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21
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 18 (11 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.
Subword Complexity of Profinite Words and Subgroups of Free Profinite Semigroups
 Internat. J. Algebra Comput
, 2003
"... We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated implicit operators, subword complexity and the associated entropy. ..."
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Cited by 14 (6 self)
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We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated implicit operators, subword complexity and the associated entropy.
Dynamics of implicit operations and tameness of pseudovarieties of groups
 Trans. Amer. Math. Soc
, 2002
"... Abstract. This work gives a new approach to the construction of implicit operations. By considering “higherdimensional ” spaces of implicit operations and implicit operators between them, the projection of idempotents back to onedimensional spaces produces implicit operations with interesting prop ..."
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Cited by 11 (4 self)
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Abstract. This work gives a new approach to the construction of implicit operations. By considering “higherdimensional ” spaces of implicit operations and implicit operators between them, the projection of idempotents back to onedimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiĭ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of pgroups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V, if it can be decided whether a finitely generated subgroup of the free group with the proV topology is dense, then V is tame. 1.
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 deci ..."
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Cited by 8 (8 self)
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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...
Globals Of Pseudovarieties Of Commutative Semigroups: The Finite Basis Problem, Decidability, And Gaps
 Proc. Edinburgh Math. Soc
"... . Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or !. Nevertheless, on these pseudovarieties, the operat ..."
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Cited by 7 (2 self)
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. Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or !. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains order isomorphic to the reals. AMS subject classification numbers (1990): 20M07, 20M05 1. Introduction Building on ideas of J. Rhodes and others [15, 16], Tilson [17] introduced categories and semigroupoids (categories without local identities) as a tool for studying semidirect products of semigroups. Weil and the first author [10] integrated into Tilson's theory the profinite perspective culminating in the description of a basis of pseudoidentities for a semidirect product V W of pseudovarieties of semigroups depending on a bas...
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 ..."
Profinite groups associated with weakly primitive substitutions, Fundam
 Prikl. Mat
, 2007
"... Abstract. A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword which is a factor of all its infinite factors, that is one which lies ..."
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Cited by 4 (1 self)
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Abstract. A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword which is a factor of all its infinite factors, that is one which lies
Complete reducibility of systems of equations with respect to R
"... It is shown that the pseudovariety R of all finite Rtrivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of ..."
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Cited by 3 (3 self)
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It is shown that the pseudovariety R of all finite Rtrivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of the corresponding rational constraint and the system is verified in R, then there is some such evaluation which is “regular”, that is one in which, additionally, the pseudowords only involve multiplications and ωpowers.
THE EQUATIONAL THEORY OF ωTERMS FOR FINITE Rtrivial Semigroups
, 2005
"... A new topological representation for free profinite Rtrivial semigroups in terms of spaces of vertexlabeled complete binary trees is obtained. Such a tree may be naturally folded into a finite automaton if and only if the element it represents is an ωterm. The variety of ωsemigroups generated by ..."
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Cited by 3 (3 self)
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A new topological representation for free profinite Rtrivial semigroups in terms of spaces of vertexlabeled complete binary trees is obtained. Such a tree may be naturally folded into a finite automaton if and only if the element it represents is an ωterm. The variety of ωsemigroups generated by all finite Rtrivial semigroups, with the usual interpretation of the ωpower, is then studied. A simple infinite basis of identities is exhibited and a lineartime solution of the word problem for relatively free ωsemigroups is presented. This work is also compared with recent work of Bloom and Choffrut on transfinite words.