# What do you call such a relation between subsets in a poset

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).

• In the paper arxiv.org/abs/1907.07801 we used the notation $B\angle A$ for this, but I have not seen that anywhere else. Note that $A\angle\emptyset$ and $\emptyset\angle B$ for vacuous reasons, so transitivity fails as well as reflexivity. Apr 14, 2021 at 18:06
• @NeilStrickland do you mind explaining why $A\angle\emptyset$? Wouldn't that depend on the initial order on $X$?
– tsm
Apr 14, 2021 at 20:04
• To say that $A\angle\emptyset$ means that for all $a\in A$ and $b\in\emptyset$ we have $a\leq b$. But there are no possible cases for $b$, so this is vacuously satisfied. Apr 14, 2021 at 20:15
• I would just say that it is an induced relation, although not an induced order, and although induced may have other meanings. You may also call it a derived 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. Apr 15, 2021 at 0:50
• @MatthieuLatapy I am comparing the relation in question to the strong-set order (SSO). I am wondering if it has a commonly used name, so that I avoid confusion for the reader.
– tsm
Apr 15, 2021 at 4:32