= Solution
<Idempotence> concerns applying an operation to the same \i[whole value]. It says more than saying that the equal-entry pairs are unchanged. Choose the <idempotent> <binary operation> `min`. Then
$$
[1,2]\,X_{\min}\,[1,2]=[1,1,1,2]\ne[1,2].
$$
More generally, a finite <list> of length $m$ paired with itself produces $m^2$ entries, so length alone contradicts <idempotence> when $m>1$. Therefore
$$
\boxed{\text{False: the lifted operation need not be idempotent.}}
$$
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