О многообразии m-групп N с субнормальными скачками

Аннотация

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 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 N is an idempotent of M and $N=\bigvee\limits_{n \in \mathbb{N}}\mathcal{A}^{n},$ where $\mathcal{A}$ is the variety of all abelian m-groups.