4 research outputs found
Some recent results and open problems on sets of lengths of Krull monoids with finite class group
Some of the fundamental notions related to sets of lengths of Krull monoids
with finite class group are discussed, and a survey of recent results is given.
These include the elasticity and related notions, the set of distances, and the
structure theorem for sets of lengths. Several open problems are mentioned
A characterization of class groups via sets of lengths
Let be a Krull monoid with class group such that every class contains
a prime divisor. Then every nonunit can be written as a finite
product of irreducible elements. If , with
irreducibles , then is called the length of the
factorization and the set of all possible is called the set
of lengths of . It is well-known that the system depends only on the class group . In the present
paper we study the inverse question asking whether or not the system is characteristic for the class group. Consider a further Krull monoid
with class group such that every class contains a prime divisor and
suppose that . We show that, if one of the
groups and is finite and has rank at most two, then and are
isomorphic (apart from two well-known pairings).Comment: The current version is close to the one to appear in J. Korean Math.
Soc., yet it contains a detailed proof of Proposition 2.4. The content of
Chapter 4 of the first version had been split off and is presented in ' A
characterization of Krull monoids for which sets of lengths are (almost)
arithmetical progressions' by the same authors (see hal-01976941 and
arXiv:1901.03506
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)]{set minus}{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. © 2013 Elsevier Ltd