Monotone Catenary Degree in Numerical Monoids

TL;DR

This paper introduces the monotone catenary degree for numerical monoids, compares it with the traditional catenary degree, and explores classes where they are equal or differ significantly, revealing unbounded differences.

Contribution

It defines the monotone catenary degree, compares it with catenary degree across classes of numerical monoids, and shows that their difference can be arbitrarily large.

Findings

In arithmetical numerical monoids, monotone catenary degree equals catenary degree.

In certain embedding dimension 3 monoids, the degrees differ strictly.

The difference between the degrees can grow without bound.

Abstract

Recent investigations on the catenary degrees of numerical monoids have demonstrated that this invariant is a powerful tool in understanding the factorization theory of this class of monoids. Although useful, the catenary degree is largely not sensitive to the lengths of factorizations of an element. In this paper, we study the monotone catenary degree of numerical monoids, which is a variant of catenary degree that requires chains run through factorization lengths monotonically. In general, the monotone catenary is greater than or equal to the catenary degree. We begin by providing an important class of monoids (arithmetical numerical monoids) for which monotone catenary degree is equal to the catenary degree. Conversely, we provide several classes of embedding dimension 3 numerical monoids where monotone catenary degree is strictly greater. We conclude by showing that this difference can grow arbitrarily large.