How Do You Measure Primality?

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.