Perfect Pairs of Ideals and Duals in Numerical Semigroups

This paper considers numerical semigroups $S$ that have a non-principal relative ideal $I$ such that ${\mu }_S(I){\mu }_S(S-I)={\mu }_S(I+(S-I))$. We show the existence of an infinite family of such which $I+(S-I)=S\mathrm{\backslash }\{0\}$. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them.