# Factorization Theory in Commutative Monoids

## TL;DR

This survey explores the structure and properties of various classes of monoids in factorization theory, focusing on their arithmetical finiteness properties and examples illustrating these concepts.

## Contribution

It provides a comprehensive overview of factorization properties across multiple monoid classes, highlighting new examples and the boundaries of finiteness properties.

## Key findings

- Descriptions of sets of lengths and their unions
- Examples of monoids with and without finiteness properties
- Analysis of catenary degrees in different monoid classes

## Abstract

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.