On the variety generated by all nilpotent latticeordered groups
Authors:
V. V. Bludov and A. M. W. Glass
Journal:
Trans. Amer. Math. Soc. 358 (2006), 51795192
MSC (2000):
Primary 06F15, 20F18, 20F12
Published electronically:
July 25, 2006
MathSciNet review:
2238913
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In 1974, J. Martinez introduced the variety of weakly Abelian latticeordered 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
(99m:20074)
 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
(2003j:06020), http://dx.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
(2006h:20053), http://dx.doi.org/10.1016/j.jalgebra.2005.05.014
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M. W. Glass, Partially ordered groups, Series in Algebra,
vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999.
MR
1791008 (2001g:06002)
 5.
A.
M. W. Glass, Weakly abelian latticeordered
groups, Proc. Amer. Math. Soc.
129 (2001), no. 3,
677–684. MR 1801994
(2002a:06022), http://dx.doi.org/10.1090/S0002993900057063
 6.
S.
A. Gurchenkov, About varieties of weakly abelian
𝑙groups, Math. Slovaca 42 (1992),
no. 4, 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, Latticeordered locally nilpotent groups, Algebra
i Logika 14 (1975), no. 4, 407–413 (Russian).
MR
0401583 (53 #5410)
 9.
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M. Kopytov and N.
Ya. Medvedev, The theory of latticeordered groups,
Mathematics and its Applications, vol. 307, Kluwer Academic Publishers
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(97k:06028)
 10.
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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
(97d:20001)
 11.
Jorge
Martinez, Varieties of latticeordered groups, Math. Z.
137 (1974), 265–284. MR 0354483
(50 #6961)
 12.
Roberta
Botto Mura and Akbar
Rhemtulla, Orderable groups, Marcel Dekker, Inc., New
YorkBasel, 1977. Lecture Notes in Pure and Applied Mathematics, Vol. 27.
MR
0491396 (58 #10652)
 13.
Norman
R. Reilly, Nilpotent, weakly abelian and Hamiltonian lattice
ordered groups, Czechoslovak Math. J. 33(108) (1983),
no. 3, 348–353. MR 718919
(85m:06035)
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Derek
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Graduate Texts in Mathematics, vol. 80, SpringerVerlag, New York,
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1357169 (96f:20001)
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Robert
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Mathematics, Vol. 513, SpringerVerlag, BerlinNew York, 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).
 1.
 V. V. Bludov, On locally nilpotent groups, Trudy Inst. Mat. Sobolev 30 (1996), 2647 (in Russian): English translation, Siberian Advances in Math. 8 (1998) 4979. 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), 225239. MR 1958028 (2003j:06020)
 3.
 V. V. Bludov, A. M. W. Glass and A. H. Rhemtulla, On centrally ordered groups, J. Algebra 291 (2005), 129143. 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 latticeordered groups, Proc. American Math. Soc. 129 (2001), 677684; corrigendum: ibid 130 (2001), 925926. MR 1801994 (2002a:06022)
 6.
 S. A. Gurchenkov, On varieties of weakly Abelian groups, Mat. Slovaca 42 (1992), 437441. MR 1195037 (94a:20067)
 7.
 P. Hall, Nilpotent Groups, Lectures given at the Canadian Mathematical Congress, University of Alberta, 1957.
 8.
 V. M. Kopytov, Latticeordered locally nilpotent groups, Algebra i Logika 14 (1975), 407413 (in Russian); English translation Algebra and Logic 14 (1975), 249251. MR 0401583 (53:5410)
 9.
 V. M. Kopytov and N. Ya. Medvedev, The Theory of Latticeordered 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, th edition, Novosibirsk, 1995. MR 1392713 (97d:20001)
 11.
 J. Martinez, Varieties of latticeordered groups, Math. Z. 137 (1974), 265284. 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 latticeordered groups, Czech. Math. J. 33 (1983), 348353. MR 0718919 (85m:06035)
 14.
 D. J. S. Robinson, A Course in the Theory of Groups (second edition), Graduate Texts in Math. 80, SpringerVerlag, Heidelberg, 1996. MR 1357169 (96f:20001)
 15.
 R. Warfield, Nilpotent Groups, Lecture Notes in Math. 513, SpringerVerlag, 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
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:
http://dx.doi.org/10.1090/S0002994706038827
PII:
S 00029947(06)038827
Keywords:
Nilpotent group,
residually torsionfreenilpotent,
variety,
quasivariety,
commutator calculus,
latticeordered 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. 030100320
Dedicated:
To Valerie Kopytov on his sixtyfifth birthday
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
