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Let $H$ be a multiplicatively written monoid with identity $1_H$ (inparticular, a group). We denote by $\mathcal P_{\rm fin,\times}(H)$ the monoidobtained by endowing the collection of all finite subsets of $H$ containing aunit with the operation of setwise multiplication $(X,Y) \mapsto \{xy: x \in X,y \in Y\}$; and study fundamental features of the arithmetic of this andrelated structures, with a focus on the submonoid, $\mathcalP_{\text{fin},1}(H)$, of $\mathcal P_{\text{fin},\times}(H)$ consisting of allfinite subsets $X$ of $H$ with $1_H \in X$.  Among others, we prove that $\mathcal{P}_{\text{fin},1}(H)$ is atomic (i.e.,each non-unit is a product of irreducibles) iff $1_H \ne x^2 \ne x$ for every$x \in H \setminus \{1_H\}$. Then we obtain that$\mathcal{P}_{\text{fin},1}(H)$ is BF (i.e., it is atomic and every element hasfactorizations of bounded length) iff $H$ is torsion-free; and show how totransfer these conclusions to $\mathcal P_{\text{fin},\times}(H)$.  Next, we introduce "minimal factorizations" to account for the fact thatmonoids may have non-trivial idempotents, in which case standard definitionsfrom Factorization Theory degenerate. Accordingly, we obtain conditions for$\mathcal P_{\text{fin},\times}(H)$ to be BmF (meaning that each non-unit hasminimal factorizations of bounded length); and for$\mathcal{P}_{\text{fin},1}(H)$ to be BmF, HmF (i.e., a BmF-monoid where allthe minimal factorizations of a given element have the same length), orminimally factorial (i.e., a BmF-monoid where each element has an essentiallyunique minimal factorization). Finally, we prove how to realize certainintervals as sets of minimal lengths in $\mathcal P_{\text{fin},1}(H)$.  Many proofs come down to considering sumset decompositions in cyclic groups,so giving rise to an intriguing interplay with Arithmetic Combinatorics.

On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups