3.21 (1969)

Let $\mathfrak{K}$ be the class of one-based models of signature $\sigma$, $\vartheta$ a property that makes sense for models in $\mathfrak{K}$, and $\mathfrak{K}_0$ the class of two-based models whose first base is a set $M$ taken from $\mathfrak{K}$ and whose second base consists of all submodels of $M$ with property $\vartheta$ and whose signature consists of the symbols in $\sigma$ together with $\in$ and $\subseteq$ in the usual set-theoretic sense. The elementary theory of $\mathfrak{K}_0$ is called the element-$\vartheta$-submodel theory of $\mathfrak{K}$. Is any of the following theories decidable:
$\qquad$ a) the element-pure-subgroup theory of abelian groups?
$\qquad$ b) the element-pure-subgroup theory of abelian torsion-free groups?
$\qquad$ c) the element-$\vartheta$-submodel theory of abelian groups, when the set of $\vartheta$-subgroups is linearly ordered by inclusion?