Inductive purities in Abelian groups

In the paper we study purities $\omega$ in categories of Abelian groups having the property that the union of an increasing chain of $\omega$-pure subgroups of an Abelian group $G$ is itself an $\omega$-pure subgroup of $G$. Such purities are called inductive. For every prime number $p$ we set $A\subseteq_{\eta_p}B$ if for $A\ni a=p^kb$, $b\in B$, there is an $a'\in A$ and an $l\geqslant0$ such that $p^la=p^{k+l}a'$. Head purities are defined as purities of the form $\eta_\Pi=\bigcap_{p\in\Pi}\eta_p$, where $\Pi$ is a set of prime numbers. Head purities and $\varepsilon$-purities, evidently, are inductive. In the paper we show that every inductive purity in the category of all torsion-free Abelian groups is a certain $\Pi$-servancy, every inductive purity in the category of all periodic Abelian groups is a certain $\varepsilon$-purity, and every inductive purity in the category of all Abelian groups is the intersection of a certain $\varepsilon$-purity and a certain Head purity.
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