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 $A\le B$ 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 $A_1,\dots,A_n$ (that are essential parts of $P$ in the sense that $P\not\le P\sm A_i$).

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