Representation theory of finite semigroups, semigroup radicals and formal language theory

Abstract. Work of Clifford, Munn and Ponizovskiȋ parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskiȋ result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.

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“…[2,3]. This paper, like [25,26,19,13,1], aims to reconcile semigroup representation theory with representation theory at large.…”

Section: Introductionmentioning

confidence: 99%

On the irreducible representations of a finite semigroup

Ganyushkin

Mazorchuk

Steinberg

2009

Proc. Amer. Math. Soc.

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“…(1) ⊓ ⊔ k S (2) and hence S ∈ W. Now we show that if (S, A) is a finitary pg-pair with S ∈ W, then (S, A) ∈ V. The proof is by induction on the construction of S from elements of W by means of block products. If S ∈ W, then (S, A) ∈ pgp(W) = V by definition.…”

Section: ⊓ ⊔mentioning

confidence: 93%

Algebraic Characterization of Logically Defined Tree Languages

Ésik

Weil

2010

Int. J. Algebra Comput.

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We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindström quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable.

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“…. k, are semigroups, whence this last theorem implies Theorem 7.3 of [1], but it does not yield the linear bound (n − 1) provided in Theorem 2.6 of [3]. Furthermore, we do not know if Corollary 18 is more general, since it is not known whether the fact that I i \ {0}, for i = 1, .…”

Section: Lemma 14mentioning

confidence: 94%

“…For more information on synchronizing automata we refer the reader to Volkov's survey [21]. In this paper we follow a representation theoretic approach to theČerný conjecture and synchronizing automata initially pursued in [1,3,5,19]. However, our approach has a more ring theoretic flavor making use of the well-known Wedderburn-Artin theory for semisimple rings.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we show how the Wedderburn-Artin Theory may be used to factorize the problem of finding "short" radical words into the sub-problem of finding "short" words that are zero in the projections into the simple components. This approach works for instance in case the 0-minimal ideals in the factor monoids do not contain zero H-classes similarly to [1,3]. In Section 5 we introduce cyclic ideal languages and we characterize them.…”

Section: Introductionmentioning

confidence: 99%

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Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Almeida¹,

Rodaro²

2014

Developments in Language Theory

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Abstract. We present a ring theoretic approach toČerný's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those inČerný's series. Semisimplicity gives also the advantage of "factorizing" the problem of finding a synchronizing word into the sub-problems of finding "short" words that are zeros into the projection of the simple components in the WedderburnArtin decomposition. In the general case this last problem is related to the search of radical words of length at most (n − 1) 2 where n is the number of states of the automaton. We show that the solution of this "Radical Conjecture" would give an upper bound 2(n−1) 2 for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases andČerný's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

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Representation theory of finite semigroups, semigroup radicals and formal language theory

Cited by 67 publications

References 65 publications

On the irreducible representations of a finite semigroup

On the irreducible representations of a finite semigroup

Algebraic Characterization of Logically Defined Tree Languages

Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

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