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 ${\mathcal{P}}_{\mathrm{fin}}(H)$ 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.

arXiv comment34 pp., no figures. Final version to appear in Journal of Algebra

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Cited by in corpus (27)

• On strongly primary monoids, with a focus on Puiseux monoids
• Structural properties of subadditive families with applications to factorization theory
• Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• An Abstract Factorization Theorem and Some Applications
• On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups
• Long sets of lengths with maximal elasticity
• Factorization under Local Finiteness Conditions
• Abstract Factorization Theorems with Applications to Idempotent Factorizations
• Lattices over Bass rings and graph agglomerations
• On noncommutative bounded factorization domains and prime rings
• A conjecture by Bienvenu and Geroldinger on power monoids
• On half-factoriality of transfer Krull monoids
• On the finiteness of certain factorization invariants
• On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups
• On power monoids and their automorphisms
• On the arithmetic of power monoids
• Sidon set systems
• Semigroups of ideals and isomorphism problems
• Factorization in monoids and rings
• A reciprocity on finite abelian groups involving zero-sum sequences
• Torsion groups and the Bienvenu--Geroldinger conjecture
• A realization result for systems of sets of lengths
• Factorization Theory in Commutative Monoids
• On the automorphisms of the power semigroups of a numerical semigroup
• On the ideal of some sumset semigroups
• Factorization in Finitely-Presented Monoids