On the arithmetic of power monoids

Abstract

Given a monoid $H$ (written multiplicatively), the family ${\mathcal{P}}_{\mathrm{fin},1}(H)$ of all non-empty finite subsets of $H$ containing the identity element $1_H$ is itself a monoid, called the reduced finitary power monoid of $H$, under the operation of setwise multiplication induced by $H$. We investigate the arithmetic of ${\mathcal{P}}_{\mathrm{fin},1}(H)$ from the perspective of minimal factorizations into irreducibles, paying particular attention to the potential presence of non-trivial idempotents. Among other results, we provide necessary and sufficient conditions on $H$ for ${\mathcal{P}}_{\mathrm{fin},1}(H)$ to admit unique minimal factorizations. Our results generalize and shed new light on recent developments on the topic.

Version accepted for publication on Journal of Algebra

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Cited by in corpus (3)

• On the automorphisms of the power semigroups of a numerical semigroup
• Torsion groups and the Bienvenu--Geroldinger conjecture
• Factorization in Finitely-Presented Monoids