Atoms of the relative block monoid

Abstract

Let G be a finite abelian group with subgroup H and let F(G) denote the free abelian monoid with basis G. The classical block monoid B(G) is the collection of sequences in F(G) whose elements sum to zero. The relative block monoid BH(G), defined by Halter-Koch, is the collection of all sequences in F(G) whose elements sum to an element in H. We use a natural transfer homomorphism θ : BH(G) → B(G/H) to enumerate the irreducible elements of BH(G) given an enumeration of the irreducible elements of B(G/H). MSC 2000: 11P70, 20M14