On Ordered Divisible Groupso)

Introduction and remarks. In the theory of r)a-sets three main theorems stand out: that an r/0-set is universal for totally ordered sets of power not exceeding Xa, that any two 7/a-sets of power Na are isomorphic and that 7/a-sets of power Xa exist provided Ka is a regular cardinal number and 2~2i<<> 2t*sS&a.; L. Gillman and M. Jerison [6] have shown that a real closed (totally ordered) field which is an T7„-set is universal for totally ordered fields of power not exceeding N„, provided a>0. Further, P. Erdos, L. Gillman and M. Henriksen [3] have shown that any two real closed fields which are Tja-sets of power fc<„ are isomorphic, provided a>0. They also showed that real closed fields which are 7ja-sets of power Na exist, assuming the continuum hypothesis, if a=l. The question of existence for a>l, even assuming N„ to be regular and 23«<« 2k0 an 7/«-group (i.e., a totally ordered Abelian divisible group which is an 7/a-set) is universal for totally ordered Abelian groups of power not exceeding K„, that any two 7/a-groups of power K« are isomorphic and finally that, given an 7/a-set of power Ka, 7/a-groups of power Na exist. Background. Let a be an ordinal number. By IF(a) is meant the set of all ordinal numbers 5 such that 5<a. Let Na be a cardinal number. By w„ is meant the least ordinal number such that IF(co„) is of power K„. A cardinal number N„ is called regular if given an ordinal 7r<coa and given a family of sets iSi)s<T, each of power less than i$a, the power of Uj<x Ss is less than K«. Let P and T' he totally ordered sets and let/be a mapping of P into P'. We will call/order preserving (order reversing) if x, yET and xSy implies fix) Sfiy), ifix) ^/(y)) and strictly order preserving (strictly order reversing) if / is order preserving (order reversing) and one-to-one. Let H and K be subsets of P. We will write H<K if h<k for every hEH and kEK. Note, according to this definition, 0<K and K<0 for all subsets K of P including 0. T is said to be an na-set if, given subsets 77 and K of P of power less than Na such that 77<K then there exists tET such that 77< {t} <K. Let P be an r/„-set and let 77 be a subset of P of power less than Na. By definition, 0<77 and H<0. Thus there exist u, vET such that 0<{«} <77 and