FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 2, PAGES 335-356

**The variety $$****N**_{3}**N**_{2} of
commutative alternative nil-algebras of
index $3$ over a field of
characteristic $3$

A. V. Badeev

Abstract

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```
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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k02/k022/k02202t.htm.

Last modified: November 26, 2002