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} :=
[[x_{1}, x_{2}], x_{3}]
R(x_{4}) ...
R(x_{n-2})
[x_{n-1}, x_{n}],
n = 4k+2, 4k+3;

g_{n} := [x_{1}, x_{2}]R(x_{3}) ...
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.

Location: http://mech.math.msu.su/~fpm/eng/k01/k013/k01316h.htm

Last modified: December 23, 2001