Power monoids and their arithmetic: a survey

TL;DR

This survey explores the arithmetic properties of power monoids, which are formed by finite subsets of monoids, highlighting recent advances and their implications for factorization theory in complex algebraic structures.

Contribution

It provides a comprehensive overview of recent developments in the study of power monoids and their unique arithmetic properties, emphasizing new perspectives in non-cancellative and non-commutative factorizations.

Findings

Power monoids exhibit unusual arithmetic properties.

Recent research has expanded understanding of factorizations in non-cancellative settings.

The survey summarizes key developments and related aspects in the field.

Abstract

The non-empty finite subsets of a multiplicatively written monoid form a monoid under setwise multiplication. The same holds for finite subsets containing the identity element. Partly due to their unusual arithmetic properties, these structures, generically known as power monoids, have attracted increasing attention in recent years, stimulating new perspectives in the study of factorizations in non-cancellative or non-commutative settings. We survey these developments and briefly review some related aspects.