A Class of Strongly Decomposable Abelian Groups

Abstract

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 ν of G such that (G,ν) has a recursively enumerable sequence of elements g _i ∈ G _i , then G is called a strongly decomposable group. Let pi, i∈ω, be some sequence of primes whose denominators are degrees of a number p _i and let $\mathop \oplus \limits_{i \in \omega } Q_{Pi}$. A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that $p_{i_1 } = ... = p_{i_k } = p$ for some numbers i ₁,...,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.