On the set of catenary degrees of finitely generated cancellative commutative monoids (Q2814682)

This paper studies the catenary degree of a finitely generated commutative cancellative monoid \(S\). This invariant has received much attention in the recent literature of nonunique factorizations. The catenary degree of an element \(n\) is a nonnegative integer that gives a measure of the distance between any two factorizations of \(n\). The catenary degree of a monoid \(S\) is the supremum of the catenary degrees of the elements of \(S\). The goal of this paper is to understand more than just this supremum. The authors study the set \(C(S)\) of catenary degree of elements of \(S\).NEWLINENEWLINEA finitely generated monoid \(S\) has a special set of elements called Betti elements, and it is known that the maximum element of \(C(S)\) is equal to the maximum catenary degree achieved by a Betti element of \(S\) [\textit{S. T. Chapman} et al., Manuscr. Math. 120, No. 3, 253--264 (2006; Zbl 1117.20045)]. A main result of this paper is to show that the minimum nonzero value of \(C(S)\) occurs as the catenary degree of a Betti element of \(S\). The authors give an example of a monoid \(S\) and an element \(x\) of \(C(S)\) such that no Betti element has catenary degree equal to \(x\). The authors also give several nice examples of computations of \(C(S)\) for families of numerical monoids. They show that \(|C(S)|\) can be arbitrarily large for numerical monoids generated by three elements.