Sets of Lengths Do Not Characterize Numerical Monoids

Abstract

We study the sets of lengths and unions of sets of lengths of numerical monoids. Our paper focuses on a numerical monoid S generated by an arithmetic progression of positive integers. First, we determine exact solutions for the length sets of S and then use these formulas to enumerate the Vn(S) sets. Next, we determine necessary and sufficient conditions for two such numerical monoids to have identical sequences of Vn(S) sets. Finally, we determine necessary and sufficient conditions for two such numerical monoids to have equal length sets