1. Ordered groups. Suppose r is a simply ordered set and for each y in r let Ry be an additive group. Consider the set H=H(r, RY) of all vectors ( * * *, ry, * * * ) for which ry is in Ry and almost every ry is zero (i.e., the set of y's for which ry - 0 is inversely well ordered). H is a group w.r.t. (with respect to) vector addition, called the r-sum of the RY. If the RY are ordered groups (notation o-group), then H is an o-group w.r.t. the following definition of order: h in H is positive if h 5 O and the nonzero component with greatest y is positive. For the rest of this section let G denote an additive o-group. The set rG of all pairs of convex subgroups GY, Gy of G where Gy covers Gy is simply ordered by inclusion, and each GY/Gy is o-isomorphic to a subgroup of the additive group of real numbers. Thus G determines a rG-SUm H= H(rG, GY/Gy), and H is an abelian o-group. The value V(g) of g O in G is the y in rG for which g is in GY but not in G. If a and b are elements of H for which by=ay for all y>,B, be =O for all ,y <,B, and a#$O, then b is called the f,th head of a. LEMMA 1.1. There exists a 1-1 order and value-preserving mapping r of the set G into the set H such that for every g in G: (a) If V(g) =,y, then (g7r)y=G+g. 07r=O. (b) If a is the f3th head of g7r, then there exists an h in G for which