Free ℓ-groups

Abstract

Because the class of lattice-ordered groups is a variety, it follows from universal algebra that there exist free objects for the class [Co]; in fact, the same observation holds for any variety of lattice-ordered groups, and in particular for the variety of abelian l-groups. (In the next chapter we shall consider varieties of l-groups in some detail.) However it took considerable work by several mathematicians before a satisfactory construction of free l-groups was obtained. Birkhoff [42] was the first to mention free l-groups; he described explicitly the free l-group on one generator. Baker, in his Ph.D. thesis (Harvard, 1966), and Weinberg [63], [65] first constructed the free abelian l-group over a set of generators, and more generally, over a partially ordered abelian group (see also Henriksen and Isbell [62], whose work in a category of lattice-ordered rings predated Weinberg’s). Work of Birkhoff [B] , Baker [68] and Beynon [74] has led to a more intuitively satisfying description of the free abelian l-group on a set of generators. Bernau [70] first constructed nonabelian free l-groups, using an equivalence relation on the set of all non-empty finite subsets of the partially ordered group. Conrad [70a] then generalized Weinberg’s construction to the nonabelian case, thus providing a more understandable representation for the free l-group. Recently McCleary [85a], [85b] has used permutation group methods to considerably add to our understanding of arbitrary free l-groups; we will briefly discuss his results at the end of this chapter.