DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

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Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

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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

[1] R.H. Bruck, A survey of binary systems, Springer-Verlag, Berlin 1958.
[2] O. Chein, H.O. Pflugfelder and J.D.H. Smith, (eds.) Quasigroups and Loops: Theory and Applications, Heldermann Verlag, Berlin 1990.
[3] V.T. Filippov, n-Lie algebras (Russian), Sibirsk. Mat. Zh. 26 (1985), no. 6, 126-140.
[4] S. Lang, Algebra, Addison-Wesley Publ. Co., Reading, MA, 1965.
[5] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, (second revised edition), Dover Publ., New York 1976.
[6] Yu.A. Medvedev, Finite basis property of varieties with binomial identities (Russian), Algebra i Logika 17 (1978), 705-726.
[7] Yu.A. Medvedev, Example of a variety of alternative at algebras over a field of characteristic two, that does not have a finite basis of identities (Russian), Algebra i Logika 19 (1980), 300-313.
[8] The Dniester Notebook: Unsolved problems in the theory of rings and modules (Russian), Third edition; Akad. Nauk SSSR Sibirsk Otdel., Inst. Mat., Novosibirsk 1982.
[9] A. Thedy, Right alternative rings, J. Algebra 37 (1975), 1-43.
[10] N.I. Sandu, Centrally nilpotent commutative Moufang loops (Russian), Mat. Issled. No. 51 (1979), (Quasigroups and loops), 145-155.
[11] N.I. Sandu, Infinite irreducible systems of identities of commutative Moufang loops and of distributive Steiner quasigroups (Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 51 (1987), 171-188.
[12] N.I. Sandu, On the Bruck-Slaby theorem for commutative Moufang loops (Russian), Mat. Zametki 66 (1999), 275-281; Eglish transl.: Math. Notes 66 (1999), 217-222.
[13] N.I. Sandu, About the embedding of Moufang loops into alternative algebras, to appear.
[14] U.U. Umirbaev, The Specht property of a variety of solvable alternative algebras (Russian), Algebra i Logika 24 (1985), 226-239.
[15] K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, and A.I. Shirshov, Rings that are nearly associative, Academic Press, New York 1982.

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


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