Equational description of pseudovarieties of homomorphisms

Kunc, Michal

doi:10.1051/ita:2003018

Cited by 28 publications

(21 citation statements)

References 4 publications

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“…If QA is considered as an lm-variety, this result can be improved as follows (see [5]): Proposition 6.3 The lm-variety QA is defined by the single identity…”

Section: Identities Of Mal'cev Productsmentioning

confidence: 99%

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Some results onC-varieties

Straubing

2005

RAIRO-Theor. Inf. Appl.

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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 · · · a k ) + , where a1, . . . , a k 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. RésuméDans un article antérieur, le second auteur avait proposé une extension de la théorie des variétés d'Eilenberg enétablissant une correspondance entre certaines classes de morphismes de monoïdes et certaines classes de langages rationnels. Nous complétons cette théorie dans plusieurs directions. Nous commençons parétendre le théorème de Reiterman relatif a la définition des variétés par identités. Nous illustrons ce résultat en décrivant les identités attachées aux langages de la forme (a1a2 · · · a k ) + , où a1, . . . , a k 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 proposentégalement une approche unifiée pour des cas qui nécessitaient jusqu'ici un traitement séparé.

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“…If QA is considered as an lm-variety, this result can be improved as follows (see [5]): Proposition 6.3 The lm-variety QA is defined by the single identity…”

Section: Identities Of Mal'cev Productsmentioning

confidence: 99%

“…Independently of us, Kunc [5] also developed the equational theory for C-varieties. Kunc worked with the original definition of these varieties, but as a drawback, his identities must be interpreted in a non-standard manner.…”

Section: Introductionmentioning

confidence: 99%

Some results onC-varieties

Straubing

2005

RAIRO-Theor. Inf. Appl.

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“…Then one can define an appropriate notion of variety of stamps, which corresponds bijectively to C-varieties [39]. Reiterman's theorem can also be extended to this setting [19,28]. First observe that every morphism from A * to B * is uniformly continuous for the profinite metric and thus extends uniquely to a continuous morphism f : A * → B * .…”

Section: C-varietiesmentioning

confidence: 99%

“…Recall that AC 0 is the set of unbounded fan-in, polynomial size, constant-depth Boolean circuits. Straubing [4,39] has shown that the regular languages recognised by a circuit in AC 0 form a m-variety defined by the m-identity (x ω−1 y) ω = (x ω−1 y) ω+1 [19,28]. Both examples have a nice interpretation in logic, but we need to enrich our vocabulary by introducing some new predicate symbols, called the modular predicate symbols.…”

Section: C-varietiesmentioning

confidence: 99%

Equational Descriptions of Languages

2012

Int. J. Found. Comput. Sci.

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This paper is a survey on the equational descriptions of languages. The first part is devoted to Birkhoff's and Reiterman's theorems on equational descriptions of varieties. Eilenberg's variety theorem and its successive generalizations form the second part. The more recent results on equational descriptions of lattices of languages are presented in the third part of the paper.Equations have been used for a long time in mathematics to provide a concise description of various mathematical objects. This article roughly follows a historical approach to present such equational descriptions for formal languages, ranging over a period of 45 years: from Schützenberger's characterization of star-free languages [36] to the following recent result of [18]: Every lattice of languages admits an equational description.This evolution was made possible by a gradual abstraction of the notion of equation. The story really starts in 1935 with Birkhoff's theorem on equational classes [6]. It holds for any kind of universal algebra, but I will present it only for monoids.

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“…The definition of identities can be extended to profinite identities to obtain a generalisation of Reiterman's theorem to C-varieties 14,22 . It follows from the previous results that the star-height problem amounts to showing that the lp-varieties of stamps corresponding to the languages of star-height n are decidable.…”

Section: Back To the Star-height Problemmentioning

confidence: 99%

Open Problems on Regular Languages: A Historical Perspective

Chaubard

2007

Semigroups and Formal Languages

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Equational description of pseudovarieties of homomorphisms

Cited by 28 publications

References 4 publications

Some results onC-varieties

Some results onC-varieties

Equational Descriptions of Languages

Open Problems on Regular Languages: A Historical Perspective

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