On variety $\mathcal{N}$ of normal valued $m$-groups

Recall that an $m$-group is a pair $(G,_{*}),$ where $G$ is an $\ell$-group and $_{*}$ is a decreasing order two automorphism of $G$. An $m$-group can be regarded as an algebraic system of signature $m$ and it is obvious that the $m$-groups form a variety in this signature. The set $M$ of varieties of all $m$-groups is a partially ordered set with respect to the set-theoretic inclusion. Moreover, $M$ is a lattice with respect to the naturally defined operations of intersection and union of varieties of $m$-groups. In this article we study the characteristics of a variety $\mathcal{N}$ of normal valued $m$-groups which is defined by the identity $ |x||y|\wedge |y|^{2}|x|^{2}=|x||y|.$ We will prove that $\mathcal{N}$ is an idempotent of $M$ and $\mathcal{N}=\bigvee\limits_{n \in \mathbb{N}}\mathcal{A}^{n},$ where $\mathcal{A}$ is the variety of all abelian $m$-groups.