Products of two atoms in Krull monoids and arithmetical characterizations of class groups

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let sans-serifD(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most sans-serifD(G) atoms. We study this extremal case and consider the set scriptU{2,D(G)}(H) defined as the set of all l∈double-struckN with the following property: there are two atoms u,v∈H such that uv can be written as a product of l atoms as well as a product of sans-serifD(G) atoms. If G is cyclic, then scriptU{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) scriptU{2,D(G)}(H)=[2,D(G)]∖{3}. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank 2 prime power order group uniquely characterizes the group.

Monoids of modules and arithmetic of direct-sum decompositions

“…A survey of these problems can be found in [37,Sections 7.1 and 7.2]. For recent progress, see [63], [62], and [10]. In Corollary 6.8 we exhibit that for many Krull monoids stemming from the Module Theory of Sections 4 and 5, the system of sets of lengths and the behavior of absolutely irreducible elements characterizes the class group of these monoids.…”

Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group

“…These constants, especially ρ k (H) are those that received most interest so far. The sets U k (H) were introduced by Chapman and Smith [9], and the generalization U M (H) appeared in [3].…”

“…Considering U {k,ρ k (H)} (H) is thus an interesting extremal case. This problem was investigated recently by Baginski, Geroldinger, Grynkiewicz, Philipp [3], with a focus on groups of rank two; again, it is important to know the structure of minimal zero-sum sequences of maximal length.…”

“…The preceding results yield the following corollary. We mention that this corollary allows to fill what we believe to be a minor gap in the proof of [3, Theorem 5.6]; it can be invoked there instead of [3,Theorem 4.2] (that is the result we recalled as Theorem 4.13).…”

The Catenary Degree of Krull Monoids Ii

“…For recent work in this direction we refer to [28,29,6] or to [19,Section 7.3], [15] for an overview. It turns out the extremal values of (H) discussed in Theorem 1.1 are the main tool to derive an arithmetical characterization of the associated groups.…”