On a class of baric algebras
In this work, the authors prove a strict relation of inclusion between two classes of baric algebras, which are among the most usual in genetic algebra. Let \(K\) be an infinite field of characteristic not \(2\). A commutative \(K\)-algebra \(A\) endowed with a nonzero algebra homomorphism \(w:A\to K\) is said to be a \textsl{baric} algebra. In this paper, the baric algebras satisfying \((x^2)^2=w(x)x^3\) are studied, where \(x^1=x\), ..., \(x^{k+1}=x^kx\), \(k\geq 1\). It is proved (theorem 2.2) that such algebras are Jordan algebras, and also are Bernstein algebras of order two, that is, the identity \(x^8=w(x^4)x^4\) holds. Hence, it follows from the fact that \(A\) is a power-associative algebra that \(x^3x=(x^2)^2\) and therefore every baric algebra satisfying the identity \((x^2)^2=w(x)x^3\) also satisfies \(x^4=w(x)x^3\) (corollary 2.4). However, two counterexamples show that the converse is not true.
scientific article; zbMATH DE number 931919