Cristián Mallol

Sur les algèbres de Bernstein. II

The extension to Bernstein algebras of higher order of known results for one order is not trivial. In this paper we give some results in this direction.

Les Algèbres de Mutation

The definition of ω M-algebra is introduced and we study a special class : the basic algebras with nilpotent kernels of index 2, named mutation algebras.

Sur la Classification des Nilalgèbres Commutatives de Nilindice 3

ABSTRACT We give some structure results and recursive-like methods for constructions and classifications of commutative nilalgebras of nilindex 3.

Les Pol(n,m)-algèbres: identités polynômiales symétriques dans des algèbres

Resume Symmetric polynomial identities arise in several algebras. In this work we study nonassociative, noncommutative algebras verifying these kinds of identities. The equivalence between these algebras and algebras satisfying particular monomial relations (in which a linear form appears) is proved. We thus present simplified definitions of some classical baric algebras, for instance, Bernsteins algebras.

Train Algèbres Alternatives: Relation Entre Nilindice et Degré

We determine train polynomials for power associative algebras and for alternative train algebras. We show bonds between polynomials and nilindices of some factors of the Peirce decomposition.

Sur Les Train Algèbres De Degré Quatre: Structures Et Classifications

We study train algebras of fourth degree, give some genetic examples and explicit the plenary train identities associated. These algebras fall ultimately into four classes; the first two of them do not have 1/2 as train root and thus have idempotents. For these types we provide structure theorems, which are then used for classification purposes in small dimensions.

Algebres De Clifford Separables II

The aim of this note is to give a characterization of associative algebras with unit that can be written as quotients of Clifford algebras of free quadratic modules. In particular, we are able to say when an associative algebra with unit is the Clifford algebra of a free quadratic module.

Groupoids, idempotents and pointwise inverses in relational categories

Abstract In this paper we investigate idempotents and regular elements of Dedekind categories (these appear to be a proper axiomatic framework for establishing certain aspects of binary relational categories). Classical techniques (using preorders and Greens equivalences) are replaced by techniques which use categories with adapted objects (idempotents), unities and universes. We extensively investigate the links between the groupoid of invertible elements in an idempotent category and the ‘classical’ isomorphisms. We also investigate relational algebras in which every morphism is regular. Part of this work is devoted to morphisms between total orders. In this context we discuss Guttman-Ferrers morphisms, and we shall prove that they are regulars.

Critère d’existence d’idempotent basé sur les algèbres de Rétrocroisement

ABSTRACT We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.

Sur les identités polynomiales vérifiées par les algèbres de rétrocroisement

ABSTRACT We study the ideal of polynomial identities of a single indeterminate satisfied by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and sufficient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities satified by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8.