Employing the language of representations (of posets, see \textit{D. M. Arnold}'s book [Abelian groups and representations of finite partially ordered sets (2000; Zbl 0959.16011)]), for a fixed prime \(p\), the category \(CD^1(T,p)\) which consists of all pairs \(V=(C,D)\) such that \(C\) is completely decomposable of finite rank with critical type \(T\) and \(D\) is a completely decomposable subgroup of \(C\) such that \(p^nC\subseteq D\) for some \(n\geq 1\). The main result is Theorem 4: If the set \(T\) of critical types is an antichain, then indecomposable objects in \(CD^1(T,p)\) have rank 1. If \(T\) is not an antichain, for each finite dimensional \(\mathbb{Z}/p\mathbb{Z}\)-algebra \(A\) there is an object \(V\in CD^1(T,p)\) such that there is a ring epimorphism \(\varphi\colon\text{End}(V)\to A\). Moreover, \(\varphi\) is the `modulo \(p\)' map followed by an epimorphism with nilpotent kernel. Corollary. The stacked basis theorem holds in \(CD^1(T,p)\) if and only if \(T\) is an antichain.