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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the relationship of backcrossing algebras with mutation algebras and algebras satisfyingω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identityfthere is a backcrossing algebra satisfyingf. 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. [ABSTRACT FROM AUTHOR]

We define and study the properties of baric algebras defined by ω-polynomial identities, called ω-PI algebras. We show that every finite dimensional baric algebra is ω-PI. Next we introduce the study of ω-PI algebras of degree 4 with one indeterminate. By gametization we reduce their study to four types. We study the first type corresponding to algebras that are neither barycentric nor invariant by gametization. We show that the variety of these algebras is partitioned into only two subvarieties admiting an unique ponderation, an idempotent, and verifying ω-monomial identities of degrees > 4. [ABSTRACT FROM AUTHOR]

In this article we study the algebras satisfying the ω-polynomial identity x2x2 - x4 = δ(x2 - x) with δ ≠ 0 but do not satisfy any monomial identities of degree ≤4. We show that there exist such algebras for all δ ≠ 0 and they have a unique baric function. We give conditions for the existence of idempotents of weight 0 or 1, and we construct the three Peirce decompositions associated to these idempotent elements. [ABSTRACT FROM AUTHOR]

In this article we prove that, if (U, ω) is a finite dimensional baric algebra of (γ, δ) type over a field F of characteristic ≠ 2,3,5 such that γ2 - δ2 + δ = 1 and δ ≠ 0,1, then rad(U) = R(U) ∩ (bar(U))2, where R(U) is the nilradical (maximal nil ideal) of U. [ABSTRACT FROM AUTHOR]