# Realizable sets of catenary degrees of numerical monoids

## TL;DR

This paper characterizes which finite sets of nonnegative integers can be realized as the catenary degrees of numerical monoids, showing that almost all such sets are achievable by these monoids.

## Contribution

It provides a classification of realizable catenary degree sets for numerical monoids, establishing that nearly all finite subsets are attainable.

## Key findings

- Most finite subsets of nonnegative integers are realizable as catenary degree sets.
- The paper identifies one exceptional subset that cannot be realized.
- It advances understanding of factorization invariants in numerical monoids.

## Abstract

The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb Z_{\ge 0}$ occur as the set of catenary degrees of a numerical monoid (i.e., a co-finite, additive submonoid of $\mathbb Z_{\ge 0}$). In particular, we show that, with one exception, every finite subset of $\mathbb Z_{\ge 0}$ that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.