Factorization invariants in numerical monoids

TL;DR

This survey reviews key factorization invariants in numerical monoids, summarizing recent results and highlighting open questions in the study of nonunique factorizations.

Contribution

It provides a comprehensive overview of major factorization invariants in numerical monoids, consolidating current knowledge and identifying open problems.

Findings

Summarizes recent advances in length set, elasticity, delta set, ω-primality, and catenary degree invariants.

Highlights key open questions in the study of factorization invariants.

Provides a unified overview of the literature on numerical monoid factorizations.

Abstract

Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of the natural numbers), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, $\omega$-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain.