Sets of Lengths

TL;DR

This paper introduces the concept of sets of lengths in algebraic structures, exploring their properties and significance in understanding non-unique factorizations in rings of integers and transfer Krull monoids.

Contribution

It provides a largely self-contained introduction to the known results about systems of sets of lengths in algebraic number theory and transfer Krull monoids.

Findings

Explains the structure of sets of lengths in algebraic number fields.

Describes the role of transfer Krull monoids in factorization theory.

Summarizes key properties and known results in the area.

Abstract

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.