Combinatorial Factorization Theory and Zero-sum Theory

Let $H$ be a Krull monoid (e.g., a Dedekind domain) with finite class group $G$ and suppose that every class contains a prime divisor. Then every non-unit $a \in H$ can be written as a product of irreducible elements, say $a = u_1 \cdot \ldots \cdot u_k$. The number of irreducible factors is called the length of the factorization, and $\mathsf L(a)=\{ k \mid a \ \text{has a factorization of length\} k}\subset\mathbb N$ denotes the {\it set of lengths} of $a$. Then $\mathcal L (H) = \{\mathsf L (a)\mid a \in H \}$ is the system of all sets of lengths of $H$. Here are some simple observations:

(1).$H$ is a factorial monoid if and only if $|G| = 1$.

(2). (Carlitz 1960) $|G| \le 2$ if and only if $|\mathsf L (a)| = 1$ for all non-units $a \in H$.

(3). If $|G| \ge 3$, then all sets of lengths are finite, and \newline for every $m \in \mathbb N$, there is an $a_m \in H$ with $|\mathsf L (a_m)| = m$.

Sets of lengths in $H$ can be studied in the monoid of zero-sum sequences $\mathcal B (G)$ associated to $H$. A zero-sum sequence means a finite unordered sequence of group elements from $G$ (repetition is allowed) which sums up to zero. The set  of zero-sum sequences forms a monoid where the operation is just the juxtaposition of sequences. This monoid $\mathcal B (G)$ is a Krull monoid again and its system of sets of lengths coincides with the system of our starting monoid $H$. Indeed, we have $\mathcal L(H)=\mathcal L\big(\mathcal B(G)\big)=: \mathcal L (G)$. The system $\mathcal L (G)$ is studied with  methods from Additive Combinatorics. In this talk, we will give a brief introduction to Factorization Theory and talk about arithmetic invariants of $\mathcal L (G)$ and Characterization Problem.