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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]

In this paper, we firstly study a commutative algebra E over a field F of Char(F) ⵐ 2 that satisfying dim(Ꜫ ² ) = 1. We show that, such an algebra is an evolution algebra. Afterwards, we pay attention to commutative duplicate of a commutative algebra E. We find necessary and sufficient condition in which the duplicate D(E) is an evolution algebra. And, we finish by studying an evolution algebra that is a Bernstein algebra. We classify that algebras, up to isomorphism, in dimension ≤ 4. [ABSTRACT FROM AUTHOR]

The Bernstein algebra DnGα of simple Mendelian inheritance is the n-fold duplicate of the gametic algebra Gα with basis A1,..., Aα and structure constants [image omitted]. We simplify and generalize results of Bernad et al. and Peresi by showing that every identity for Gα follows from Costa's shape identity C, and that every identity for the zygotic algebra DGα follows from the recombination identity R. We use computer algebra to determine the identities of degree ≤7 for the copular algebra D2Gα. [ABSTRACT FROM AUTHOR]

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated. [ABSTRACT FROM AUTHOR]

We extend the concept of the "join" to the case of infinitely many weighted algebras. We study the problem of its uniqueness (up to weighted isomorphism) which gives rise to a natural notion of homogeneous weighted algebras. We show that several classes of weighted algebras coming from genetics are homogeneous and that homogeneity is preserved by duplication. Finally, we examine some well-known weighted algebras satisfying identities, as Bernstein, train, and evolution algebras. [ABSTRACT FROM AUTHOR]