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

Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 3, pp. 849-871

Résumé

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 =[[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 =[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.