Geodesic
The variety $\mathbf N_3\mathbf N_2$ of commutative alternative nil-algebras of index 3 over a field of characteristic $3$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 335-356 
A. V. Badeev. The variety $\mathbf N_3\mathbf N_2$ of commutative alternative nil-algebras of index 3 over a field of characteristic $3$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 335-356. http://geodesic.mathdoc.fr/item/FPM_2002_8_2_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A variety is called a Specht variety if every algebra in this variety has a finite basis of identities. In 1981 S. V. Pchelintsev defined the topological rank of a Specht variety. Let $\mathbf N_k$ be the variety of commutative alternative algebras over a field of characteristic 3 with nilpotency class not greater than $k$. Let $\mathbf D$ be the variety $\mathbf N_3\mathbf N_2$ of nil-algebras of index 3, i.e. the commutative alternative algebras with identities $$ x^3=0,\quad [(x_1x_2)(x_3x_4)](x_5x_6)=0. $$ In the paper we prove that the varieties $\mathbf N_k\mathbf N_l$ are Specht varieties. Moreover, a base of the space of polylinear polynomials in the free algebra $F(\mathbf D)$ is built and the topological rank $\mathrm r_{\mathrm t}(\mathbf D_n)=n+2$ of varieties $$ \mathbf D_n=\mathbf D\cap\mathrm{Var}((xy\cdot zt)x_1\ldots x_n) $$ is found. This implies that the topological rank of the variety $\mathbf D$ is infinite.

[1] Zhevlakov K. A., Slinko A. M., Shestakov I. P., Shirshov A. I., Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR | Zbl

[2] Dnestrovskaya tetrad, Novosibirsk, 1982

[3] Medvedev Yu. A., “Konechnaya baziruemost mnogoobrazii s dvuchlennym tozhdestvom”, Algebra i logika, 17:6 (1978), 705–726 | MR

[4] Medvedev Yu. A., “Primer mnogoobraziya razreshimykh alternativnykh algebr nad polem kharakteristiki 2, ne imeyuschego konechnogo bazisa tozhdestv”, Algebra i logika, 19:3 (1980), 300–313 | MR | Zbl

[5] Pchelintsev S. V., “Razreshimye indeksa 2 mnogoobraziya algebr”, Mat. sbornik, 115 (1981), 179–203 | MR | Zbl

[6] Iltyakov A. V., “Reshetka podmnogoobrazii mnogoobraziya dvustupenchato razreshimykh alternativnykh algebr”, Algebra i logika, 21:2 (1982), 170–177 | MR

[7] Umirbaev U. U., “Shpekhtovost mnogoobraziya razreshimykh alternativnykh algebr”, Algebra i logika, 24:2 (1985), 226–239 | MR | Zbl

[8] Shestakov I. P., “Superalgebry i kontrprimery”, Sib. mat. zhurn., 32:6 (1991) | MR

[9] Drensky V. S., Rashkova T. G., “Varieties of metabelian Jordan algebras”, Serdica Bulgarical mathematical publications, 15 (1989), 293–301 | MR | Zbl