Sets of minimal distances and characterizations of class groups of Krull monoids

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set L(a) of all possible factorization lengths k is called the set of lengths of a. There is a constant M ∈ N such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ * (H ), where * (H ) denotes the set of minimal distances of H . We study the structure of * (H ) and establish a characterization when * (H ) is an interval. The system L(H ) = {L(a) | a ∈ H } of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system L(H ) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to C r n with r, n ∈ N and * (H ) is not an interval.

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

“…The first and so far only groups, for which the Characterization Problem was solved whereas the Davenport constant is unknown, are groups of the form , where r , n ∈ ℕ and 2 r < n −2 (this is done by Geroldinger and Zhong [ 12 ] and Zhong [ 27 ]), which use a deep characterization of the structure of Δ *( G ). In this paper, we go on to study groups of the form and obtain the following theorem.…”

“…5. If Δ 1 ( G ) is an interval, then by 4 which implies that Δ *( G ) is an interval by Zhong [ 27 , Theorem 1.1.2]. If Δ *( G ) is an interval, then Δ 1 ( G ) is an interval by 1..…”

“…If Δ *( G ) is an interval, then Δ 1 ( G ) is an interval by 1.. It follows by Zhong [ 27 , Theorem 1.1.2] that Δ *( G ) is an interval if and only if r ( G )+ k ≥exp( G )−1 or .…”

Long Sets of Lengths With Maximal Elasticity

“…The set ∆(G) is an interval by [22], but ∆ 1 (G) is far from being an interval ( [32]). A characterization when ∆ 1 (G) is an interval can be found in [37]. We have max ∆ 1 (G) = max{r(G) − 1, exp(G) − 2} (for |G| ≥ 3, by [24]).…”

On congruence half-factorial Krull monoids with cyclic class group

“…The investigations so far mainly focused on the first type of questions, due to the fact that a (partial) solution to it is a precondition for even beginning to consider the second one; see [24,35] for some initial results. Moreover, we point out that it was typical -we do so as well -to focus on (relatively) large d; mainly, since they are the more interesting ones in understanding the arithmetic and since they are more relevant in applications, for example, to the problem of giving arithmetic characterizations of the class group (see, e.g., [25,36,23,44]).…”