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Discussiones Mathematicae General Algebra and Applications 24(1) (2004) 5-30
DOI: https://doi.org/10.7151/dmgaa.1072

INFINITE INDEPENDENT SYSTEMS OF IDENTITIES OF ALTERNATIVE COMMUTATIVE ALGEBRA OVER A FIELD OF CHARACTERISTIC THREE

Nicolae Ion Sandu

Tiraspol State University
The author's home address:
Deleanu str 1, Apartment 60
Kishinev MD-2071, Moldova
e-mail: sandumn@yahoo.com

Abstract

Let \frak A₃ denote the variety of alternative commutative (Jordan) algebras defined by the identity x³ = 0, and let \frak S₂ be the subvariety of the variety \frak A₃ of solvable algebras of solviability index 2. We present an infinite independent system of identities in the variety \frak A₃ Ç\frak S₂\frak S₂. Therefore we infer that \frak A₃Ç\frak S₂\frak S₂ contains a continuum of infinite based subvarieties and that there exist algebras with an unsolvable words problem in \frak A₃ Ç\frak S₂ \frak S₂.

It is worth mentioning that these results were announced in 1999 in works of the international conference ``Loops'99'' (Prague).

Keywords: infinite independent system of identities, alternative commutative algebra, solvable algebra, commutative Moufang loop.

2000 Mathematics Subject Classification: 17D05, 20N05.

Bibliografie

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Received 27 August 2001
Revised 8 December 2002
Revised 16 June 2004
Revised 14 July 2004