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13
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.
Tameness of pseudovariety joins involving R
, 2004
"... In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all Jtrivial (resp. Rtrivial) semigroups. In particular, we show that the pseudovariety V ∨ ..."
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Cited by 7 (6 self)
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In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all Jtrivial (resp. Rtrivial) semigroups. In particular, we show that the pseudovariety V ∨ W is (completely) κtame when V is a subpseudovariety of J and W is (completely) κtame. Moreover, if W is a κtame pseudovariety which satisfies the pseudoidentity x1 · · · xry ω+1 zt ω = x1 · · · xryzt ω, then we prove that R ∨ W is also κtame. In particular the joins R ∨ Ab, R ∨ G, R ∨ OCR, and R ∨ CR are decidable.
Complete reducibility of systems of equations with respect to
"... 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 ..."
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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.
Complete reducibility of the pseudovariety LSl
 Int. J. Algebra Comput
"... In this paper we prove that the pseudovariety LSl of local semilattices is completely κreducible, where κ is the implicit signature consisting of the multiplication and the ωpower. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudo ..."
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In this paper we prove that the pseudovariety LSl of local semilattices is completely κreducible, where κ is the implicit signature consisting of the multiplication and the ωpower. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudowords of the system over LSl implies the existence of a solution by κwords of the system over LSl satisfying the same constraints.
Characterization of group radicals with an application to Mal’cev products, in preparation
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COMPLETE REDUCIBILITY OF PSEUDOVARIETIES
"... Abstract. The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility ..."
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Abstract. The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility for specific pseudovarieties. 1.
A counterexample to a conjecture concerning concatenation hierarchies
"... Abstract. We give a counterexample to the conjecture which was originally formulated by Straubing in 1986 concerning a certain algebraic characterization of regular languages of level 2 in the StraubingThérien concatenation hierarchy of starfree languages. 1. ..."
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Abstract. We give a counterexample to the conjecture which was originally formulated by Straubing in 1986 concerning a certain algebraic characterization of regular languages of level 2 in the StraubingThérien concatenation hierarchy of starfree languages. 1.
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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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
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.
SEMIDIRECT PRODUCT WITH AN ORDERCOMPUTABLE PSEUDOVARIETY AND TAMENESS
"... Abstract. The semidirect product of pseudovarieties of semigroups with an ordercomputable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning t ..."
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Abstract. The semidirect product of pseudovarieties of semigroups with an ordercomputable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning the behavior of the operation with respect to various kinds of tameness properties are obtained as applications.