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, \({\mathcal L}_n \not\subseteq {\mathcal E}_n\).
Similar content being viewed by others
References
Darnel, M.R.: Theory of Lattice-Ordered Groups. Marcel Dekker (1995)
Holland, W.C., Kopperman, R., Pajoohesh, H.: Intrinsic generalized metrics. Algebra Univers. 67, 1–18 (2012)
Holland, W.C., Mekler, A., Reilly, N.: Varieties of lattice-ordered groups in which prime powers commute. Algebra Univers. 23, 196–214 (1986)
Holland, W.C., Reilly, N.: Metabelian varieties of ℓ-groups which contain no nonabelian o-groups. Algebra Univers. 24, 202–223 (1987)
Reilly, N.: Varieties of lattice-ordered groups that contain no non-abelian o-groups are solvable. Order 3, 287–297 (1986)
Smith, J.E.: A new family of łgroup vareties. Houst. J. Math. 7, 551–570 (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-012-9251-7