#### 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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