On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups

TL;DR

This paper explores the arithmetic properties of power monoids formed from finite subsets of monoids, focusing on factorizations, atomicity, and minimal factorizations, with significant connections to sumset decompositions in cyclic groups.

Contribution

It introduces and analyzes the concepts of atomicity, bounded factorizations, and minimal factorizations in power monoids, extending factorization theory to structures with idempotents and linking to arithmetic combinatorics.

Findings

$ ext{P}_{ ext{fin},1}(H)$ is atomic iff $1_H e x^2 e x$ for all $x$ in $H\setminus\{1_H\}$

$ ext{P}_{ ext{fin},1}(H)$ is BF iff $H$ is torsion-free

Conditions for $ ext{P}_{ ext{fin}, imes}(H)$ to be BmF, HmF, or minimally factorial

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.