FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

The variety $$N₃N₂ 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 $$N₃N₂ of nil-algebras of index $3$, i.e. the commutative alternative algebras with identities

In the paper we prove that the varieties $$N_kN_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

is found. This implies that the topological rank of the variety $$D is infinite.

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Last modified: November 26, 2002