This paper continues work on the problem to determine classes of finite rank Butler groups up to quasi-isomorphism and near-isomorphism, in order to find out their representation type. This leads to representation theory of posets over rings. The authors use this category equivalence in order to carry out their work on the rep-side. The following category turns out to be crucial: Let \(R\) be the ring of integers localized at some prime \(p\). Then \(\text{rep}_{fpure}(S,R)\) for some finite poset \(S\) consists of representations \((U_0,U_i:i\in S)\) with \(U_0\) a finitely generated \(R\)-module and \(U_i\subseteq_*U_0\) (summands) of \(U_0\). If \(j\in\mathbb{N}\), then \(\text{rep}_{fpure}(S,R,j)\) is the full subcategory with \(p^jU_0\subseteq\sum_{i\in S} U_i\).
In the first part of the paper wildness is shown in many cases (e.g. for any \(\text{width}(S)\geq 3\)). The case \(\text{width}(S)=2\) is not fully settled. These rep-results are applied to Butler groups and complicated calculations illustrate the case \(S_3\) of antichains of width 3 or posets of width 2.