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Duplication of algebras. III. (English) Zbl 0712.17001

This paper follows two other publications of the same author [cf. Indagationes Math. 44, 121–125 (1982; Zbl 0488.17002); 50, 235–244 (1988; Zbl 0661.17004)]. Here the structure of the duplicate algebra of a given algebra is investigated in a more general way.
The duplicate of a nonassociative algebra is either associative or not associative; an algebra of the first type is called an \(\alpha\)-algebra, an algebra of the second type a \(\beta\)-algebra. The characteristic property is: the duplicate of an algebra is associative if and only if this algebra \(A\) satisfies one of the following conditions: (i) \(A^4=(0)\); (ii) \((A^2)^2=(0)\); (iii) \(A^2\) is a one-dimensional ideal in \(A\).
Theorems about \(\beta\)-algebras are then proved; for instance: Each nonassociative algebra with a left or right regular element is a \(\beta\)-algebra. The author studies \(\beta\)-algebras in which \(A^2\) is associative, and \(\beta\)-algebras in which \(A^2\) is nonassociative.
The last part is devoted to the properties of \(n\)-prod-associative (\(n\)-PA) algebras. The following theorem is proved: If \(A\) is a strictly \(n\)-PA algebra with \(n\ge 5\), the duplicate algebra is \(n\)-associative if \(n\) is odd and \(k\) is even, \((n-1)\)-associative if \(n\) is even and \(k\) is even or \(n\) is odd and \(k\) is odd, \((n-2)\)-associative if \(n\) is even and \(k\) is odd, \((k=[n/2+])\). In each case, several examples are given.

MSC:

17A30 Nonassociative algebras satisfying other identities
17D92 Genetic algebras
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References:

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