Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics

Abstract

We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of Baeth and Smertnig's work on the factorization theory of non-commutative, but cancellative monoids [J. Algebra 441 (2015), 475-551]. Then, we bring in power monoids and, applying the abstract machinery developed in the first part, we undertake the study of their arithmetic. More in particular, let $H$ be a multiplicatively written monoid. The set $\mathcal P_{\rm fin}(H)$ of all non-empty finite subsets of $H$ is naturally made into a monoid, which we call the power monoid of $H$ and is non-cancellative unless $H$ is trivial, by endowing it with the operation $(X,Y) \mapsto \{xy: (x,y) \in X \times Y\}$. Power monoids are, in disguise, one of the primary objects of interest in arithmetic combinatorics, and here for the first time we tackle them from the perspective of factorization theory. Proofs lead to consider various properties of finite subsets of $\mathbf N$ that can or cannot be split into a sumset in a non-trivial way, which gives rise to a rich interplay with additive number theory.