- Start:
- 7.3.2016
- End:
- 6.7.2019
- Project leader:
- Alfred Geroldinger
- FWF project number:
- P 28864-N35

# Participants

Professor

- E-Mail:
- alfred.geroldinger@uni-graz.at
- Phone:
- +43 (0)316 380 - 5154
- Office:
- Heinrichstraße 36, Room 526 (4th floor)
- Homepage:
- http://imsc.uni-graz.at/geroldinger

Postdoc

- E-Mail:
- qinghai.zhong@uni-graz.at
- Phone:
- +43 (0)316 380 - 5155
- Office:
- Heinrichstraße 36, Room 528 (4th floor)
- Homepage:
- http://qinghai-zhong.weebly.com

All participants are located at the

University of Graz

Heinrichstraße 36

8010 Graz

Austria

# 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.

# Publications

Sets of lengths.

Amer. Math. Monthly, 123 (2016), 960-988.-

Systems of Sets of Lengths: Transfer Krull Monoids versus Weakly Krull Monoids.

In Rings, Polynomials, and Modules, 191-235. Springer, 2017. -

Long sets of lengths with maximal elasticity.

Canadian Journal of Mathematics, to appear.

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

The Ramanujan Journal, to appear.-

SETS OF LENGTHS IN ATOMIC UNIT-CANCELLATIVE FINITELY PRESENTED MONOIDS.

Colloquium Math., to appear.

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

Communications in Algebra, to appear.-

A realization theorem for sets of lengths in numerical monoids.

Forum Math., to appear. -

Which sets are sets of lengths in all numerical monoids.

submitted.