Valuations of Lattice-Ordered Groups Original Research Article

In this paper, we introduce the concept of a valuation mapping of anl-groupGonto a distributive lattice and use such valuations to investigate the structure ofG. Then we examine the maximal immediate extensions ofGwith respect to these valuations. For the natural valuation, these are the archimedean extensions (a-extensions) first investigated by[12]and [30]. This leads to new results and new proofs of old results abouta-extensions. Then we obtain new structure theorems for Δ-extensions ofl-groups. In another paper, it will be shown that this valuation theory determinesallthe torsion classes ofl-groups that have invariant torsion radicals. This includes most of the well known and interesting torsion classes that are notl-varieties.