Project leader:
Alfred Geroldinger
FWF project number:
P 28864-N35


Alfred Geroldinger, Ao.Univ.-Prof. Mag. Dipl.-Ing. Dr.
+43 (0)316 380 - 5154
Heinrichstraße 36, Room 526 (4th floor)
Qinghai Zhong, Dr.
+43 (0)316 380 - 5155
Heinrichstraße 36, Room 528 (4th floor)

All participants are located at the

Project summary

Let $H$ be a Krull monoid with finite class group $G$ such that each class contains a prime divisor (this setting includes holomorphy rings in global fields). Then every nonunit $a \in H$ can be written as a finite product of atoms (irreducible elements). If $a = u_1 \cdot \ldots \cdot u_k$ with atoms $u_1, \ldots, u_k \in H$, then $k$ is called the length of the factorization. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$, and this is a finite set of positive integers. We consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H\}$ of sets of lengths of $H$ (for convenience, we set $\mathsf L (a) = \{0\}$ if $a$ is a unit in $H$). The Krull monoid $H$ is factorial if and only if the class group $G$ is trivial. Furthermore, $H$ is half-factorial (i.e., $|L|=1$ for all $L \in \mathcal L (H)$) if and only if $|G| \le 2$. If $|G| \ge 3$, then for every $N \in \mathbb N$ there is a set $L \in \mathcal L (H)$ such that $|L| \ge N$.

The system $\mathcal L (H)$ depends only on the class group $G$. To make this precise, consider the set $\mathcal B (G)$ of zero-sum sequences over $G$. By a sequence over $G$, we mean a finite sequence of terms from $G$ where repetition is allowed and the order is disregarded, and we say that $S$ is a zero-sum sequence if its terms sum up to zero. Defining an operation as the concatenation of sequences we obtain a monoid structure on $\mathcal B (G)$. Indeed, $\mathcal B (G)$ is a Krull monoid with class group isomorphic to $G$ (provided that $|G| \ge 3$) and every class contains a prime divisor. Moreover, the systems of sets of lengths of $H$ and that of $\mathcal B (G)$ coincide. Thus $\mathcal L (H) = \mathcal L \big( \mathcal B (G) \big)$ can be studied with methods from Zero-Sum Theory, a flourishing subfield of Additive Combinatorics.

The Structure Theorem for Sets of Lengths states that every $L \in \mathcal L (G)$ is an almost arithmetical multiprogression with universal bounds for all parameters controlling these multiprogressions. It is a main goal of the present project to study the involved parameters (such as the possible differences of the multiprogressions) in terms of classical zero-sum invariants (such as the Davenport constant of $G$) or even in terms of the group invariants. All work on these parameters will be done with a view towards the Characterization Problem, a main open question in this area. Indeed, let $G$ and $G'$ be two finite abelian groups with $|G|\ge 4$ and $|G'|\ge 4$ and suppose that $\mathcal L \big( \mathcal B (G) \big) = \mathcal L \big( \mathcal B (G') \big)$. Does it follow that $G$ and $G'$ are isomorphic? The answer is affirmative, among others, if $G$ or $G'$ has rank at most two. However, the Characterization Problem is far open in general, and it will be in the focus of the the present project.


  1. A. Geroldinger
    Sets of lengths.
    Amer. Math. Monthly, 123 (2016), 960-988.
  2. A. Geroldinger, W. Schmid and Qinghai Zhong
    Systems of Sets of Lengths: Transfer Krull Monoids versus Weakly Krull Monoids.
    In Rings, Polynomials, and Modules, 191-235. Springer, 2017.
  3. A. Geroldinger and Qinghai Zhong
    Long sets of lengths with maximal elasticity.
    Canadian Journal of Mathematics, to appear.
  4. Qinghai Zhong
    Sets of minimal distances and characterizations of class groups of Krull monoids.
    The Ramanujan Journal, to appear.
  5. A. Geroldinger and Emil Daniel Schwab
    Colloquium Math., to appear.
  6. Qinghai Zhong
    A characterization of finite abelian groups via sets of lengths in transfer Krull monoids.
    Communications in Algebra, to appear.
  7. A. Geroldinger and W. Schmid
    A realization theorem for sets of lengths in numerical monoids.
    Forum Math., to appear.
  8. A. Geroldinger and W. Schmid
    Which sets are sets of lengths in all numerical monoids.