Let $\mathcal{A} \subset \{ X \in \mathcal P \left({Z}\right) | X = X + C \}$ and $B \in \{ X \in \mathcal P \left({Z}\right) | X = X + C \}$ where + is the usual Minkowski sum.
Define the supremum of a family to be $$\text{sup} \mathcal{A} = \bigcup_{A\in \mathcal{A}} A$$
Then show that the following are equivalent:
$\text{(a) sup} (\mathcal{A}+B) = \text{sup} (\mathcal{A})+B$ $\text{(b) For each } A, B \in \{ X \in \mathcal P \left({Z}\right) | X = X + C \}$, the set $\{ D \in \{ X \in \mathcal P \left({Z}\right) | X = X + C \} | B + D \subset A\}$ has a greatest element with respect to $\subset$.
I found this question in the paper "Set Optimization—A Rather Short Introduction" by Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff and Carola Schrage.
The (a) $\implies$ (b) is provided in the text, but the converse direction got me stumped. Any ideas on how to approach the converse direction would be greatly appreciated. Thanks!