  This article concerns monomial ideals fixed by differential operators of
affine semi-group rings over $\mathbb{C}$. We give a complete characterization
of when this happens. Perhaps surprisingly, every monomial ideal is fixed by an
infinite set of homogeneous differential operators and is in fact determined by
them. This opens up a new tool for studying monomial ideals. We explore
applications of this to (mixed) multiplier ideals and other variants as well as
give examples of detecting ideal membership in integrally closed powers and
symbolic powers of squarefree monomial ideals.
