Realizable sets of catenary degrees of 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.