, Volume 41, Issue 4, pp 274-283

A Class of Strongly Decomposable Abelian Groups

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Let G be a completely decomposable torsion-free Abelian group and G= ⊕ Gi, 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.