On the Specht Property of Varieties of Commutative Alternative Algebras over a Field of Characteristic 3 and Commutative Moufang Loops

Abstract

A variety of algebras is a Specht variety if each of its subvarieties is finitely based. The problem of whether the variety of soluble alternative algebras is a Specht variety was formulated by A. M. Slin′ko in “Dnestrovskaya Tetrad′” [1, Problem 129]. This problem was solved in the affirmative in the case of a field of characteristic other than 2 and 3. As regards index 2 soluble algebras, this follows from the results of Yu. A. Medvedev’s article [2]. Furthermore, in [3] Yu. A. Medvedev exhibited a variety of soluble alternative algebras over a field of characteristic 2 without a finite basis of identities. Next, let Nk denote the variety of alternative algebras of nilpotency class at most k and A, the variety of algebras with zero multiplication. S. V. Pchelintsev [4] proved that every soluble alternative algebra A over a field of characteristic other than 2 and 3 belongs to the variety NkA ∩N3Nm; i.e., (A) = (A) = 0