FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2001, VOLUME 7, NUMBER 3, PAGES 849-871
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.
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Last modified: December 23, 2001