A Class of Strongly Decomposable Abelian Groups

Let G be a completely decomposable torsion-free Abelian group and G= G_i, where G_i is a rank 1 group. If there exists a strongly constructive numbering of G such that (G,) has a recursively enumerable sequence of elements g_i G_i, then G is called a strongly decomposable group. Let pi, i, be some sequence of primes whose denominators are degrees of a number p_i and let . A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that for some numbers i₁,...,i_k. We bring in the concept of a quasihyperhyperimmune set, and specify a necessary and sufficient condition on the characteristic of A subject to which the group in question is strongly decomposable. Also, it is proved that every hyperhyperimmune set is quasihyperhyperimmune, the converse being not true.