Abstract
ABSTRACTLet H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a∈H can be written as a product of irreducible elements, say , and the number of factors k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H′ be a further transfer Krull monoid over a finite abelian group G′ and suppose that ℒ(H) = ℒ(H′). We prove that, if with r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G′ are isomorphic.
Highlights
Introduction and main resultLet H be an atomic unit-cancellative monoid
Each nonunit a ∈ H can be written as a product of irreducible elements, say a = u1 . . . uk, and the number of factors k is called the length of the factorization
It is classical that the system L(H) = {L(a) | a ∈ H} of all sets of lengths depends only on the group G, and a standing conjecture states that the system L(H) is characteristic for the group G
Summary
Let H be an atomic unit-cancellative monoid. each non-unit a ∈ H can be written as a product of atoms, and if a = u1 . . . uk with atoms u1, . . . , uk of H, k is called the length of the factorization. Note that the system of sets of lengths of H depends only on the class group G. Zero-sum theoretical invariants (such as the Davenport constant or the cross number) and the associated inverse problems play a crucial role. It is not surprising that most a rmative answers to the Characterization Problem so far have been restricted to those groups where we have a good understanding of the Davenport constant. Let H be a transfer Krull monoid over a nite abelian group G with D(G) ≥ 4. Suppose G ∼= Cnr with r, n ∈ N and L(H) = L(H′), where H′ is a further transfer Krull monoid over a nite abelian group G′. Throughout the paper, let G ∼= Cn1 ⊕ . . . ⊕ Cnr be a nite abelian group with D(G) ≥ 4, where r, n1, . . . , nr ∈ N and 1 < n1 | . . . | nr
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