Non-Abelian varieties of lattice-ordered groups in which every solvable \(\ell\)-group is Abelian. (Russian) Zbl 0574.06012

Kopytov, V. M.

Mat. Sb., N. Ser. 126(168), No. 2, 247-266 (1985).

From the author’s introduction: ”In the present paper linearly ordered groups (l.o. groups) are constructed which possess a number of interesting properties and a variety of o-approximable \(\ell\)-groups in which solvable non-abelian \(\ell\)-groups do not exist. In particular, the l.o. group F constructed in §§ 2 and 3 possesses the following properties: F is non-abelian with the property \(e<a\ll b\) implies \(a\ll b^{-1}ab\); the factor group H/N of a subgroup H of F modulo a convex and normal (with respect to H) subgroup N is either abelian or non- solvable. The variety of \(\ell\)-groups \(\ell\)-var F generated by F is non-abelian and consists of o-approximable \(\ell\)-groups and each of its solvable \(\ell\)-groups is abelian. This variety is a new example of a minimal cover of the \(\ell\)-variety of the abelian \(\ell\)-groups in the lattice of varieties of \(\ell\)-groups. The description of properties of the l.o. group F or the variety \(\ell\)-var F is given in §3 or 4 resp.”

Reviewer: F.Šik

MSC:

06F15	Ordered groups
08B15	Lattices of varieties
06B20	Varieties of lattices
20F60	Ordered groups (group-theoretic aspects)
20E10	Quasivarieties and varieties of groups

Keywords:

linearly ordered groups; o-approximable \(\ell\)-groups; variety of \(\ell\)- groups; solvable \(\ell\)-groups; minimal cover; lattice of varieties of \(\ell\)-groups