Relating Subsets of a Poset,and a Partition Theorem for WQOs

We point out some basic properties of the partial ordering which a poset P induces on its power set, defining AB to mean that every element of A lies below some element of B. One result is that if P is a WQO then P decomposes uniquely into finitely many indivisible sets A₁,...,A_n (that are essential parts of P in the sense that PPA_i).