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Solvable groups and varieties of $$l$$-groups. (Russian, English) Zbl 1101.06011
Algebra Logika 44, No. 3, 355-367 (2005); translation in Algebra Logic 44, No. 3, 197-204 (2005).
Summary: A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l.o.) group are abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l.o. group are contained in a group variety $$\mathcal V$$. In particular, for every solvable l.o. group $$G$$ of solvability index $$n$$, $$n\geq 2$$, factors of a system of normal convex subgroups are solvable l.o. groups of solvability index at most $$n-1$$. It is proved that the variety $$\mathcal R$$ of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with the variety generated by all solvable l.o. groups. It is shown that if $$\mathcal V$$ is any $$o$$-approximable variety of $$l$$-groups and if every identity in the group signature is not identically true in $$\mathcal V$$, then $$\mathcal V$$ contains free l.o. groups.

##### MSC:
 06F15 Ordered groups 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 20F16 Solvable groups, supersolvable groups 20F60 Ordered groups (group-theoretic aspects) 20E10 Quasivarieties and varieties of groups
##### Keywords:
variety of $$l$$-groups; solvable group
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