On differential operators of numerical semigroup rings

If $S=<d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_{\nu }>$ is a numerical semigroup, we call the ring $\text{C}[S]=\text{C}[t^{d_1},\mathrm{.}\mathrm{.}\mathrm{.},t^{d_{\nu }}]$ the semigroup ring of $S$. We study the ring of differential operators on $\text{C}[S]$, and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe in case $S$ is a so called Arf semigroup. If $I$ is an ideal in $\text{C}[S]$ that is generated by monomials, we also give some results on $\text{der}(I,I)$ (the set of derivations which map $I$ into $I$).