Nilpotent lattice-ordered groups

Abstract

A lattice-ordered groupG isweakly abelian if for each 0<x∈G andg∈G,x ^g=−g+x+g≦2x. The class of weakly abelian lattice-ordered groups is a variety which we denote byW. Each nilpotent class 2 lattice-ordered group is necessarily weakly abelian. The main theorem is that each subvariety ofW containing a non-abelian lattice-ordered group contains the variety generated by the free nilpotent class 2 product of two copies ofZ, the additive group of integers with the usual order. Moreover, a reasonably precise description is given of the free products of copies ofZ in varieties of nilpotent lattice-ordered groups. It is further shown that the lattice-subgroup generated by the commutator subgroup of the free lattice-ordered group inW on two generators is convex; this is false for three or more generators.