Let $\mathcal P_{{\rm fin},0}(\mathbf N)$ be the monoid of all finite subsets of $\mathbf N$ containing $0$ with the operation of set addition $$ (X, Y) \mapsto X + Y := \{x+y: x \in X \text{ and }y \in Y\}. $$ We call $\mathcal P_{{\rm fin},0}(\mathbf N)$ the restricted power monoid of $(\mathbf N, +)$, and denote by $\mathcal A(\mathcal P_{{\rm fin},0}(\mathbf N))$ the set of all $X \in \mathcal P_{{\rm fin},0}(\mathbf N) \setminus \bigl\{\{0\}\bigr\}$ for which there do not exist $A, B \in \mathcal P_{{\rm fin},0}(\mathbf N)$ with $|A|,|B| \ge 2$ and $X = A+B$. In arithmetic combinatorics, the elements of $\mathcal A(\mathcal P_{{\rm fin},0}(\mathbf N))$ are usually called primitive (or irreducible) sets, and they are just the atoms (or irreducible elements) of the monoid $\mathcal P_{{\rm fin},0}(\mathbf N)$ in the sense of factorization theory. There are deep conjectures related to primitive sets in finite fields. However, my question here is much more basic:
Q. Is it true that for every $A \in \mathcal A(\mathcal P_{{\rm fin},0}(\mathbf N))$ there exist $B, C \in \mathcal A(\mathcal P_{{\rm fin},0}(\mathbf N))$ such that $B \ne C$, but $A+B = A+C$ (resp., $A \ne B$ and $A+C=B+C$)?
This emerged from a failed attempt I was making with a colleague to prove, by using a certain property of weak transfer homomorphisms with respect to the catenary degree, that each member in a certain class of (non-cancellative) monoids are not transfer Krull monoids in the sense of Baeth and Smertnig [J. Algebra 441 (2015), 475–551]. In the end, we have been lucky and found a different (and almost trivial) proof, but in principle I'm still interested in the above question.
