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

Nicolae Ion Sandu

Discussiones Mathematicae - General Algebra and Applications (2004)

  • Volume: 24, Issue: 1, page 5-30
  • ISSN: 1509-9415

Abstract

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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).

How to cite

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Nicolae Ion Sandu. "Infinite independent systems of identities of alternative commutative algebra over a field of characteristic three." Discussiones Mathematicae - General Algebra and Applications 24.1 (2004): 5-30. <http://eudml.org/doc/287763>.

@article{NicolaeIonSandu2004,
abstract = {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).},
author = {Nicolae Ion Sandu},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {nfinite independent system of identities; alternative commutative algebra; solvable algebra; commutative Moufang loop},
language = {eng},
number = {1},
pages = {5-30},
title = {Infinite independent systems of identities of alternative commutative algebra over a field of characteristic three},
url = {http://eudml.org/doc/287763},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Nicolae Ion Sandu
TI - Infinite independent systems of identities of alternative commutative algebra over a field of characteristic three
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2004
VL - 24
IS - 1
SP - 5
EP - 30
AB - 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).
LA - eng
KW - nfinite independent system of identities; alternative commutative algebra; solvable algebra; commutative Moufang loop
UR - http://eudml.org/doc/287763
ER -

References

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  1. [1] R.H. Bruck, A survey of binary systems, Springer-Verlag, Berlin 1958. Zbl0081.01704
  2. [2] O. Chein, H.O. Pflugfelder and J.D.H. Smith, (eds.), Quasigroups and Loops: Theory and Applications, Heldermann Verlag, Berlin 1990. Zbl0719.20036
  3. [3] V.T. Filippov, n-Lie algebras (Russian), Sibirsk. Mat. Zh. 26 (1985), no. 6, 126-140. 
  4. [4] S. Lang, Algebra, Addison-Wesley Publ. Co., Reading, MA, 1965. 
  5. [5] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, (second revised edition), Dover Publ., New York 1976. Zbl0362.20023
  6. [6] Yu.A. Medvedev, Finite basis property of varieties with binomial identities (Russian), Algebra i Logika 17 (1978), 705-726. 
  7. [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. [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. [9] A. Thedy, Right alternative rings, J. Algebra 37 (1975), 1-43. 
  10. [10] N.I. Sandu, Centrally nilpotent commutative Moufang loops (Russian), Mat. Issled. No. 51 (1979), (Quasigroups and loops), 145-155. Zbl0439.20050
  11. [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. Zbl0615.20055
  12. [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. Zbl0962.20050
  13. [13] N.I. Sandu, About the embedding of Moufang loops into alternative algebras, to appear. Zbl1197.20060
  14. [14] U.U. Umirbaev, The Specht property of a variety of solvable alternative algebras (Russian), Algebra i Logika 24 (1985), 226-239. Zbl0576.17016
  15. [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. Zbl0487.17001

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