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12
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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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.
Möbius functions and semigroup representation theory. II. Character formulas and multiplicities
 Adv. Math
"... Abstract. We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota’s theory of Möbius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idemp ..."
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Cited by 16 (5 self)
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Abstract. We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota’s theory of Möbius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigaire, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing
A topological approach to inverse and regular semigroups
 Pacific J. Math
, 2000
"... Work of Ehresmann and Schein shows that an inverse semigroup can be viewed as a groupoid with an order structure; this approach was generalized by Nambooripad to apply to arbitrary regular semigroups. This paper introduces the notion of an ordered 2complex and shows how to represent any ordered gro ..."
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Cited by 9 (3 self)
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Work of Ehresmann and Schein shows that an inverse semigroup can be viewed as a groupoid with an order structure; this approach was generalized by Nambooripad to apply to arbitrary regular semigroups. This paper introduces the notion of an ordered 2complex and shows how to represent any ordered groupoid as the fundamental groupoid of an ordered 2complex. This approach then allows us to construct a standard 2complex for an inverse semigroup presentation. Our primary applications are to calculating the maximal subgroups of an inverse semigroup which, under our topological approach, turn out to be the fundamental groups of the various connected components of the standard 2complex. Our main results generalize results of Haatja, Margolis, and Meakin giving a graph of groups decomposition for the maximal subgroups of certain regular semigroup amalgams. We
The spectra of lamplighter groups and Cayley machines, Geom. Dedicata
"... Abstract. We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr Z, with G a finite group by calculating the Kestenvon NeumannSerre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generali ..."
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Cited by 5 (3 self)
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Abstract. We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr Z, with G a finite group by calculating the Kestenvon NeumannSerre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and ˙ Zuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by automata coincides with the Kestenvon NeumannSerre spectral measure. 1.
Volkov, Modular and threshold subword counting and matrix representations of finite monoids
 in “Words 2005, 5 th International Conference on Words, 13–17 September 2005, Acts”, eds. S. Brlek, C. Reutenauer, Publications du Laboratoire de Combinatoire et d’ Informatique Mathématique, UQAM 36
, 2005
"... Recall that a word u over a finite alphabet Σ is said to be a subword of a word v ∈ Σ ∗ if, for some n ≥ 1, there exist words u1,..., un, v0, v1,..., vn ∈ Σ ∗ such that u = u1u2 · · · un and v = v0u1v1u2v2 · · · unvn. (1.1) The subword relation reveals interesting combinatorial properties and pl ..."
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Cited by 2 (2 self)
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Recall that a word u over a finite alphabet Σ is said to be a subword of a word v ∈ Σ ∗ if, for some n ≥ 1, there exist words u1,..., un, v0, v1,..., vn ∈ Σ ∗ such that u = u1u2 · · · un and v = v0u1v1u2v2 · · · unvn. (1.1) The subword relation reveals interesting combinatorial properties and plays a prominent role in formal language theory. For instance, recall that languages consisting of all words over Σ having a given word u ∈ Σ ∗ as a subword serve as a generating system for the Boolean algebra of socalled piecewise testable languages. It was a deep study of combinatorics of the subword relation that led Simon [20,21] to his elegant algebraic characterization of piecewise testable languages. Further, the natural idea to put certain rational constraints on the factors v0, v1,..., vn that may appear in a decomposition of the form (1.1) gave rise to the useful notion of a marked product of languages studied from the algebraic viewpoint by Schützenberger [18], Reutenauer [10], Straubing [23], Simon [22], amongst others. Yet another natural idea is to count how many times a word v ∈ Σ ∗ contains a given word u as a subword, that is, to count different decompositions of the form (1.1). Clearly, if one wants to stay within the realm of rational languages, one can only count up to a certain threshold and/or modulo a certain number. For instance, one may consider Boolean combinations of languages consisting of all words over Σ having t modulo p occurrences of a given word u ∈ Σ ∗ (where p is a given prime number). This class of languages also admits a nice algebraic characterization, see [5, Sections VIII.9 and VIII.10] and also [25]. Combining modular counting with rational constraints led to the idea of marked products with modular counters explored, in particular, by Weil [27] and Peladeau [7]. The most natural version of threshold counting is formalized via the notion of an unambiguous marked product in which one considers words v ∈ Σ ∗
Characterization of group radicals with an application to Mal’cev products, in preparation
"... Abstract. Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups. 1. ..."
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Abstract. Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups. 1.
AUTOMATA OVER A BINARY ALPHABET GENERATING FREE GROUPS OF EVEN RANK
, 2006
"... Abstract. We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free produc ..."
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Abstract. We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata. 1.
A MODERN APPROACH TO SOME RESULTS OF STIFFLER
"... ABSTRACT. We give a modern proof of Stiffler’s classical results describing the pseudovarieties of Rtrivial semigroups and locally Rtrivial semigroups as the wreath product closures of semilattices, respectively semilattices and right zero semigroups. Our proof uses the derived category of a funct ..."
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Cited by 1 (1 self)
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ABSTRACT. We give a modern proof of Stiffler’s classical results describing the pseudovarieties of Rtrivial semigroups and locally Rtrivial semigroups as the wreath product closures of semilattices, respectively semilattices and right zero semigroups. Our proof uses the derived category of a functor developed by the author with B. Tilson. We prove a more general result describing functors between finite categories which are injective on coterminal Requivalent elements. 1.
POINTLIKE SETS WITH RESPECT TO R AND J
"... Abstract. We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite Rtrivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperi ..."
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Abstract. We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite Rtrivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute Jpointlike sets, where J denotes the pseudovariety of all finite Jtrivial semigroups. We finally show that, in contrast with the situation for R, the natural adaptation of Henckell’s algorithm to J computes pointlike sets, but not all of them. 1.