On the arithmetic of power monoids and sumsets in cyclic groups

Abstract

Let $H$ be a multiplicatively written monoid with identity $1_H$ (in particular, a group). We denote by $\mathcal P_{\rm fin,\times}(H)$ the monoid obtained by endowing the collection of all finite subsets of $H$ containing a unit 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 and related structures, with a focus on the submonoid, $\mathcal P_{\text{fin},1}(H)$, of $\mathcal P_{\text{fin},\times}(H)$ consisting of all finite 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 has factorizations of bounded length) iff $H$ is torsion-free; and show how to transfer these conclusions to $\mathcal P_{\text{fin},\times}(H)$. Next, we introduce "minimal factorizations" to account for the fact that monoids may have non-trivial idempotents, in which case standard definitions from Factorization Theory degenerate. Accordingly, we obtain conditions for $\mathcal P_{\text{fin},\times}(H)$ to be BmF (meaning that each non-unit has minimal factorizations of bounded length); and for $\mathcal{P}_{\text{fin},1}(H)$ to be BmF, HmF (i.e., a BmF-monoid where all the minimal factorizations of a given element have the same length), or minimally factorial (i.e., a BmF-monoid where each element has an essentially unique minimal factorization). Finally, we prove how to realize certain intervals 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.