators and be infinite for n 2; there are exactly 2 jAj J classes and they are easily characterized for n = 1 but there are infinitely many J classes and they are difficult to c...
Profinite Identities for Finite Semigroups Whose Subgroups Belong to a Given Pseudovariety
"... We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite proidentity basis for the pseudovariety H of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of ..."
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We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite proidentity basis for the pseudovariety H of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of finite groups.
On the Burnside Semigroups . . .
, 1995
"... In this paper we prove that the congruence classes of A associated to the Burnside semigroup with jAj generators defined by the equation x n = x n+m , for n 4 and m 1, are recognizable. This problem was originally formulated by Brzozowski in 1969 for m = 1 and n 2. De Luca and Varricchio ..."
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In this paper we prove that the congruence classes of A associated to the Burnside semigroup with jAj generators defined by the equation x n = x n+m , for n 4 and m 1, are recognizable. This problem was originally formulated by Brzozowski in 1969 for m = 1 and n 2. De Luca and Varricchio solved the problem for n 5 in 90. A little later, McCammond extended the problem for m 1 and solved it independently in the cases n 6 and m 1. Our work, which is based on the techniques developed by de Luca and Varricchio, extends both these results. We effectively construct a minimal generator \Sigma of our congruence. We introduce an elementary concept, namely the stability of productions, which allows to eliminate all hypothesis related to the values of n and m. A substantial part of our proof consists of showing that all productions in \Sigma are stable, for n 4 and m 1. We also show that \Sigma is a ChurchRosser rewriting system, thus solving the word problem, and sho...
Tameness Of Pseudovarieties Of Semigroups
, 2000
"... Tameness is a property introduced in 1997 by Steinberg and the author in connection with the KrohnRhodes complexity problem in order to establish the decidability of semidirect products of pseudovarieties of semigroups. Since then a number of works have been dedicated to proving tameness of pseudov ..."
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Tameness is a property introduced in 1997 by Steinberg and the author in connection with the KrohnRhodes complexity problem in order to establish the decidability of semidirect products of pseudovarieties of semigroups. Since then a number of works have been dedicated to proving tameness of pseudovarieties. This paper is a survey of work in this area.
On a problem of Brzozowski and Fich
"... The problem of decidability of membership in the semidirect product Sl L of the pseudovarieties Sl of all finite semilattices and L of all finite L trivial semigroups has been around since 1984. This paper discusses several developments motivated by this question and leading to a proof that set ..."
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The problem of decidability of membership in the semidirect product Sl L of the pseudovarieties Sl of all finite semilattices and L of all finite L trivial semigroups has been around since 1984. This paper discusses several developments motivated by this question and leading to a proof that settles it affirmatively. 1 Introduction The historical importance of certain "test problems" in the development of Mathematics is probably easily acknowledged in view of examples such as the squaring of the circle or, more recently, Fermat's "last theorem" or the fourcolour map conjecture. In all such examples, one may say that the developments they led to have far reaching consequences, even if the results themselves are not much more than curiosities. Each researcher tends to have a personal set of test problems in mind guiding contributions to the theories of her/his own interest. When a problem is chosen, one usually thinks that one should be able to solve it. Sometimes, one sees immed...