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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 18 (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...
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 10 (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...
Algorithmic problems in groups, semigroups and inverse semigroups
 Semigroups, Formal Languages and Groups
, 1995
"... ..."
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 6 (0 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.
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...
Algebraic and Topological Theory of Languages
"... : A language is torsion (resp. bounded torsion, aperiodic, bounded aperiodic) if its syntactic monoid is torsion (resp. bounded torsion, aperiodic, bounded aperiodic). We generalize the regular language theorems of Kleene, Schutzenberger and Straubing to describe the classes of torsion, bounded tors ..."
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: A language is torsion (resp. bounded torsion, aperiodic, bounded aperiodic) if its syntactic monoid is torsion (resp. bounded torsion, aperiodic, bounded aperiodic). We generalize the regular language theorems of Kleene, Schutzenberger and Straubing to describe the classes of torsion, bounded torsion, aperiodic and bounded aperiodic languages. These descriptions involve taking limits of sequences of languages and automata for certain topologies defined by filtrations of the free monoid. A theorem for arbitrary languages over finite alphabets is also stated and proved. AMS Mathematics Subject Classification: 68Q45, 68Q70, 20M35. * Both authors gratefully acknowledge support from the first author's National Science Foundation grant DMS8803362. The second author was also supported in part by the Projet de Recherche Coordonn'ee "Math'ematiques et Informatique". J. Rhodes and P. Weil Introduction The aim of this paper is to generalize the central results of the theory of rational, o...
ITERATED PERIODICITY OVER FINITE APERIODIC SEMIGROUPS
"... Abstract. This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that can be described from the free generators using only the operations of multiplication and ωpower. A necessary and sufficient condition for this property to hold turns out t ..."
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Abstract. This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that can be described from the free generators using only the operations of multiplication and ωpower. A necessary and sufficient condition for this property to hold turns out to be given by the conjunction of two rather simple finiteness conditions: the nonexistence of infinite antichains of factors and the rationality of the language of McCammond normal forms of ωterms that define factors of the given pseudoword. The relationship between pseudowords with this property and arbitrary pseudowords is also investigated. 1.
ωTERMS OVER FINITE APERIODIC SEMIGROUPS
"... Abstract. This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that are given by ωterms, that is that can be obtained from the free generators using only multiplication and the ωpower. A necessary and sufficient condition for this property ..."
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Abstract. This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that are given by ωterms, that is that can be obtained from the free generators using only multiplication and the ωpower. A necessary and sufficient condition for this property to hold turns out to be given by the conjunction of two rather simple finiteness conditions: the nonexistence of infinite antichains of factors and the rationality of the language of McCammond normal forms of ωterms that define factors. 1. Introduction and