How Do You Measure Primality?

TL;DR

This paper reviews the concept of -primality in commutative monoids, highlighting recent developments, especially in numerical monoids, and presents open problems for further research.

Contribution

It offers a comprehensive overview of -primality, including new examples, recent results, and open questions, advancing understanding in the factorization theory of monoids.

Findings

Detailed examples of -primality in monoids

Recent surprising results in numerical monoids

Open problems accessible to students

Abstract

In commutative monoids, the $\omega$-value measures how far an element is from being prime. This invariant, which is important in understanding the factorization theory of monoids, has been the focus of much recent study. This paper provides detailed examples and an overview of known results on $\omega$-primality, including several recent and surprising contributions in the setting of numerical monoids. As many questions related to $\omega$-primality remain, we provide a list of open problems accessible to advanced undergraduate students and beginning graduate students.