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

DOI:
https://doi.org/10.1090/S0002-9947-06-03882-7

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 of weakly Abelian lattice-ordered groups; it is defined by the identity

**1.**V. V. Bludov,*On locally nilpotent groups [translation of Trudy Instituta Matematiki, Vol. 30, 26–47, Izdat. Ross. Akad. Nauk, Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1996]*, Siberian Adv. Math.**8**(1998), no. 1, 49–79. MR**1651902****2.**V. V. Bludov, A. M. W. Glass, and Akbar H. Rhemtulla,*Ordered groups in which all convex jumps are central*, J. Korean Math. Soc.**40**(2003), no. 2, 225–239. MR**1958028**, https://doi.org/10.4134/JKMS.2003.40.2.225**3.**V. V. Bludov, A. M. W. Glass, and A. H. Rhemtulla,*On centrally orderable groups*, J. Algebra**291**(2005), no. 1, 129–143. MR**2158514**, https://doi.org/10.1016/j.jalgebra.2005.05.014**4.**A. M. W. Glass,*Partially ordered groups*, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR**1791008****5.**A. M. W. Glass,*Weakly abelian lattice-ordered groups*, Proc. Amer. Math. Soc.**129**(2001), no. 3, 677–684. MR**1801994**, https://doi.org/10.1090/S0002-9939-00-05706-3**6.**S. A. Gurchenkov,*About varieties of weakly abelian 𝑙-groups*, Math. Slovaca**42**(1992), no. 4, 437–441. MR**1195037****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), no. 4, 407–413 (Russian). MR**0401583****9.**V. M. Kopytov and N. Ya. Medvedev,*The theory of lattice-ordered groups*, Mathematics and its Applications, vol. 307, Kluwer Academic Publishers Group, Dordrecht, 1994. MR**1369091****10.**V. D. Mazurov and E. I. Khukhro (eds.),*Unsolved problems in group theory. The Kourovka notebook*, Thirteenth augmented edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 1995. MR**1392713****11.**Jorge Martinez,*Varieties of lattice-ordered groups*, Math. Z.**137**(1974), 265–284. MR**0354483**, https://doi.org/10.1007/BF01214370**12.**Roberta Botto Mura and Akbar Rhemtulla,*Orderable groups*, Marcel Dekker, Inc., New York-Basel, 1977. Lecture Notes in Pure and Applied Mathematics, Vol. 27. MR**0491396****13.**N. R. Reilly,*Nilpotent, weakly Abelian and Hamiltonian lattice-ordered groups*, Czech. Math. J.**33**(1983), 348-353. MR**0718919 (85m:06035)****14.**Derek J. S. Robinson,*A course in the theory of groups*, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR**1357169****15.**Robert B. Warfield Jr.,*Nilpotent groups*, Lecture Notes in Mathematics, Vol. 513, Springer-Verlag, Berlin-New York, 1976. MR**0409661****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

Email:
bludov@math.isu.ru

**A. M. W. Glass**

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

Email:
amwg@dpmms.cam.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-06-03882-7

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.