Algebra and Logic

, Volume 41, Issue 4, pp 274–283 | Cite as

A Class of Strongly Decomposable Abelian Groups

  • N. G. Khisamiev
Article

Abstract

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 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.

strongly decomposable Abelian group hyperhyperimmune set quasihyperhyperimmune set 

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© Plenum Publishing Corporation 2002

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  • N. G. Khisamiev

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