On the set of elasticities in numerical monoids

In an atomic, cancellative, commutative monoid S, the elasticity of an element provides a coarse measure of its non-unique factorizations by comparing the largest and smallest values in its set of factorization lengths (called its length set). In this paper, we show that the set of length sets $${\mathcal {L}}(S)$$L(S) for any arithmetical numerical monoid S can be completely recovered from its set of elasticities R(S); therefore, R(S) is as strong a factorization invariant as $${\mathcal {L}}(S)$$L(S) in this setting. For general numerical monoids, we describe the set of elasticities as a specific collection of monotone increasing sequences with a common limit point of $$\max R(S)$$maxR(S).