Closure of varieties of languages under products with counter

AbstractWe characterize the varieties of rational languages closed under products with counter. They are exactly the varieties that correspond via Eilenberg's theorem to the varieties of monoids closed under inverse LGsol-relational morphisms. This yields some decidability results for certain classes of rational languages

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

We present an extension of Eilenberg's variety theorem, a well-known result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids.The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat Politècnica de València

Languages associated with saturated formations of groups

International audienceIn a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.Dans un article précédent, les auteurs avaient montré comment étendre le théorème des variétés d'Eilenberg à des structures plus générales, les formations. Dans cet article, nous donnons une méthode générale pour décrire les langages correspondant à des formations saturées de groupe, qui sont beaucoup étudiées en théorie des groupes. Nous retrouvons de cette façon les résultats connus sur les langages correspondant aux classes des groupes nilpotents, résolubles, et superrésolubles. Notre méthode s'applique aussi à de nouveaux exemples, telles que la classe des groupes ayant une tour de Sylow

Languages associated with saturated formations of groups

In a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The first author acknowledges support from MEC. The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat Politècnica de València

Two algebraic approaches to variants of the concatenation product

AbstractWe extend an existing approach of the bideterministic concatenation product of languages aiming at the study of three other variants: unambiguous, left deterministic and right deterministic. Such an approach is based on monoid expansions. The proofs are purely algebraic and use another approach, based on properties on the kernel category of a monoid relational morphism, without going through the languages. This gives a unified fashion to deal with all these variants and allows us to better understand the connections between these two approaches. Finally, we show that local finiteness of an M-variety is transferred to the M-varieties corresponding to these variants and apply the general results to the M-variety of idempotent and commutative monoids

Syntactic semigroups

This chapter gives an overview on what is often called the algebraic theory of finite automata. It deals with languages, automata and semigroups, and has connections with model theory in logic, boolean circuits, symbolic dynamics and topology

Pseudovariedades de grupos e variedades de linguagens associadas

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2008O principal objectivo deste trabalho consiste em dar uma descrição das linguagens reconhecidas pelos grupos super-resolúveis finitos. Essa descrição será feita de dois modos distintos: através de produtos modulares concatenados, mostrando que uma tal linguagem pertence à álgebra de Boole gerada por produtos modulares concatenados de linguagens comutativas elementares e, através de transdutores, provando que essas linguagens são combinações Booleanas de linguagens da forma rτ−1, em que p é um número primo, r ∈ Zp e τ : A∗ → Zp é uma função realizada por algum transdutor na forma triangular estrita.Com vista a esse estudo, faremos uma análise detalhada da pseudovariedade dos grupos super-resolúveis e também de outras pseudovariedades de grupos, em particular, das pseudovariedades dos p-grupos e dos grupos abelianos cujo expoente divide um dado natural n. Caracterizaremos também o produto de pseudo variedades e daremos especial atenção à pseudovariedade Gp ∗ Abp−1. Estudaremos as variedades de linguagens associadas às pseudovariedades de grupos consideradas e iremos demonstrar o Princípio do Produto em Coroa de Straubing, o qual nos fornece uma descrição das linguagens reconhecidas pelo produto em coroa de dois monóides. Além disso, apresentaremos uma versão deste princípio para variedades de linguagens. Será ainda considerado o produto de linguagens com contador e descrita a operação entre monóides que lhe está associada.The main subject of this work is to give a description of the languages recognized by finite super-soluble groups. That description will be done in two distinct ways. The first one uses the modular concatenation product, more precisely, we will prove that such a language is in the Boolean algebra generated by the concatenated modular products of elementary commutative languages. In the second one we prove that the languages recognized by super-soluble groups are Boolean combinations of languages that take the form of rτ−1, where p is a prime number, r ∈ Zp and τ : A∗ → Zp is a function realized by some transductor in the strict triangular form. In view of that study, we will analyse in detail the pseudovarieties of super-soluble groups as well as other pseudovarieties of groups, in particular we will consider the pseudovariety of p-groups and the pseudovariety of abelian groups whose exponent divides a given natural n. We will also characterize the product of pseudovarieties, dedicating particular attention to the pseudovariety Gp ∗ Abp−1.We will study the varieties of languages associated with the pseudovarieties of groups considered and will prove the Straubing's Wreath Product Principle, which gives us a description of the languages recognized by the wreath product of two monoids. In addition, we will present a version of this principle applied to varieties of languages. The product of languages with counter will also be considered and the associated operation between monoids will be described

Products of languages with counter

AbstractIt is well known that varieties of rational languages are in one-to-one correspondence with varieties of finite monoids. This correspondence often extends to operations on languages and on monoids. We investigate the special case of the product of languages with counter, and describe the associated operations on monoids and varieties