Let $P$ be a poset and define $Q$ to be the poset of all finite multisets containing elements of $P$ where $A \leq B$ iff there exists an injection $f:A \rightarrow B$ such that for all $a \in A$, $a \leq f(a)$. Is there a name for this?

For example, take the natural numbers with the normal ordering. Then the new poset would be the Young lattice.



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