  We consider the Noetherian properties of the ring of differential operators
of an affine semigroup algebra. First we show that it is always right
Noetherian. Next we give a condition, based on the data of the difference
between the semigroup and its scored closure, for the ring of differential
operators being anti-isomorphic to another ring of differential operators.
Using this, we prove that the ring of differential operators is left Noetherian
if the condition is satisfied. Moreover we give some other conditions for the
ring of differential operators being left Noetherian. Finally we conjecture
necessary and sufficient conditions for the ring of differential operators
being left Noetherian.
