Uniform decision problems for automatic semigroups

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which are necessary to make the problem well defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse.