Khisamiev, N.G. Algebra and Logic (2002) 41: 274. doi:10.1023/A:1020112806274

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.

strongly decomposable Abelian grouphyperhyperimmune setquasihyperhyperimmune set