The bar-radical in power-associative \(n^{\text{th}}\)-order Bernstein algebras (Q699737)

The \(n\)-th order Bernstein algebras \((A,\omega)\) are among the few nonassociative algebras with applications in science: they arise in genetics in the study of populations that reach equilibrium after \(n\) generations. Such an algebra \(A\) is defined over a field \(K\) of characteristic \(\neq 2\). If \(\omega: A\to K\) is a non-zero homomorphism, then \((A,\omega)\) is also called a baric algebra over \(K\), where \(\omega\) is called its weight function. In this case, it is denoted by \(\text{bar}(A)\). There is a class of \((A,\omega)\) algebras that are power-associative, that is, they satisfy the condition that \(x^ix^j = i^{i+j}\) for every \(i,j\geq 1\) and \(x\in A\). A commutative \(n\)-th order Bernstein algebra is also called a Jordan algebra if it satisfies the identity: \((x^2y)x = x^2(xy)\) for all \(x,y\in A\). The author proves that if \((A,\omega)\) is power-associative, then the bar-radical of \((A,\omega)\) is contained in \((\text{bar}(A))^2\). And if \((A,\omega)\) is Jordan, then the bar radical is \((\text{bar}(A))^2\).