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).

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    $\begingroup$ 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. $\endgroup$ Apr 14, 2021 at 18:06
  • $\begingroup$ @NeilStrickland do you mind explaining why $A\angle\emptyset$? Wouldn't that depend on the initial order on $X$? $\endgroup$
    – tsm
    Apr 14, 2021 at 20:04
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    $\begingroup$ 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. $\endgroup$ Apr 14, 2021 at 20:15
  • $\begingroup$ 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. $\endgroup$ Apr 15, 2021 at 0:50
  • $\begingroup$ @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. $\endgroup$
    – tsm
    Apr 15, 2021 at 4:32


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