$P$-spectra of Abelian groups

We consider four types of subgroups of Abelian groups: arbitrary subgroups ($s$-subgroups), algebraically closed subgroups ($a$-subgroups), pure subgroups ($p$-subgroups), and elementary subgroups ($e$-subgroups). A language $L(X)$ is an extension of a language $L$ by a set $X$ of constants. A language $L_P$ is an extension of $L$ by one unary predicate symbol $P$. For $i\in\{s,a,p,e\}$ let $\Delta_i$ consist of sentences in $L_P$ , where $L$ is the language of Abelian groups, expressing the fact that a predicate $P$ defines a subgroup of type $i$. For a complete theory $T$ of Abelian groups and for $i\in\{s,a,p,e\}$, a cardinal function assigning a cardinal $\lambda$ the supremum of the number of completions of sets $(T^*\cup\{P(a)\mid a\in X\}\cup\Delta_i)$ in the language $(L(X))_P$ for complete extensions $T^*$ of $T$ in the language $L(X)$ for sets $X$ of cardinality $\lambda$ is called the $(P,i)$- spectrum of the theory $T$. For each $i\in\{s,a,p,e\}$, we describe all possible $(P,i)$-spectra of complete theories of Abelian groups.