Weakly abelian lattice-ordered groups
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- by A. M. W. Glass PDF
- Proc. Amer. Math. Soc. 129 (2001), 677-684 Request permission
Corrigendum: Proc. Amer. Math. Soc. 130 (2002), 925-926.
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
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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
- Received by editor(s): May 21, 1999
- Published electronically: September 20, 2000
- Communicated by: Stephen D. Smith
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1801994
Dedicated: Respectfully dedicated (with gratitude) to W. Charles Holland on his 65th Birthday