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Peirce-evanescent baric identities are polynomial identities verified by baric algebras such that their Peirce polynomials are the null polynomial. In this paper procedures for constructing such homogeneous and non homogeneous identities are given. For this we define an algebraic system structure on the free commutative nonassociative algebra generated by a set T which provides for classes of baric algebras satisfying a given set of identities similar properties to those of the varieties of algebras. Rooted binary trees with labeled leaves are used to explain the Peirce polynomials. It is shown that the mutation algebras satisfy all Peirce-evanescent identities, it results from this that any part of the field K can be the Peirce spectrum of a K-algebra satisfying a Peirce-evanescent identity. We end by giving methods to obtain generators of homogeneous and non-homogeneous Peirce-evanescent identities that are applied in several univariate and multivariate cases.

International audience; We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras.

We generalize the gametization process introduced in [5] . For this we use not necessarily convex linear combinations of a baric algebra ( A , ω ) with the gametic algebra defined by the weight ω , we call these combinations homogametizations. After establishing a ponderability condition for a homogametization, we introduce the homogametization operators group and we define two actions of this group on the class of baric algebras and the non-commutative and non-associative algebra K 〈 X 〉 of polynomials with variables in X = { x 1 , … , x n } . When an algebra is defined by an identity, we show that this property is preserved after the group action on the algebra and the identity. We give explicit constructions of elements in K 〈 X 〉 which are invariants or universal invariants for this group action.

International audience; 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.

We continue the study of the algebras satisfying an identity of the shape aX2X2 + bX4 − λX3 − μX2 − νX, we introduce here the case a + b = 1, ab ≠ 0, λ = 2, and μ + ν = −1. By studying a 26 × 27 linear system, we show that this class admits a partition into six parts among which four correspond to algebras not verifying other identity of degree ≤4. In this article we study these four algebras. We show they satisfy principal and plenary train identities of rank > 4, we determine the algebras admitting an idempotent and we give their Peirce decompositions in the two cases: with and without idempotent.

The gametization process reduces the study of non-commutative and non-associative algebras satisfying non-homogeneous polynomial identities with variables in X = { x 1 , … , x n } to algebras verifying simpler identities. However after a gametization, certain identities remain invariant and other identities, said universal invariant, are invariant for every gametization. Now in the case n = 1 , for all algebras satisfying a universal invariant polynomial identity studied until now, we know that the existence of an idempotent is not certain. Using an action of the gametization operators group on the non-commutative and non-associative algebra of polynomials K 〈 X 〉 , we give all identities which are invariant and universal invariant by gametization.

International audience; 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.

International audience; We give the identities generating the spaces of degree 4 multilinear identities satisfied by the non commutative 0th-order Bernstein algebras and the non commutative mutation algebras. For it, we use a method reducing the search for multilinear identities to the resolution of linear systems.

The existence of idempotent elements in baric algebras defined by $\omega$-polynomial identities ($\omega$-PI algebras) is an important problem for the study of genetic algebras. We conjecture here two criteria on the existence of an idempotent. These criteria are based on the existence of 1/2 as double root of a polynomial built from the identity defining a $\omega$-PI algebra. We show that these criteria are true in all the algebras studied until now and for which we have results concerning the existence of idempotent elements.

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.

International audience; In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body's ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from R n , n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.

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.

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

Resume Nous etudions l'algebre Der( A ) d'une algebre de Bernstein a l'aide des transformations issues des decompositions de Peirce. Nous etudions la structure des derivations quand les morphismes de Peirce sont des automorphismes. Nous construisons Aut( A ) et Der( A ) pour quelques types de ces algebres, en particulier dans la dimension 4.

We show that if A2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. Also, if A2 is a Bernstein algebra of order n and period p, the commutative duplicate of A is Bernstein of order n + 1 and period p.

International audience

Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'algèbres à idempotents, commutatives, non associatives, définies par des relations. On y trouve une structure qui n'est en général ni pondérée ni de Jordan et où, cependant, ces conditions sont équivalentes.Following a previous paper, we study here a class of commutative and non associative algebras with idempotents and defined by relations. We find a structure which is not, in general, baric or Jordan but where these notions are equivalent.

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.

Resume On introduit la notion de gametisation d'une algebre ponderee comme un outil simple qui facilite l'etude de structures qui verifient des identites ω-polynomiales. Concernant les train-identites d'ordre quelconque, on etablit l'existence des identites invariantes par gametisation et leur forme y est precisee. Des resultats de classification y sont donnes pour les algebres d'ordre plus petit ou egal a 4.

Resume Nous etudions le dictionnaire de passage entre deux decompositions de Peirce de ce type d'algebre et les morphismes ad hoc qui entrent en jeu.