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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k01/k013/k01316t.htm.

Last modified: December 23, 2001