Varieties of lattice ordered groups that contain no non-Abelian o-groups are solvable.

*(English)*Zbl 0616.06016The author extends the recent progress in describing the lattice of varieties of lattice-ordered groups and in particular the lattice of subvarieties in the family of varieties \(\{L_ n\}\), one for each positive integer n, where \(L_ n\) is defined by the identity \(x^ ny^ n=y^ nx^ n\). It is shown that if n is a product of k not necessarily distinct prime numbers, then \(L_ n\subseteq A^{k+1}\) the \((k+1)st\) power of the variety A of all Abelian lattice-ordered groups. Thus, the variety \(L_ n\) is solvable of class \(k+1\). Let R denote the variety generated by all totally ordered groups. That \(L_ n\cap R=A\) was known and the author shows that if the variety V satisfies \(V\cap R=A\), then \(V\subseteq L_ n\) for some n and V is a variety of solvable \(\ell\)- groups. Hence, if V is any variety of \(\ell\)-groups that contains no non- abelian o-groups, then V is solvable. It therefore follows by a result of S. A. Gurchenkov that the only covers of A not in R are the Scrimger varieties \(\{S_ p\}\), p some prime number. The author notes that this last result was obtained independently by V. M. Kopytov and M. Darnel.

Reviewer: S.P.Hurd

##### Keywords:

lattice of varieties; lattice-ordered groups; Abelian lattice-ordered groups; totally ordered groups; variety of solvable \(\ell \)-groups; covers; Scrimger varieties
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