The Conrad-Harvey-Holland Representation

Abstract

In 1907 H. Hahn [07] proved that every abelian totally ordered group could be represented as a group of real-valued functions on a totally ordered set. His proof is long and difficult, and is one of the first successful uses of the technique of transfìnite induction. It was not until after the Second World War that simplified proofs of Hahn’s theorem appeared, by Conrad [53] and Clifford [54]. In 1963, Conrad, Harvey and Holland [63] were able to generalize Hahn’s theorem to the class of lattice-ordered groups, using the idea of Banaschewskì’s proof [57] of the original theorem; their theorem states that any abelian ℓ-group can be represented as a group of real-valued functions on a partially ordered set (in fact, on a root system). In this chapter we shall present Wolfenstein’s proof [66] (or see [C]) of the Conrad-Harvey-Holland theorem.