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Totally orthogonal Bernstein algebras. (English) Zbl 0724.17026
Let K be a infinite field of characteristic not 2. A finite dimensional, commutative, nonassociative k-algebra A, with a nonzero homomorphism w: \(A\to K\) is called a Bernstein algebra if \((x^ 2)^ 2=w^ 2(x)x^ 2\) for all \(x\in A\). Such algebras always have an idempotent e such that \(w(e)=1\). In this case there is a decomposition of A, say \(A=Ke\oplus U\oplus V\), where \(U=\{x\in Ker w\); \(ex= x\}\) and \(V=\{x\in Ker W\); \(ex=0\}.\)
In this note we show by an example that the concept of orthogonal Bernstein algebra depends on the choice of the idempotent in the Peirce decomposition. We introduce the concept of total orthogonality and give necessary and sufficient conditions for this condition. We prove also that for Jordan-Bernstein algebras, dim \(U^ 3\) is an invariant.

17D92 Genetic algebras
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