On the set of catenary degrees of finitely generated cancellative commutative monoids

TL;DR

This paper studies the set of catenary degrees in finitely generated cancellative commutative monoids, providing methods to compute extremal elements and exploring the structure of these sets with examples and open questions.

Contribution

It introduces a method to compute the smallest nonzero catenary degree in such monoids, answering an open question and analyzing the set of all catenary degrees.

Findings

Provided a method to compute the smallest nonzero element of (S)

Demonstrated extremal behaviors of (S) through examples

Presented open questions for future research

Abstract

The catenary degree of an element $n$ of a cancellative commutative monoid $S$ is a nonnegative integer measuring the distance between the irreducible factorizations of $n$. The catenary degree of the monoid $S$, defined as the supremum over all catenary degrees occurring in $S$, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set $\mathsf C(S)$ of catenary degrees achieved by elements of $S$ as a factorization invariant, focusing on the case where $S$ in finitely generated (where $\mathsf C(S)$ is known to be finite). Answering an open question posed by Garc\'ia-S\'anchez, we provide a method to compute the smallest nonzero element of $\mathsf C(S)$ that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for $\mathsf C(S)$, and present some open questions for further study.