  We give explicit descriptions of rings of differential operators of toric
face rings in characteristic $0$. For quotients of normal affine semigroup
rings by radical monomial ideals, we also identify which of their differential
operators are induced by differential operators on the ambient ring. Lastly, we
provide a criterion for the Gorenstein property of a normal affine semigroup
ring in terms of its differential operators.
  Our main technique is to realize the k-algebras we study in terms of a
suitable family of their algebra retracts in a way that is compatible with the
characterization of differential operators. This strategy allows us to describe
differential operators of any k-algebra realized by retracts in terms of the
differential operators on these retracts, without restriction on char(k).
