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 $X \in \mathcal P_{{\rm fin},0}(\mathbf N)$ 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*, but they are just the atoms of $\mathcal P_{{\rm fin},0}(\mathbf N)$ in the sense of [factorization theory][1]. There are [deep conjectures][2] related to primitive sets in finite fields, but 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 $A+B = A+C$?

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.


  [1]: https://www.crcpress.com/Non-Unique-Factorizations-Algebraic-Combinatorial-and-Analytic-Theory/Geroldinger-Halter-Koch/p/book/9781584885764
  [2]: http://dx.doi.org/10.4064/aa155-1-4