We explore the connections between clusters and least common ancestors (LCAs) in directed acyclic graphs (DAGs), focusing on the interplay between so-called ℐ-lca-relevant DAGs and DAGs with the ℐ-lca-property. Here, ℐ denotes a set of integers. In ℐ-lca-relevant DAGs, each vertex is the unique LCA for some subset A of leaves of size |A|∈ℐ, whereas in a DAG with the ℐ-lca-property there exists a unique LCA for every subset A of leaves satisfying |A|∈ℐ. We elaborate on the difference between these two properties and establish their close relationship to pre-ℐ-ary and ℐ-ary set systems. This, in turn, generalizes results established for (pre-) binary and k-ary set systems. Moreover, we build upon recently established results that use a simple operator ⊖, enabling the transformation of arbitrary DAGs into ℐ-lca-relevant DAGs. This process reduces unnecessary complexity while preserving key structural properties of the original DAG. The set ℭ_G consists of all clusters in a DAG G, where clusters correspond to the descendant leaves of vertices. While in some cases ℭ_H=ℭ_G when transforming G into an ℐ-lca-relevant DAG H, it often happens that certain clusters in ℭ_G do not appear as clusters in H. To understand this phenomenon in detail, we characterize the subset of clusters in ℭ_G that remain in H for DAGs G with the ℐ-lca-property. Furthermore, we show that the set W of vertices required to transform G into H=G⊖W is uniquely determined for such DAGs. This, in turn, allows us to show that the “shortcut-free” version of the transformed DAG H is always a tree or a galled-tree whenever ℭ_G represents the clustering system of a tree or galled-tree and G has the ℐ-lca-property. In the latter case ℭ_H=ℭ_G always holds.