Metabelian varieties ofl-groups which contain no non-abeliano-groups

Abstract

The aim of this paper is to describe the lattice of all varieties of lattice ordered groups which are solvable class two (that is,metabelian) as groups and which have the property that they contain no non-abeliano-groups. In doing so we obtain a fairly explicit description of the structure of any finitely generated subdirectly irreducible member of such a variety and show that each such member generates a variety determined by a single identity.

This continues the work of [HR] and [HMR]. In [S], Scrimger discovered an infinite set of varieties which cover the abelian variety, each of which is metabelian and contains only abeliano-groups. In [HR] the sub-directly irreducible members of the Scrimger varieties were characterized and from that a basis of identities was obtained. In [HMR] and independently in [G], those results were extended to obtain a description of the subvarieties of the varietyℒ _p of metabelianl-groups satisfying the identityx ^p y ^p=y^px^p (p a prime).