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

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.