Hoare domination preorder ($S{\le }_{H}T$)

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.

Finite subsets of a well-quasi-ordering, compared by Hoare domination preorder, form a well-quasi-ordering. Enumerate each finite subset as a word and apply Higman lemma: a subsequence embedding with increased letters supplies a domination witness for every source element. This does not assert the same result for the full power set.