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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

Abstract:

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity $x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2$. 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.


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

  • [1] A. M. W. Glass, Partially Ordered Groups, Series in Algebra vol. 7, World Scientific Pub. Co., Singapore, 1999.
  • [2] S. A. Gurchenkov, About varieties of weakly abelian 𝑙-groups, Math. Slovaca 42 (1992), no. 4, 437–441. MR 1195037
  • [3] Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959. MR 0103215
  • [4] J. Martinez, Varieties of lattice-ordered groups, Math. Z. 137(1974), 265-284.
  • [5] Norman R. Reilly, Nilpotent, weakly abelian and Hamiltonian lattice ordered groups, Czechoslovak Math. J. 33(108) (1983), no. 3, 348–353. MR 718919
  • [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