Atoms of the relative block monoid

Abstract

Let $G$ be a finite abelian group with subgroup $H$ and let $\mathcal{ℱ}\left(G\right)$ denote the free abelian monoid with basis $G$. The classical block monoid $\mathcal{ℬ}\left(G\right)$ is the collection of sequences in $\mathcal{ℱ}\left(G\right)$ whose elements sum to zero. The relative block monoid ${\mathcal{ℬ}}_H\left(G\right)$, defined by Halter-Koch, is the collection of all sequences in $\mathcal{ℱ}\left(G\right)$ whose elements sum to an element in $H$. We use a natural transfer homomorphism $\theta :{\mathcal{ℬ}}_H\left(G\right)\to \mathcal{ℬ}\left(G∕H\right)$ to enumerate the irreducible elements of ${\mathcal{ℬ}}_H\left(G\right)$ given an enumeration of the irreducible elements of $\mathcal{ℬ}\left(G∕H\right)$.