Rado order ($R$)

The Rado order has domain $R=\{(m,n)\in {\mathbb{N}}^2:m<n\}$ and $(m,n){\le }_R(k,l)$ iff either $m=k$ and $n\le l$, or $n<k$. It is a well-quasi-ordering: either one row recurs infinitely in a sequence, or the first coordinates are unbounded and yield a cross-row comparison. Its infinite rows form an infinite antichain under Hoare domination preorder, showing that arbitrary-subset lifting need not preserve well-quasi-ordering.