Algebra and Logic

, Volume 41, Issue 4, pp 274–283

A Class of Strongly Decomposable Abelian Groups


  • N. G. Khisamiev

DOI: 10.1023/A:1020112806274

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


Let G be a completely decomposable torsion-free Abelian group and G= ⊕ Gi, where Gi 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 giGi, then G is called a strongly decomposable group. Let pi, i∈ω, be some sequence of primes whose denominators are degrees of a number pi 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 i1,...,ik. 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

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© Plenum Publishing Corporation 2002