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: \begin{align*} f_n &\prisv [[x_1, x_2], x_3]R(x_4) \ldots R(x_{n-2})\cdot [x_{n-1}, x_n], \quad n = 4k+2, 4k+3;\\ g_n &\prisv [x_1, x_2]R(x_3) \ldots R(x_{n-2})\cdot [x_{n-1}, x_n], \quad n = 4k, 4k+3. \end{align*} 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. 

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Last modified: December 23, 2001