FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 3, PAGES 849-871

The structure of weak identities on the Grassman envelopes of central-metabelian alternative superalgebras of superrank $1$ over a field of characteristic $3$

S. V. Pchelintsev

Abstract

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The work is devoted to clarify the structure of weak identities of central-metabelian alternative Grassmann algebras over a field of characteristic $3$. Canonical systems of weak identities $\{f$_n} and $\{g$_n} are constructed:

f_n := [[x₁, x₂], x₃] R(x₄) ... R(x_n-2) [x_n-1, x_n],    n = 4k+2, 4k+3;
g_n := [x₁, x₂]R(x₃) ... R(x_n-2) [x_n-1, x_n],   n = 4k, 4k+3.

It is proved that for any infinitie system of nonzero weak identity there is number $n$₀, since which each of identities of the given system of a degree $n\; >\; n$₀ is equivalent to one of canonical identities $f$_n or $g$_n.

As consequence the variety of alternative algebras with unit over a field of characteristic $3$ which has not final bases of identities is specified.

It is proved also, that the class of weak identities of a rather high degree coinside with the class of mufang functions.

All articles are published in Russian.

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Last modified: December 23, 2001