 Hoare domination preorder

For subsets of a preorder, Hoare domination means every source element has some greater or equal target element: $S{\le }_{H}T⟺\forall s\in S\exists t\in T(s\le t)$. It is reflexive and transitive. If the base is a well-quasi-ordering, its finite subsets are well-quasi-ordered by this relation; arbitrary subsets need not be, as the Rado order shows.