Opérations polynomiales et hiérarchies de concaténation

RésuméSoit C une classe de langages. Notons Pol(C) la fermeture polynomiale de C. Pol(C) est la plus petite classe de langages contenant C et fermée par union finie et produit marqué LaL' où a est une lettre. Nous déterminons les clôtures polynomiales de diverses classes de langages rationnels puis nous étudions les propriétés des fermetures polynomiales. Par exemple, si C est fermée par quotients (resp. quotients et morphisme inverse), alors il en est de même de Pol(C). Notre résultat principal montre que si C est une algèbre de Boole fermée par résiduels alors Pol(C) est fermée par intersection. Comme application, nous affinons la hiérarchie de concaténation introduite par Straubing et nous prouvons la décidabilité des niveaux 12 et 32 de cette hiérarchie.AbstractGiven a class C of languages, let Pol(C) be the polynomial closure of C, that is, the smallest class of languages containing C and closed under the operations union and marked product LaL', where a is a letter. We determine the polynomial closure of various classes of rational languages and we study the properties of polynomial closures. For instance, if C is closed under quotients (resp. quotients and inverse morphism) then Pol(C) has the same property. Our main result shows that if C is a boolean algebra closed under quotients then Pol(C) is closed under intersection. As an application, we refine the concatenation hierarchy introduced by Straubing and we show that the levels 12 and 32 of this hierarcy are decidable

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

Trees, congruences and varieties of finite semigroups

AbstractA classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups ♦u(V) is associated to each tree u. In this paper, starting with the congruence γA generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence u (γA) in such a way to generate ♦u(V) for A. We give partial results on the problem of comparing the congruences u (γA) or the pseudovarieties ♦u(V). We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties

Trees, Congruences and Varieties of Finite Semigroups

A classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups &#x25CAu(V) is associated to each tree u. In this paper, starting with the congruence &#x03B3A generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence &#x2261u (&#x03B3A) in such a way to generate &#x25CAu(V) for A. We give partial results on the problem of comparing the congruences &#x2261u (&#x03B3A) or the pseudovarieties &#x25CAu(V). We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties

Familles infinies de boucles incassables

Une boucle incassable est une boucle ne contenant aucune sous-boucle propre. Ce mémoire propose la construction de quatre familles de boucles incassables caractérisées par leur commutativité et par leur groupe de multiplication. La première famille est tout simplement une famille de boucles incassables de tout ordre. La seconde famille est une famille de boucles incassables commutatives d'ordre premier dont le groupe de multiplication est le groupe symétrique. La troisième famille élargit la seconde à tous les ordres impairs. La dernière famille est une famille de boucles incassables commutatives d'ordre impair dont le groupe de multiplication est le groupe alterné