Weakly abelian lattice-ordered groups

Author:
A. M. W. Glass

Journal:
Proc. Amer. Math. Soc. **129** (2001), 677-684

MSC (2000):
Primary 06F15, 20E05, 20F19; Secondary 20F18, 08B15

DOI:
https://doi.org/10.1090/S0002-9939-00-05706-3

Published electronically:
September 20, 2000

Corrigendum:
Proc. Amer. Math. Soc. 130 (2002), 925--926.

MathSciNet review:
1801994

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Abstract | References | Similar Articles | Additional Information

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity . In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Problem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also *many* weakly Abelian ordered free groups on finitely many generators. If *every* weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpotent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov's problem is even more intriguing.

**[1]**A. M. W. Glass,*Partially Ordered Groups*, Series in Algebra vol. 7, World Scientific Pub. Co., Singapore, 1999.**[2]**S. A. Gurchenkov,*On varieties of weakly Abelian -groups*, Mat. Slovaca**42**(1992), 437-441. MR**94a:20067****[3]**M. Hall,*The Theory of Groups*, Macmillan Co., New York, 1959. MR**21:1996****[4]**J. Martinez,*Varieties of lattice-ordered groups*, Math. Z.**137**(1974), 265-284.**[5]**N. R. Reilly,*Nilpotent, weakly Abelian and Hamiltonian lattice-ordered groups*, Czech. Math. J.**33**(1983), 348-353. MR**85m:06035****[6]**A. A. Vinogradov,*Non-axiomatizability of lattice-ordered groups*, Siberian Math. J.**13**(1971), 331-332.

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Additional Information

**A. M. W. Glass**

Affiliation:
Department of Pure Mathematics & Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB, England

Email:
amwg@dpmms.cam.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-00-05706-3

Received by editor(s):
May 21, 1999

Published electronically:
September 20, 2000

Dedicated:
Respectfully dedicated (with gratitude) to W. Charles Holland on his 65th Birthday

Communicated by:
Stephen D. Smith

Article copyright:
© Copyright 2000
American Mathematical Society