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2004 | 24 | 1 | 5-30
Tytuł artykułu

Infinite independent systems of identities of alternative commutative algebra over a field of characteristic three

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let 𝔄₃ denote the variety of alternative commutative (Jordan) algebras defined by the identity x³ = 0, and let 𝔖₂ be the subvariety of the variety 𝔄₃ of solvable algebras of solviability index 2. We present an infinite independent system of identities in the variety 𝔄₃ ∩ 𝔖₂𝔖₂. Therefore we infer that 𝔄₃ ∩ 𝔖₂𝔖₂ contains a continuum of infinite based subvarieties and that there exist algebras with an unsolvable words problem in 𝔄₃ ∩ 𝔖₂𝔖₂.
It is worth mentioning that these results were announced in 1999 in works of the international conference "Loops’99" (Prague).
Rocznik
Tom
24
Numer
1
Strony
5-30
Opis fizyczny
Daty
wydano
2004
otrzymano
2001-08-27
poprawiono
2002-12-08
poprawiono
2004-06-16
poprawiono
2004-07-14
Twórcy
  • Tiraspol State University, The author's home address: Deleanu str 1, Apartment 60, Kishinev MD-2071, Moldova
Bibliografia
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1072
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