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The Relationship of Partial Metric Varieties and Commuting Powers Varieties

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Abstract

Holland et al. (Algebra Univers 67:1–18, 2012) considered varieties \({\mathcal E}_n\) of lattice-ordered groups defined by partial metrics, and showed for all n that \({\mathcal E}_n\) is contained within the variety \({\mathcal L}_n\) defined by x n y n = y n x n. They also showed that if n were prime, then \({\mathcal E}_n = {\mathcal L}_n\). Letting \({\mathcal A}^2\) denote the metabelian variety (defined at the beginning of Section 2), this article continues their work, showing that for all n, \({\mathcal L}_n \cap {\mathcal A}^2 \subseteq {\mathcal E}_n\) while showing that if n is not prime, \({\mathcal L}_n \not\subseteq {\mathcal E}_n\).

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References

  1. Darnel, M.R.: Theory of Lattice-Ordered Groups. Marcel Dekker (1995)

  2. Holland, W.C., Kopperman, R., Pajoohesh, H.: Intrinsic generalized metrics. Algebra Univers. 67, 1–18 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Holland, W.C., Mekler, A., Reilly, N.: Varieties of lattice-ordered groups in which prime powers commute. Algebra Univers. 23, 196–214 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Holland, W.C., Reilly, N.: Metabelian varieties of ℓ-groups which contain no nonabelian o-groups. Algebra Univers. 24, 202–223 (1987)

    Article  MathSciNet  Google Scholar 

  5. Reilly, N.: Varieties of lattice-ordered groups that contain no non-abelian o-groups are solvable. Order 3, 287–297 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Smith, J.E.: A new family of łgroup vareties. Houst. J. Math. 7, 551–570 (1981)

    MATH  Google Scholar 

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Correspondence to Michael R. Darnel.

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Darnel, M.R., Holland, W.C. & Pajoohesh, H. The Relationship of Partial Metric Varieties and Commuting Powers Varieties. Order 30, 403–414 (2013). https://doi.org/10.1007/s11083-012-9251-7

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  • DOI: https://doi.org/10.1007/s11083-012-9251-7

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