This paper by R. P. Dilworth focuses on a decomposition theorem for partially ordered sets (posets). The main theorem states that if every set of \( k + 1 \) elements in a poset \( P \) is dependent while at least one set of \( k \) elements is independent, then \( P \) can be expressed as a set sum of \( k \) disjoint chains. The author provides a proof for the finite case and extends it to the general case using transfinite arguments. The theorem has applications in proving the Radó-Hall theorem on representatives of sets and an embedding theorem for finite distributive lattices. The proof involves analyzing the structure of independent subsets and disjoint chains within the poset, demonstrating that the maximal number of independent elements in certain subsets is less than \( k \), leading to the desired decomposition.This paper by R. P. Dilworth focuses on a decomposition theorem for partially ordered sets (posets). The main theorem states that if every set of \( k + 1 \) elements in a poset \( P \) is dependent while at least one set of \( k \) elements is independent, then \( P \) can be expressed as a set sum of \( k \) disjoint chains. The author provides a proof for the finite case and extends it to the general case using transfinite arguments. The theorem has applications in proving the Radó-Hall theorem on representatives of sets and an embedding theorem for finite distributive lattices. The proof involves analyzing the structure of independent subsets and disjoint chains within the poset, demonstrating that the maximal number of independent elements in certain subsets is less than \( k \), leading to the desired decomposition.