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16
Representation theory of finite semigroups, semigroup radicals and formal language theory
 in preparation. COUNTING AND MATRIX REPRESENTATIONS 11
"... Abstract. In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier ..."
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Cited by 38 (17 self)
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Abstract. In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, twosided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Čern´y’s conjecture for an important class of automata.
Predicting nonlinear cellular automata quickly by decomposing them into linear ones
 Physica D
, 1998
"... We show that a wide variety of nonlinear cellular automata (CAs) can be decomposed into a quasidirect product of linear ones. These CAs can be predicted by parallel circuits of depthO(log 2 t) using gates with binary inputs, orO(log t) depth if “sum mod p ” gates with an unbounded number of inputs ..."
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Cited by 24 (8 self)
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We show that a wide variety of nonlinear cellular automata (CAs) can be decomposed into a quasidirect product of linear ones. These CAs can be predicted by parallel circuits of depthO(log 2 t) using gates with binary inputs, orO(log t) depth if “sum mod p ” gates with an unbounded number of inputs are allowed. Thus these CAs can be predicted by (idealized) parallel computers much faster than by explicit simulation, even though they are nonlinear. This class includes any CA whose rule, when written as an algebra, is a solvable group. We also show that CAs based on nilpotent groups can be predicted in depth O(log t) or O(1) by circuits with binary or “sum mod p ” gates respectively. We use these techniques to give an efficient algorithm for a CA rule which, like elementary CA rule 18, has diffusing defects that annihilate in pairs. This can be used to predict the motion of defects in rule 18 in O(log 2 t) parallel time. PACS Keywords: 02.10, 02.70, 05.45, 46.10 1
Famous trails to Paul Erdős
 MATHEMATICAL INTELLIGENCER
, 1999
"... The notion of Erdős number has floated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientific research has become a very collaborative process in the twentieth century, not an activity engaged in solely by ..."
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Cited by 23 (0 self)
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The notion of Erdős number has floated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientific research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. In this paper we explore some (fairly short) collaboration paths that one can follow from Paul Erdős to researchers inside and outside of mathematics. In particular, we find that all the Fields Medalists up through 1998 have Erdős numbers less than 6, and that over 60 Nobel Prize winners in physics, chemistry, economics, and medicine have Erdős numbers less than 9.
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 18 (10 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 14 (6 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.
Profinite semigroups and applications
 IN STRUCTURAL THEORY OF AUTOMATA, SEMIGROUPS, AND UNIVERSAL ALGEBRA
, 2003
"... Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which ..."
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Cited by 11 (2 self)
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Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial problems on rational languages translate into algebraictopological problems on profinite semigroups. The aim of these lecture notes is to introduce these topics and to show how they intervene in the most recent developments in the area.
Equational complexity of the finite algebra membership problem
"... In Celebration of the Accomplishments of Béla Csákány We associate to each variety of algebras of finite signature a function on the positive integers called the equational complexity of the variety. This function is a measure of how much of the equational theory of a variety must be tested to deter ..."
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Cited by 6 (2 self)
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In Celebration of the Accomplishments of Béla Csákány We associate to each variety of algebras of finite signature a function on the positive integers called the equational complexity of the variety. This function is a measure of how much of the equational theory of a variety must be tested to determine whether a finite algebra belongs to the variety. We provide general methods for giving upper and lower bounds on the growth of equational complexity functions and provide examples using algebras created from graphs and from finite automata. We also show that finite algebras which are inherently nonfinitely based via the shift automorphism method cannot be used to settle an old problem of Eilenberg and Schützenberger.
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.
Some results on Cvarieties
, 2004
"... In an earlier paper, the second author generalized Eilenberg’s variety theory by establishing a basic correspondence between certain classes of monoid morphisms and families of regular languages. We extend this theory in several directions. First, we prove a version of Reiterman’s theorem concerning ..."
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In an earlier paper, the second author generalized Eilenberg’s variety theory by establishing a basic correspondence between certain classes of monoid morphisms and families of regular languages. We extend this theory in several directions. First, we prove a version of Reiterman’s theorem concerning the definition of varieties by identities, and illustrate this result by describing the identities associated with languages of the form (a1a2 · · · ak) +, where a1,..., ak are distinct letters. Next, we generalize the notions of Mal’cev product, positive varieties, and polynomial closure. Our results not only extend those already known, but permit a unified approach of different cases that previously required separate treatment. Résumé Dans un article antérieur, le second auteur avait proposé une extension de la théorie des variétés d’Eilenberg en établissant une correspondance entre certaines classes de morphismes de monoïdes et certaines classes de langages rationnels. Nous complétons cette théorie dans plusieurs directions. Nous commençons par étendre le théorème de Reiterman relatif à la définition des variétés par identités. Nous illustrons ce résultat en décrivant les identités attachées aux langages de la forme (a1a2 · · · ak) +, où a1,..., ak sont des lettres distinctes. Ensuite, nous généralisons les notions de produit de Mal’cev, de variétés positives et de fermeture polynomiale. Nos résultats permettent non seulement d’étendre les résultats déjà connus, mais proposent également une approche unifiée pour des cas qui nécessitaient jusqu’ici un traitement séparé. 1