In this talk, we first explore the underlying mathematical structure of edge subsets on a finite directed acyclic graph in using a lattice-theoretic approach. We prove that a collection of edge subsets with certain conditions, associated with the corresponding “cut-separating” partial order, forms a (semi-)lattice. The bottom and top thus derived generalize the concept of the primary minimum cut introduced by Guang and Yeung (2018) and hence we provide a new way from a lattice-theoretic point of view to understand the primary minimum cut and to justify its existence and uniqueness. We further develop efficient algorithms to find the top and the bottom, whose computational complexity is in a linear time of the number of edges in the graph. The introduced concepts and obtained results regarded as a bridge connect graph theory and lattice theory, which appear to be of fundamental interest in graph theory, lattice theory, and even beyond. In addition, by applying the approach of the edge-subset (semi-)lattice, we obtain an improved upper bound on the minimum required field size for the existence of linear network error correction (LNEC) codes. In LNEC coding, the minimum required field size for the existence of LNEC (MDS) codes is an open problem not only of theoretical interest but also of practical importance. We quantify the improvement over the existing results by both theoretical analysis and numerical simulations and thus show that the improvement is in general significant.