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 $$**N**_{k}
be the variety of commutative alternative
algebras over a field of characteristic $3$ with nilpotency class not
greater than $k$.
Let $$**D**
be the variety $$**N**_{3}**N**_{2}
of nil-algebras of index $3$, i.e.
the commutative alternative algebras with identities

$x3=0,\; [(x$_{1}x_{2})(x_{3}x_{4})](x_{5}x_{6})=0.
In the paper we prove that the varieties $$**N**_{k}**N**_{l}
are Specht varieties.
Moreover, a base of the space of polylinear polynomials in the
free algebra $F($**D**)
is built and the topological rank $r$_{t}(**D**_{n}) = n+2
of varieties

$$**D**_{n} =
**D** Ç
Var((xy × zt)x_{1}¼ x_{n})
is found.
This implies that the topological rank of the variety $$**D** is infinite.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k02/k022/k02202h.htm

Last modified: November 26, 2002