ABSTRACT
We shall write a < {3 if a ~ {3 and a :F {3. An element a E f is positive, resp. strictly positive, if a ~ e, resp. a> e, where e is the neutral element of f. We write
f+ ={aEf,e~a}, r~'Y ={aEf,a~'Y} r<'Y ={aEf,a<'Y}
similar for r ~"Y' r >'Y' A subgroup l::i. of f is called convex if for a E t:1 such that a ~ 0 we have that any '1 E f such that 0 ~ '1 ~ a must necessarily be in A. The normal convex subgroups of f are precisely the kernels of the order-homomorphisms r --+ fl between totally ordered groups. The set of (normal) convex subgroups of
r is itself a totally ordered set by inclusion. Let C(f) denote this totally ordered set of convex subgroups a :/: r, including {e} in C(r). The order type of C(r) is called the rank of r and it is denoted by rk(r). For a normal convex subgroup a of r we have rkr = rka + rk(r/ a) where r / a is equiped with the total ordering induced by ~ on r in the natural way.
