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Varieties of lattice ordered groups that contain no non-Abelian o-groups are solvable. (English) Zbl 0616.06016
The 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

##### MSC:
 06F15 Ordered groups 08B15 Lattices of varieties 20E10 Quasivarieties and varieties of groups
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##### References:
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