  We present algorithms for basic computations with monoids in finitely
generated abelian groups such as monoid membership testing and computing an
element of the conductor ideal. Applying them to Mori dream spaces, we obtain
algorithms to test whether a Weil divisor class of a given Mori dream space is
base point free, to compute generators of the monoid of base point free Cartier
divisor classes and to test whether a Mori dream space with known canonical
class fulfills Fujita's base point free conjecture or not.
