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On the Specht property of varieties of commutative alternative algebras over a field of characteristic \(3\) and commutative Moufang loops. (English. Russian original) Zbl 0972.17020

Sib. Math. J. 41, No. 6, 1027-1041 (2000); translation from Sib. Mat. Zh. 41, No. 6, 1252-1268 (2000).
A variety of algebras is called a Specht variety if each of its subvarieties is finitely based. The problem of whether the variety of solvable alternative algebras of characteristic \(\neq 2,3\) is a Specht variety was solved by U. U. Umirbaev [Algebra Logika 24, No. 2, 226-239 (1985; Zbl 0576.17016)]. Yu. A. Medvedev [Algebra Logika 19, No. 3, 300-313 (1980; Zbl 0468.17005)] exhibited a variety of solvable alternative algebras of characteristic \(2\) without a finite basis of identities. S. V. Pchelintsev has recently constructed an example of an infinite system of identities irreducible in the variety of centrally metabelian (noncommutative) alternative algebras of characteristic \(3\).
Let \(N_k\) and \(A\) be the varieties of nilpotent alternative algebras of index \(\leq k\) and algebras with zero multiplication. By the result of U. U. Umirbaev, the variety \(N_kA\cap N_3N_m\) (of characteristic \(\neq 2,3\)) is a Specht variety. In the article under review, the author proves that the variety \(N_kA\cap N_3N_m\) of commutative alternative algebras of characteristic \(3\) is a Specht variety. The author also constructs infinite independent systems of identities of commutative alternative algebras of characteristic \(3\) (in particular, an example is given of a non-Specht variety of solvable commutative alternative algebras of characteristic \(3\)).
An example of infinite independent systems of identities for a commutative Moufang loop was given by N. I. Sandu [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 1, 171-188 (1987; Zbl 0615.20055)]. In the article under review, the author gives another example of such a system whose construction is simpler.

MSC:

17D05 Alternative rings
17C05 Identities and free Jordan structures
20N05 Loops, quasigroups
08B15 Lattices of varieties
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