1. Easily proven observations

(1) $P$ is well partially ordered if and only if every infinite sequence in $P$ has a (weakly) increasing subsequence.

(2) If $P = P_1 \cup P_2$ and $P_1, P_2$ are well partially ordered, then so is $P$.

(3) If $P = P_1 \times P_2$ and $P_1, P_2$ are well partially ordered, then so is $P$.

(4) If $P$ is well partially ordered and $\phi: P \rightarrow Q$ is an isotone map of $P$ onto $Q$, then $Q$ is well partially ordered.

(5) If $P$ is well partially ordered, then Downsets$(P)$ has d.c.c. Thus any collection of downsets has a minimal member.