Kunz languages for numerical semigroups are context sensitive

Abstract

There is a one-to-one and onto correspondence between the class of numerical semigroups of depth $n$, where $n$ is an integer, and a certain language over the alphabet $\{1,\ldots,n\}$ which we call a Kunz language of depth $n$. The Kunz language associated with the numerical semigroups of depth $2$ is the regular language $\{1,2\}^*2\{1,2\}^*$. We prove that Kunz languages associated with numerical semigroups of larger depth are context-sensitive but not regular.