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On the variety generated by all nilpotent lattice-ordered groups

Authors: V. V. Bludov and A. M. W. Glass
Journal: Trans. Amer. Math. Soc. 358 (2006), 5179-5192
MSC (2000): Primary 06F15, 20F18, 20F12
Published electronically: July 25, 2006
MathSciNet review: 2238913
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1974, J. Martinez introduced the variety $ {\mathcal W}$ of weakly Abelian lattice-ordered groups; it is defined by the identity

$\displaystyle x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2.$

References [Enhancements On Off] (What's this?)

  • 1. V. V. Bludov, On locally nilpotent groups, Trudy Inst. Mat. Sobolev 30 (1996), 26-47 (in Russian): English translation, Siberian Advances in Math. 8 (1998) 49-79. MR 1651902 (99m:20074)
  • 2. V. V. Bludov, A. M. W. Glass and A. H. Rhemtulla, Ordered groups in which all convex jumps are central, J. Korean Math. Soc. 40 (2003), 225-239. MR 1958028 (2003j:06020)
  • 3. V. V. Bludov, A. M. W. Glass and A. H. Rhemtulla, On centrally ordered groups, J. Algebra 291 (2005), 129-143. MR 2158514 (2006h:20053)
  • 4. A. M. W. Glass, Partially Ordered Groups, World Scientific Pub. Co., Singapore, 1999. MR 1791008 (2001g:06002)
  • 5. A. M. W. Glass, Weakly Abelian lattice-ordered groups, Proc. American Math. Soc. 129 (2001), 677-684; corrigendum: ibid 130 (2001), 925-926. MR 1801994 (2002a:06022)
  • 6. S. A. Gurchenkov, On varieties of weakly Abelian $ \ell$-groups, Mat. Slovaca 42 (1992), 437-441. MR 1195037 (94a:20067)
  • 7. P. Hall, Nilpotent Groups, Lectures given at the Canadian Mathematical Congress, University of Alberta, 1957.
  • 8. V. M. Kopytov, Lattice-ordered locally nilpotent groups, Algebra i Logika 14 (1975), 407-413 (in Russian); English translation Algebra and Logic 14 (1975), 249-251. MR 0401583 (53:5410)
  • 9. V. M. Kopytov and N. Ya. Medvedev, The Theory of Lattice-ordered Groups, Kluwer Acad. Pub., Dordrecht, 1994. MR 1369091 (97k:06028)
  • 10. Kourovka Notebook, Unsolved Problems in Group Theory, eds. V. D. Mazurov and E. I. Khukhro, $ 13$th edition, Novosibirsk, 1995. MR 1392713 (97d:20001)
  • 11. J. Martinez, Varieties of lattice-ordered groups, Math. Z. 137 (1974), 265-284. MR 0354483 (50:6961)
  • 12. R. B. Mura and A. H. Rhemtulla, Orderable Groups, Lecture Notes in Pure and Applied Maths. 27, M. Dekker, New York, 1977. MR 0491396 (58:10652)
  • 13. N. R. Reilly, Nilpotent, weakly Abelian and Hamiltonian lattice-ordered groups, Czech. Math. J. 33 (1983), 348-353. MR 0718919 (85m:06035)
  • 14. D. J. S. Robinson, A Course in the Theory of Groups (second edition), Graduate Texts in Math. 80, Springer-Verlag, Heidelberg, 1996. MR 1357169 (96f:20001)
  • 15. R. Warfield, Nilpotent Groups, Lecture Notes in Math. 513, Springer-Verlag, Berlin, 1976. MR 0409661 (53:13413)
  • 16. The Black Swamp Problem Book is edited by W. Charles Holland (Bowling Green State University, Ohio 43403, U.S.A.) and is kept there by him (in the formerly Black Swamp region of Ohio).

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Additional Information

V. V. Bludov
Affiliation: Institute of Mathematics and Economics, Irkutsk State University, Irkutsk, 664003 Russia

A. M. W. Glass
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Rd., Cambridge CB3 0WB, England

Keywords: Nilpotent group, residually torsion-free-nilpotent, variety, quasi-variety, commutator calculus, lattice-ordered group, weakly Abelian
Received by editor(s): December 27, 2003
Published electronically: July 25, 2006
Additional Notes: The first author was supported by the Russian Foundation for Basic Research, grant no. 03-01-00320
Dedicated: To Valerie Kopytov on his sixty-fifth birthday
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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