The Relationship of Partial Metric Varieties and Commuting Powers Varieties

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, ${\mathcal L}_n \not\subseteq {\mathcal E}_n$ .