Consider a poset $(X, \geq)$. Let's define a new relation $\succsim$ on subsets of $X$: for $A, B\subseteq X$, say $A\succsim B$ if for any $a\in A$ and any $b\in B$, we have $a\geq b$.

Does such a relation already have a name? Note that the relation is not a preorder (reflexivity fails).

induced relation, although not an induced order, and althoughinducedmay have other meanings. You may also call it aderived relation, but there is a risk of confusion with derivatives. Do you need more? May you please explain why? It may help providing better answers. $\endgroup$