Sets of Lengths

Oftentimes the elements of a ring or semigroup $H$ can be written as finite products of irreducible elements, say $a=u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$, where the number of irreducible factors is distinct. The set $\mathsf{L}(a)\subset \mathbb{N}$ of all possible factorization lengths of $a$ is called the set of lengths of $a$, and the full system $\mathcal L (H) = { \mathsf L (a) \mid a \in H }$ is a well-studied means of describing the non-uniqueness of factorizations of $H$. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.