Vol. 2, No. 1, 2009

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Atoms of the relative block monoid

Nicholas Baeth and Justin Hoffmeier

Vol. 2 (2009), No. 1, 29–36
Abstract

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

Keywords
zero-sum sequences, block monoids, finite abelian groups
Mathematical Subject Classification 2000
Primary: 11P70, 20M14