Sets of Lengths

Abstract

Oftentimes the elements of a ring or semigroup $H$ can be written as finite products of irreducible elements, say $a=u_1\cdot \dots \cdot u_k=v_1\cdot \dots \cdot v_{\ell }$, 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.

to appear in the American Math. Monthly