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

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let D(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most D(G) atoms. We study this extremal case and consider the set U{2,D(G)}(H) defined as the set of all l∈N 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 D(G) atoms. If G is cyclic, then U{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) U{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.