A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Abstract

Let 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.

Which sets are sets of lengths in all numerical monoids?

“…The set of distances ∆(H) is an interval with min ∆(H) = 1 ([21]) whose maximum is unknown in general ( [22]) (this is in contrast to the fact that in finitely generated Krull monoids any finite set ∆ with min ∆ = gcd ∆ may occur as set of distances [18]). The standing conjecture is that the system of sets of lengths is characteristic for the group (see [15] for a survey, and [17,23,32,31] for recent progress). This means that L(H) = L(H ′ ) for all Krull monoids H ′ having prime divisors in all classes and class group G ′ not being isomorphic to G (here we need |G| ≥ 4).…”

A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions

“…(b) G is cyclic of order |G| ≤ 6 or isomorphic to a subgroup of C 3 2 or isomorphic to a subgroup of C 2 3 . A central topic in the study of sets of lengths is the Characterization Problem (for recent progress see [4,17,23,31,30]) which reads as follows:…”

On the algebraic and arithmetic structure of the monoid of product-one sequences II

“…It asks whether for each two transfer Krull monoids H and H ′ over groups G and G ′ their system of sets of lengths coincide if and only if G and G ′ are isomorphic. The standing conjecture is that this holds true for all finite abelian groups G (apart from two trivial exceptional pairings) and we refer to [20,24,36,35] for recent progress in this direction.…”