A Class of Strongly Decomposable Abelian Groups

Let $G$ be a completely decomposable torsion-free Abelian group and $G=\bigoplus G_i$, where $G_i$ is a rank 1 group. If there exists a strongly constructive numbering $\nu$ of $G$ such that $(G,\nu)$ has a recursively enumerable sequence of elements $g_i\in G_i$, then $G$ is called a strongly decomposable group. Let $p_i$, $i\in\omega$, be some sequence of primes whose denominators are degrees of a number $p_i$ and let $A=\bigoplus\limits_{i\in\omega}Q_{p_i}$. A characteristic of the group $A$ is the set of all pairs $\langle p,k\rangle$ of numbers such that $p_{i_1}=\ldots=p_{i_k}=p$ for some numbers $i_1,\ldots,i_k$. 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.