×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI